Carnegie Learning

Geometry Student Text 2nd Edition Chapter 1 Tools of Geometry

Carnegie Learning Geometry Student Text 2nd Edition Chapter 1 Exercise 1.3 Solution Page 27 Problem 1 Answer

Question 1.

How can we use a protractor to draw a pair of supplementary angles that share a common side, and then measure each angle?

Answer:

We need to use a protractor to draw a pair of supplementary angles that share a common side, and then measure each angle.

We know that two angles are supplementary angles if the sum of their angle measures is equal to 1800.

Then we consider a=1500then

a+b=180

150+b=180

b=180−150

b=30

So angle b=300

Hence we can draw this.

Read and learn More Carnegie Learning Geometry Student Text 2nd Edition Solutions

The answer is

Carnegie Learning Geometry Chapter 1 Page 27 Problem 2 Answer

Question 2.

How can we use a protractor to draw a pair of supplementary angles that do not share a common side, and then measure each angle?

Answer:

We need to use a protractor to draw a pair of supplementary angles that do not share a common side, and then measure each angle.

We know that two angles are supplementary angles if the sum of their angle measures is equal to 1800.

We consider a=1100.

We have

a+b=180

110+b=180

b=180−110

b=70

So angleb=700.

Hence we can draw this.

Page 28 Problem 3 Answer

Question 3.

It is given that in the figure, we need to find the measure of an angle that is supplementary to ∠KJL.

We know that two angles are supplementary if the sum of their measures is equal to 180°.

Given ∠J=22°,

Calculate the measure of the angle A that is supplementary to ∠KJL.

Answer:

It is given that

We know that two angles are supplementary if the sum of their measures is equal to 180°.

Given ∠J=22°,

Then we need to find to measure an angle that is supplementary to ∠KJL.

We know that two angles are supplementary angles if the sum of their angle measures is equal to 1800.

∠J=220

A+J=180

A+22=180

A=180−22

A=158

Hence 1580 is the angle that is supplementary to∠KJL.

1580 is the angle that is supplementary to ∠KJL.

Carnegie Learning Geometry Chapter 1 Page 28 Problem 4 Answer

Question 4.

We need to a protractor to draw a pair of complementary angles that share a common side, and then measure each triangle.

Answer:

We need to use a protractor to draw a pair of complementary angles that share a common side, and then measure each angle.

Two angles are complementary angles if the sum of their angle measures is equal to 900.

We consider a=600. Then

a+b=90

60+b=90

b=90−60

b=30​

So angleb=300.

Hence we can draw this.

The answer is

Solutions for Tools of Geometry Exercise 1.3 in Carnegie Learning Geometry Page 28 Problem 5 Answer

Question 5.

Given the following figure:

We have to find the complementary angle of ∠J.

Answer:

Given figure:

We have to find the complementary angle of angle ∠J.

To find the complementary angle of an angle we have to subtract the given angle from 90

because sum of complementary angles are 90˚.

So complementary angle of angle J is

∠J=90−62

∠J=28˚

The complementary angle of angle J is 28˚

Carnegie Learning Geometry Chapter 1 Page 29 Problem 6 Answer

Question 6.

Given that two angles are both congruent and supplementary, find the measure of each angle.

Answer:

Given statement: Two angles are both congruent and supplementary.

We have to find the measure of each angle.

Let two angles are ∠A and ∠B then

As they are congruent so ∠A=∠B−−−−−−1

Also as ∠A and ∠B are supplementary so there sum must be 180.

∠A+∠B=180−−−−−−−−−−−2

Now using equation 1 and 2 we get

∠A+∠A=180

2∠A=180

∠A=180/2

∠A=90

Thus measure of angle A  and angle B are ∠A=∠B=90˚.

So we conclude that if two angles are congruent and supplementary than they must be right angles i.e., 90˚

We can say that if two angles are congruent and supplementary than they must be right angles i.e., 90˚.

Page 29 Problem 7 Answer

Question 7.

Given that two angles are both congruent and complementary, find the measure of each angle.

Answer:

Given statement: Two angles are both congruent and complementary,

We have to find the measure of each angle.

Let two angles are ∠A and ∠B then

As they are congruent so∠A=∠B−−−−−−1

Also as ∠A and ∠B are complementary so there sum must be 90.

∠A+∠B=90−−−−−−−−−−−2

Now using equation 1 and 2  we get

∠A+∠A=90

2∠A=90

∠A=90/2

∠A=45

Thus measure of angle A  and angle B are 45 degrees.

So we conclude that if two angles are congruent and complementary than they must be 45˚.

We can say that if two angles are congruent and supplementary than they must 45˚.

Carnegie Learning Geometry Chapter 1 Page 29 Problem 8 Answer

Question 8.

The complement of an angle twice the measure of the angle. What are the measures of the angle angle and its complement?

Answer:

Given statement: The complement of an angle is twice the measure of the angle.

We have to find the measure of each angle.

Let measure of angle is x.

Then according to given statement its complement is 2x.

As we know sum of complementary angles must be equal to 90 degrees .

So

x+2x=90

3x=90

x=90/3 {dividing both sides by 3}

x=30

Thus measure of angle is 30˚ and measure of its complement is 2×30=60˚.

Thus measure of angle is 30˚and measure of its complement is 60˚.

Page 29 Problem 9 Answer

Question 9.

The supplement of an angle is half the measure of the angle. What are the measures of the angle and its supplement?

Answer:

Given statement: The supplement of an angle is half the measure of the angle.

We have to find the measure of each angle.

Let measure of angle is x. Then according to given statement its supplementary angle is x/2.

As we know sum of supplementary angles must be equal to 180˚.

Thus ​x+x/2=180

Multiply both sides by 2

2x+x=360

3x=360

dividing both sides by 3

x=360/3

x=120

Thus measure of angle is120˚and measure of its supplementary angle is 180−120=60˚.

Thus measure of angle is 120˚ and measure of its supplementary angle   is 60˚ .

Carnegie Learning Geometry Chapter 1 Page 30 Problem 10 Answer

Question 10.

Given the figure, identify all the right angles in the figure.

Answer:

Given figure :

We have to name the right angles in the given figure.

From figure we observe that following angles are of right angle

∠ADC,∠BDC and ∠DCF.

From figure we observe that following angles are of right angle

∠ADC,∠BDC and ∠DCF

Carnegie Learning Geometry 2nd Edition Exercise 1.3 Solutions Page 30 Problem 11 Answer

Question 11.

Given that AB⊥CD at point E, how many right angles are formed? Describe and name these right angles.

Answer:

To draw : AB⊥CD at point E. How many right angles are formed

The drawing is

Right angles are∠AED,∠BED,∠BEC,∠CEA

Number of right angles formed is 4

The drawing is

Number of right angles formed is4

Carnegie Learning Geometry Chapter 1 Page 30 Problem 12 Answer

Question 12.

To draw: BC ⊥ AB at point E, how many right angles are formed? Describe and name these right angles.

Answer:

To draw: BC⊥AB at point B. How many right angles are formed

The drawing is

Right angle is∠ABC

Number of right angles formed is 1

The drawing is

Number of right angles formed is1

Page 31 Problem 13 Answer

Question 13.

Give a line and a point P not on the line, construct a line perpendicular to the given line through point P.

Answer:

To construct A line perpendicular to the given line through point P.

Given :

The perpendicular line means angle is 900

The drawing is

Here AP is the perpendicular line

The drawing is

Carnegie Learning Geometry Chapter 1 Page 32 Problem 14 Answer

Question 14.

Given a line segment AG and a Point B on AG, Construct a line perpendicular to Ag through point B.

Answer:

To construct : A line perpendicular to AG through point B.

Given :

The perpendicular line means angle is900

The drawing is

Here CD is the perpendicular line through B

The drawing is

Page 32 Problem 15 Answer

Question 15.

Describe the difference between constructing a perpendicular through a point on the line and constructing a perpendicular through a point not on the line.

Answer:

We have to find that the difference between the construction of perpendicular through a point on the line and construction of perpendicular through a point not on the line.

Firstly we will write the steps to construct a perpendicular in both ways.

After we will analyze the steps to find the difference between them.

The process of construction of Perpendicular in both the ways is as following :-

Construction of a perpendicular through a point on the line :-

Use B as a center and label the intersection of points C and D.

With the same radius , draw the arcs by taking C and D as centers, above and below the line.

Draw straight line through points E and F. The line EF is perpendicular to CD​.

Construction of a perpendicular through a point not on the line :-

Use B as a center and label the intersection of points C and D.

With the same radius , draw the arcs by taking C and D as centers, above and below the line.

Draw straight line through points E and F.

The line EF is perpendicular to line CD.

The construction of perpendicular in the both the ways is completely different.

It looks like same but there are difference between both the constructions.

In the construction of perpendicular through a point on the line; we can draw the arc of any radius in first step.

Conversely in the construction of perpendicular through a point not on the line; we cannot draw the arc that does not intersect the line.

That is we have to take the radius as the distance between the line and point at least.

So the main difference between the construction of perpendicular in both ways is the choice of arc in first step.

After the process is quite similar.

The difference between the construction of perpendicular through a point on line and not on line, is the choice of arc radius in first step.

After that process is quite similar.

Carnegie Learning Geometry Chapter 1 Page 34 Problem 16 Answer

Question 16.

How do you construct a perpendicular bisector of a given line segment FG and label it as a CD?

Answer:

The given line segment is FG.

We have to construct a perpendicular bisector of this line segment and label it as CD.

Let the given line is :-

Firstly we will find mid point of line.

Then draw the perpendicular bisector.

The given line is :-

Let the mid point of line is E.

Then we have the following line :-

Now to draw perpendicular bisector :-

Firstly open the compass with radius more the half of line FG and draw the arc taking F as center as shown following :-

Now with the same radius draw an arc by taking G as center, then we have :-

Now join the points C and D, then we have :-

The line segment CD is the required perpendicular bisector of the line FG.

The required perpendicular bisector CD of the given line FG is :-

Carnegie Learning Geometry Chapter 1 Page 34 Problem 17 Answer

Question 17.

How do you label the point of intersection of the perpendicular bisector CD and the line segment FG as E?

Answer:

The given line segment is FG.

We find in previous part that CD is the perpendicular Bisector of line FG.

We have to label the point of intersection CD and FG as E.

The lines FG and its perpendicular bisector CD are as following :-

Now label the point of intersection of both lines as E, then we have :-

This is the required answer.

The required intersection point E of lines CD and FG is as following :-

Tools Of Geometry Solutions Chapter 1 Exercise 1.3 Carnegie Learning Geometry Page 34 Problem 18 Answer

Question 18.

Given that CD is perpendicular to FG, determine the measures of the angles formed by these lines. Provide the necessary conclusions about these angles.

Answer:

Given

CD is perpendicular to FG

In previous parts we find that the given lines CD and FG are as following :-

If you find the conclusions about these lines if CD⊥FG.

The given lines are as following:-

Here we have CD⊥FG.

We know that if two lines are perpendicular then the angle between them is 90∘.

Hence we can conclude that the angle ∠CEF,∠CEG,∠FED,GED

ARE OF 90∘.

If CD⊥FG, then we have :-

​∠CEF=90∘,

∠CEG=90∘,

∠FED=90∘ and

∠GED=90∘

Step-By-Step Solutions For Carnegie Learning Geometry Chapter 1 Exercise 1.3 Page 34 Problem 19 Answer

Question 19.

Given that line CD bisects line FG, what can we conclude about the segments FE and EG? Provide a detailed explanation and conclusion.

Answer:

Given

Line CD bisects line FG

The given lines CD and FG are as following :-

We have to find that what we can conclude that if the line CD bisects the line FG.

The given lines are :-

We know that if a line bisect the other then the point of intersect is a midpoint of the second line.

We have the line CD bisects the line FG.

So the point of intersection E is the midpoint of line FG.

Also as E is midpoint, we have :-

Measure of line segment FE=Measure of line segment EG.

If the line CD bisects the line FG as shown following :-

Then we have :- E is a midpoint of line FG.Measure of line FE=Measure of line EG.

Page 34 Problem 20 Answer

Question 20.

Given that line CD is the perpendicular bisector of line FG, what conclusions can we draw about the angles formed and the segments of FG? Provide a detailed explanation and conclusion.

Answer:

Given

line CD is the perpendicular bisector of line FG

The given lines CD and FG are as following :-

We have to find that what we can conclude if the line CD is the perpendicular bisector of the line FG.

The given lines are as following :-

We have the line CD is the perpendicular bisector of line FG.

That is CD is perpendicular to line FG.

Then we have the angles between them are of 90∘. That is :-

∠CEF=90∘,

∠CEG=90∘,

∠FED=90∘and ∠GED=90∘

Also the line CD bisects the line FG.

So the intersection point is the midpoint of line FG.

That is the point E is the midpoint of line FG.

As E is the point of line FG, then we have :-

Measure of line FE=Measure of line EG.

If the line CD is the perpendicular bisector of line FG, then we have :-

∠CEF=90∘,∠CEG=90∘,∠FED=90∘ and∠GED=90∘

The point E is the midpoint of line FG.Measure of line FE=Measure of line EG.

Carnegie Learning Geometry Chapter 1 Page 34 Problem 21 Answer

Question 21.

Given a line segment PQ, how do we construct the midpoint of the line segment? Explain the steps and show the construction.

Answer:

The given line PQ  is as following :-

We have to construct the mid point of line PQ.

The given line is:-

As we know the midpoint divides the line into two equal line segment.

The the midpoint of line PQ will divide the line into two equal parts.

Let the mid point of line is M. Then the M is constructed as following:-

The required mid point M of line PQ is constructed as following :-

Carnegie Learning Geometry Chapter 1 Exercise 1.3 Free Solutions Page 35 Problem 22 Answer

Question 22.

What are adjacent angles? Explain and illustrate with a diagram.

Answer:

We have to explain the adjacent angles.

The adjacent angles are explained as following :- Adjacent angle :- Adjacent angles are two angles that have a common side and a common vertex (corner point) but do not overlap in any way.

Then two adjacent angles can be shown as following :-

Here the angles A and B are adjacent as they have a common side and common vertex or corner.

Adjacent angles are two angles that have a common side and a common vertex (corner point) but do not overlap in any way.

Carnegie Learning Geometry Chapter 1 Page 35 Problem 23 Answer

Question 23.

What are adjacent angles? Explain and illustrate with a diagram.

Answer:

The given angle ∠1 is:-

We have to draw ∠2 that is adjacent to angle ∠1.

The given angle ∠1 is as following:-

We know that the angles that have a common line and vertex are know as adjacent angles.

So the adjacent angle ∠2 to ∠1 can be constructed as following :-

The required angle ∠2 adjacent to angle ∠1 can be drawn as following :-

Page 35 Problem 24 Answer

Question 24.

Is it possible to draw two angles that share a common vertex but do not share a common side? If so, draw an example and explain.

Answer:

We have to check that :-

Is it possible to draw two angles share a common vertex but do not share a common side.

If it is possible we have to draw an example and explain it.

Yes, we can draw two angles with common vertex but not common side.

Now suppose two angles ∠1 and ∠2. Let these angles have common vertex or corner say O

but not have common side.

Then these angles can be drawn as following :-

Here the angles ∠1 and ∠2 have common vertex but no common side.

These angles are not adjacent as these have no common side.

Yes, we can draw two angles with common vertex but not common side and this type of angles can be drawn as following :-

Carnegie Learning Geometry Chapter 1 Page 36 Problem 25 Answer

Question 25.

Is it possible to draw two angles that share a common side but do not share a common vertex? Draw an example.

Answer:

Given:  Is it possible to draw two angles that share a common side, but do not share a common vertexTo draw an example.

Two angles that  share a common side, but do not share a common vertex can be dawn as

Here ∠1 and ∠2 are not adjacent since they do not share a common vertex.

Hence, Two angles that  share a common side, but do not share a common vertex can be dawn as

Carnegie Learning Geometry Exercise 1.3 Student Solutions Page 36 Problem 26 Answer

Question 26.

What are linear pairs of angles? Describe them with an example.

Answer:

Given:

To Describe a linear pair of angles.

Linear pair of angles are formed when two lines intersect each other at a single point.

The angles are said to be linear if they are adjacent to each other after the intersection of the two lines.

The sum of angles of a linear pair is always equal to 180°.

From the given figure we can say that ∠1 and ∠2 forms a linear pair of angle

Hence, from above we can say that Linear pair of angles are formed when two lines intersect each other at a single point.

The angles are said to be linear if they are adjacent to each other after the intersection of the two lines.

The sum of angles of a linear pair is always equal to 180°.

Carnegie Learning Geometry Chapter 1 Page 36 Problem 27 Answer

Question 27.

Identify and name all the linear pairs in the given figure.

Answer:

Given:

To Name all linear pairs in the figure shown

Form the figure given

The linear pairs are ∠1 and ∠4, ∠4 and ∠2, ∠2 and ∠3,∠3 and ∠1

Hence, the linear pairs are ∠1 and ∠4,∠4 and ∠2,∠2 and ∠3,∠3 and ∠1

Page 37 Problem 28 Answer

Question 28.

What can we conclude if the angles that form a linear pair are congruent?

Answer:

Given:  If the angles that form a linear pair are congruent,To specify what we can concludeBy following the definition

A linear pair consists of two adjacent angles that form a straight angle so they are supplementary, having a sum of 180∘.

If the two angles are congrent, then they have the same measure and thus both measure 90∘.

Hence, each angle is a right angle.

Hence, A linear pair consists of two adjacent angles that form a straight angle so they are supplementary, having a sum of 180∘.

If the two angles are congruent, then they have the same measure and thus both measure 90∘.

Hence, each angle is a right angle.

Carnegie Learning Geometry Chapter 1 Page 37 Problem 29 Answer

Question 29.

What are vertical angles, and how can they be identified in a given figure?

Answer:

Given:

To describe vertical angle

The angles that are opposite to each other when two lines intersect each other are known as vertical angles.

In the given figure ∠1 and ∠2 are vertical angles to each other

Hence, The angles that are opposite to each other when two lines intersect each other are known as vertical angles.

Tools Of Geometry Exercise 1.3 Carnegie Learning 2nd Edition Answers Page 37 Problem 30 Answer

Question 30.

Given the figure below, draw ∠2 so that it forms a vertical angle with ∠1.

Answer:

Given:

To draw ∠2 so that it forms a vertical angle with ∠1

The angles that are opposite to each other when two lines intersect each other are known as vertical angles.

So a vertical angle can be drawn as

Here ∠1 and ∠2 are vertical angle

Hence, vertical angle can be drawn as

Carnegie Learning Geometry Chapter 1 Page 38 Problem 31 Answer

Question 31.

Given the diagram below, name all pairs of vertical angles.

Answer:

Given:

To  Name all vertical angle pairs in the diagram shown.

From the given diagram we can show the vertical angle as

Here ∠1 and ∠2. ∠3 and ∠4 are the vertical angle

Hence, in the given diagram the vertical angles are ∠1 and ∠2, ∠3 and ∠4

Page 38 Problem 32 Answer

Question 32.

Given the diagram below, measure each angle and determine their properties.

Answer:

Given:

To  Measure each angle

In the given figure,∠1 and ∠2, ∠3 and ∠4 are vertical opposite angle, so they must be equal

Also ∠3 and ∠4 are acute angle, so it must be less than 90o

Also ∠1 and ∠2 are obtuse angle and it must be greater than 90o and less than 180o Hence, in the given figure ∠1and∠2,∠3and∠4 are vertical opposite angle, so they must be equal

Also ∠3and∠4 are acute angle, so it must be less then 90∘

Also ∠1and∠2 are obtuse angle and it must be greater than 90∘ and less than 180∘

Geometry Student Text 2nd Edition Chapter 1 Tools of Geometry

Carnegie Learning Geometry Student Text 2nd Edition Chapter 1 Exercise 1.2 Solution Page 14 Problem 1 Answer

Question 1.

Tools of Geometry, it is stated that the vertex of an angle is a crucial part of defining the angle. Given the angle ∠ABC, identify the vertex of this angle and explain its significance in the context of geometric constructions.

Answer:

Given

The angle ∠ABC

The angle shown as :

The vertex of an angle is :

The vertex of an angle is

So, according to the given condition we get,

Read and learn More Carnegie Learning Geometry Student Text 2nd Edition Solutions

The vertex of an angle is ∡ABC.

The vertex of an angle is ∡ABC.

Carnegie Learning Geometry Chapter 1 Page 14 Problem 2 Answer

Question 2.

Tools of Geometry, it is explained that the sides of an angle refer to the two rays or line segments that form the angle. Given the angle ∠ABC, identify the sides of this angle and describe their role in the formation of the angle.

Answer:

The sides of an angle refer to the two rays or line segments that form the angle.

In the figure below, rays BA and BC are the sides of angle ABC.

So, according to given condition we get,

The sides of a triangle are BA,BC of ∡ABC.

The sides of an angle are BA,BC of ∡ABC.

Page 14 Problem 3 Answer

Question 3.

What is the relationship between the angles ∠ED and ∠EF in the given diagram, and why are they equal?

Answer:

Given

∠ED and ∠EF

The diagram shown as :

The specific angle of ∠ED determines.

When two lines intersect each other, then the opposite angles, formed due to intersection are called vertical angles or vertically opposite angles.

A pair of vertically opposite angles are always equal to each other.

Also, a vertical angle and its adjacent angle are supplementary angles.

The specific angle of ∠ED, ∠EF are equal and are vertically opposite angles.

The specific angle of ∠ED,∠EF are vertically opposite angles and are equal.

Carnegie Learning Geometry Chapter 1 Page 14 Problem 4 Answer

Question 4.

What is the relationship between the angles ∠FEG and ∠DEC in the given diagram, and why are they equal?

Answer:

Given

The angles ∠FEG and ∠DEC

The diagram shown as :

A specific angle in the diagram of ∠DEC.

When two lines intersect each other, then the opposite angles, formed due to intersection are called vertical angles or vertically opposite angles.

A pair of vertically opposite angles are always equal to each other.

Also, a vertical angle and its adjacent angle are supplementary angles.

The angles are equal and are vertically opposite angles.

∡FEG=∡DEC

The vertical opposite angles are equal.

∡FEG=∡DEC.

Page 15 Problem 5 Answer

Question 5.

In the given figure, under what condition can one capital letter, such as ∠D, be used to name an angle?

Answer:

Given – The figure

To find – When can one capital letter be used to name an angle

The angles can be named in following ways:

By three capital letters, with the vertex letter in the middle like∠EDC or∠FEG

By one lower case letter or number written in the middle of the angle By one capital letter∠D.

This can only be used if∠D is the only angle it could be.

One capital letter like∠D can only be used if∠D is the only angle it could be.

Carnegie Learning Geometry Chapter 1 Page 15 Problem 6 Answer

Question 6.

In the given figure, under what condition can one capital letter, such as ∠D, be used to name an angle?

Answer:

Given – The figure

To find – When can one capital letter be used to name an angle

The angles can be named in following ways:

By three capital letters, with the vertex letter in the middle like∠EDC or∠FEG

By one lower case letter or number written in the middle of the angle By one capital letter∠D.

This can only be used if∠D is the only angle it could be.

One capital letter like∠D can only be used if∠D is the only angle it could be.

Solutions For Tools Of Geometry Exercise 1.2 In Carnegie Learning Geometry Page 15 Problem 7 Answer

Question 7.

Given the figure, what is the difference between ∠FGE and ∠EGF? Explain your answer.

Answer:

Given – The figure

∠FGE and ∠EGF

To find – What is the difference between∠FGE and∠EGF

The angle can be written by three capital letters like∠FGE

The main angle is the angle of middle letter and we can shuffle the first and third letter with each other

So,∠FGE is always equal to∠EGF

So there is no difference between∠FGE and∠EGF.

There is no difference between∠FGE and∠EGF.

Carnegie Learning Geometry Chapter 1 Page 15 Problem 8 Answer

Question 8.

Given the figure, explain the difference between ∠EFG and ∠EGF. What type of angles are they?

Answer:

Given – The figure

To find – Difference between∠EFG and∠EGF

As we know right angle is an angle having corner looks like L or one angle is 900 and the angle less than 900 is called acute angle.

So,∠EFG is an right angle and∠EGF is an acute angle.

∠EFG is an right angle and∠EGF is an acute angle.

Page 15 Problem 9 Answer

Question 9.

Given the figure, provide the alternate names for ∠D. How can the angle ∠D be represented using different notations?

Answer:

Given – The figure

To find – Alternate names of∠D

The angles can be named in following ways:

By three capital letters, with the vertex letter in the middle like∠EDC or∠FEG

By one lower case letter or number written in the middle of the angle By one capital letter∠D.,

This can only be used if∠D is the only angle it could be.

So,∠D can be written as∠CDE and∠ECD.

The alternate names of∠D are∠CDE and∠ECD.

Carnegie Learning Geometry Chapter 1 Page 15 Problem 10 Answer

Question 10.

How many letters are needed to name an angle, and in what situations can a single letter be used instead of three letters?

Answer:

Given – The figure

To find – How many letters are needed to name an angle

The angles can be named in following ways:

By three capital letters, with the vertex letter in the middle like∠EDC or∠FEG

By one lower case letter or number written in the middle of the angle By one capital letter∠D.

This can only be used if∠D is the only angle it could be.

So the angles are written in either one letter if it is the only angle it could be or generally with three letters with the vertex letter in the middle.

The angles are written in either one letter if it is the only angle it could be or generally with three letters with the vertex letter in the middle.

Page 15 Problem 11 Answer

Question 11.

What is an alternate name for ∠1 in the given figure, and why is it named that way?

Answer:

Given – The figure

To find – Alternate name of∠1

The angles can be named in following ways:

By three capital letters, with the vertex letter in the middle like∠DEG

By one lower case letter or number written in the middle of the angleBy one capital letter.

This can only be used if it is the only angle it could be.

So, the alternate name of∠1 is∠CEF.

The alternate name for∠1 is∠CEF.

Carnegie Learning Geometry Chapter 1 Page 15 Problem 12 Answer

Question 12.

What is an alternate name for ∠2 in the given figure, and why is it named that way?

Answer:

Given – The figure

To find – Alternate name of∠2

The angles can be named in following ways:

By three capital letters, with the vertex letter in the middle like∠DEG

By one lower case letter or number written in the middle of the angleBy one capital letter. This can only be used if it is the only angle it could be.

So, the alternate name of∠2 is∠FEG.

The alternate name for∠2 is∠FEG.

Page 15 Problem 13 Answer

Question 13.

Do ∠3 and ∠4 share a common side in the given figure? If so, what is the common side?

Answer:

Given – The figure

To find – Do∠3 and∠4 shares a common side

As we know the common side is one line, ray, or line segment used to create two angles sharing the same vertex.

Clearly∠3 and∠4 does not have the same vertex.

But,∠3 and∠4  shares a common side, AB ∠3 and∠4 shares a common side which is AB.

Carnegie Learning Geometry Chapter 1 Page 16 Problem 14 Answer

Question 14.

What is the measure of the angle shown in the figure.

Answer:

Given: A figure

We have to measure the angle shown in the figure.

An angle is formed by the cross section of two rays. In the given figure we can see that the first ray is aligned with 0∘and second ray is aligned with 90∘, therefore the measure of the angle shown is 90∘.

The measure of the angle shown is 90∘.

Page 17 Problem 15 Answer

Question 15.

What is the measure of the angle shown in the figure

Answer:

Given: A figure

In the protractor the top of the arc shows degrees from 0º to 180º from left to right, while the bottom of the arc shows degrees from 180º to 0º from left to right.

In the given figure one ray is aligned with 0∘ on the right side of protractor, therefore we will read the measure of the angle according to the bottom arc of the protractor.

We can easily see that the 2nd ray is aligned with 130∘on bottom arc of the protractor, hence the measure of angle shown is 130∘.

The measure of the angle shown is 130∘.

Carnegie Learning Geometry Chapter 1 Page 17 Problem 16 Answer

Question 16.

How do you determine which scale to use on a protractor to measure an angle?

Answer:

In a protractor, you will notice two sets of degrees along the edge: an inner and outer scale.

Both scales go from 0 to 180, but they run in opposite directions. If the angle opens to the right side of the protractor, use the inner scale. If the angle opens to the left of the protractor, use the outer scale.

If the angle opens to the right side of the protractor, use the inner scale.

If the angle opens to the left of the protractor, use the outer scale.

Tools Of Geometry Solutions Chapter 1 Exercise 1.2 Carnegie Learning Geometry Page 17 Problem 17 Answer

Question 17.

How do you determine the measure of ∠WAR using a protractor, and what is the measurement?

Answer:

Given: A figure

Find: ∠WAR

In a protractor, you will notice two sets of degrees along the edge: an inner and outer scale. Both scales go from 0 to 180, but they run in opposite directions.

If the angle opens to the right side of the protractor, use the inner scale.

If the angle opens to the left of the protractor, use the outer scale.

In the given figure, ∠WAR opens left side of the protractor, therefore we will read the angle according to the outer scale.

Ray W is aligned with 0∘on the outer scale while ray R is aligned with 50∘on the outer scale, therefore the measure of angle ∠WAR is 50∘.

The measure of ∠WAR is 50∘.

Carnegie Learning Geometry Chapter 1 Page 17 Problem 18 Answer

Question 18.

How do you determine the measure of ∠RAX using a protractor, and what is the measurement?

Answer:

Given: A figure

Find: ∠RAX

In a protractor, you will notice two sets of degrees along the edge: an inner and outer scale.

Both scales go from 0 to 180, but they run in opposite directions. If the angle opens to the right side of the protractor, use the inner scale.

If the angle opens to the left of the protractor, use the outer scale.

In the given figure, ∠RAX opens right side of the protractor, therefore we will read the angle according to the inner scale.

Ray X is aligned with0∘on the inner scale while ray R is aligned with130∘on inner scale, therefore the measure of∠RAX is 130∘.

The measure of∠RAX is 130∘.

Carnegie Learning Geometry Chapter 1 Page 17 Problem 19 Answer

Question 19.

How do you determine the measure of ∠WAX using a protractor, and what is the measurement?

Answer:

Given: A figure

Find: ∠WAX

In the protractor the top of the arc shows degrees from 0º to 180º from left to right, while the bottom of the arc shows degrees from 180º to 0º from left to right.

In the given figure, ray W is aligned with 0∘on the left side of protractor, therefore we will read the measure of the angle according to the top arc of the protractor, we can easily see that the ray X is aligned with 180∘on the top arc of the protractor, therefore the measure of∠WAX is 180∘.

The measure of ∠WAX is 180∘.

Carnegie Learning Geometry Chapter 1 Page 18 Problem 20 Answer

Question 20.

What is the measure of angle ∠SET as determined from the diagram?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle ∠SET.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps as: As the midpoint of the protractor is already placed on the vertex E of the angle ∠SET.

One side ET of the angle is lined up with the zero line of the protractor (where you see the number 0).

Now we will read the degrees where the other side ES crosses the number scale at150.

Thus the measure of angle ∠SET=150

Hence the measure of angle ∠SET=150.

Carnegie Learning Geometry Chapter 1 Page 18 Problem 21 Answer

Question 21.

What is the measure of angle ∠QEP as determined from the diagram?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle ∠QEP.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps as:

As the midpoint of the protractor is already placed on the vertex E of the angle ∠QEP.

One side EP of the angle is lined up with the zero line of the protractor (where you see the number 0).

Now we will read the degrees where the other side EQ crosses the number scale at 400.

Thus the measure of angle ∠QEP=400.

Hence the measure of angle ∠QEP=400.

Carnegie Learning Geometry Chapter 1 Page 18 Problem 22 Answer

Question 22.

What is the measure of angle ∠REQ as determined from the diagram?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle ∠REQ.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps as: As the midpoint of the protractor is already placed on the vertex E of the angle ∠REQ.

The side EP is lined up with the zero line of the protractor (where you see the number 0).

Now we will read the degrees where the side EQ crosses the number scale with respect to side EP and it comes as 400.

The side ER crosses the number scale with respect to side EP and it comes as 650.

Now we will measure the angle ∠REQ using the measure of angles ∠REP=650 and ∠QEP=400.

Thus, the measure of the required angle is as:

∠REQ=∠REP−∠QEP

=650−400

=150

Thus the angle ∠REQ=150

Hence the measure of angle ∠REQ=150

Carnegie Learning Geometry Chapter 1 Page 18 Problem 23 Answer

Question 23.

What is the measure of angle ∠REP as determined from the diagram?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle ∠REP.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps as:

As the midpoint of the protractor is already placed on the vertex E of the angle ∠REP.

One side EP of the angle is lined up with the zero line of the protractor (where you see the number 0).

Now we will read the degrees where the other side EQ crosses the number scale with respect to side EP and it comes as 650.

Thus the measure of angle ∠REP=650.

Hence the measure of angle ∠REP=650.

Carnegie Learning Geometry Chapter 1 Page 18 Problem 24 Answer

Question 24.

What is the measure of angle ∠TEQ as determined from the diagram?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle ∠TEQ.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps as: As the midpoint of the protractor is already placed on the vertex E of the angle ∠TEQ.

One side ET of the angle is lined up with the zero line of the protractor (where you see the number 0).

Now we will read the degrees where the other side EQ crosses the number scale at 1400.

Thus the measure of angle ∠TEQ=1400.

Hence the measure of angle ∠TEQ=1400

Carnegie Learning Geometry Chapter 1 Page 18 Problem 25 Answer

Question 25.

What is the measure of angle ∠PES as determined from the diagram?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle ∠PES

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps as:

As the midpoint of the protractor is already placed on the vertex E of the angle ∠PES.

One side PE of the angle is lined up with the zero line of the protractor (where you see the number 0).

Now we will read the degrees where the other side ES crosses the number scale at 1650.

Thus the measure of angle ∠PES=1650

Hence the measure of angle ∠PES=1650

Carnegie Learning Geometry Chapter 1 Page 18 Problem 26 Answer

Question 26.

What is the measure of angle ∠SER as determined from the diagram?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle ∠SER.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps as:

As the midpoint of the protractor is already placed on the vertex E of the angle ∠SER.

The side ET is lined up with the zero line of the protractor (where you see the number 0).

Now we will read the degrees where the side ER crosses the number scale with respect to side ET and it comes as 1150.

The side ES crosses the number scale with respect to side ET and it comes as 150

Now we will measure the angle ∠SER using the measure of angles ∠TER=1150 and ∠SET=150

Thus, the measure of the required angle is as:

∠SER=∠TER−∠SET

=1150−150

=1000

Thus the angle ∠SER=1000.

Hence the measure of angle ∠SER=1000

Step-By-Step Solutions For Carnegie Learning Geometry Chapter 1 Exercise 1.2 Page 18 Problem 27 Answer

Question 27.

What is the angle measured in the given figure using the protractor?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle to the nearest degree using a protractor.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will use a protractor for measuring the angle in given figure by keeping the midpoint of protractor on the vertex of the figure as shown below:

Thus the measure of angle is 450.

Hence the measure of angle for given figure is 450

Carnegie Learning Geometry Chapter 1 Page 18 Problem 28 Answer

Question 28.

What is the angle measured in the given figure using the protractor?

Answer:

Given: A diagram is given as shown below:

To Determine: One has to use the given diagram and has to determine the measure of angle to the nearest degree using a protractor.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will use a protractor to measure the angle in the given figure by keeping the midpoint of the protractor on the vertex of the figure as shown below:

Thus the measure of angle is 1450

Hence the measure of angle in given figure is 1450

Carnegie Learning Geometry Chapter 1 Page 19 Problem 29 Answer

Question 29.

Which angle is larger among the given two figures?

Answer:

Given: There are two diagrams given as shown below:

To Determine: We have to determine which angle is larger among the given two figures.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now when we will put the protractor and the first small figure coinciding with each other as shown below:

As it is clear from the above diagram that the angle made by small figure is 900.

Now when we will put the large figure and the protractor coinciding with each other as shown below:

As it is seen from the above diagram that the angle made by larger image is as 900.

Hence the angle made by both figures is the same as 900

so we can conclude that both angles are equal and no one is larger.

Carnegie Learning Geometry Chapter 1 Page 19 Problem 30 Answer

Question 30.

How can you draw an angle with a measure of 30° using a protractor?

Answer:

Given: An angle with measure of 300 is given.

To Draw: We have to draw an angle with the given measure.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps to draw an angle with measure 300 as shown below:

Draw a line segment OA.Place the center tip of the protractor at point A such that the protractor perfectly aligns with line AO.

Start from ‘A’ on the protractor in the clockwise direction and stop at 30.

Mark it as point ‘D’. If point ‘A’ lies to the right of ‘O’, then start measuring anticlockwise and stop at 30 as shown below:

Join point ‘D’ with ‘O’. ∠AOD=30° is the required 30-degree angle as shown below:

The angle with measure 300 is drawn as below:

Carnegie Learning Geometry Exercise 1.2 Student Solutions Page 19 Problem 31 Answer

Question 31.

How do you draw an angle with a measure of 130° using a protractor?

Answer:

Given: An angle with measure of 1300.

To Draw: We have to draw an angle with the given measure.

Procedure Used:

The steps to measure an angle with a protractor are:

Place the midpoint of the protractor on the VERTEX of the angle.

Line up one side of the angle with the zero line of the protractor (where you see the number 0).

Read the degrees where the other side crosses the number scale.

Now we will follow certain steps to draw an angle with measure 1300 as shown below:

Draw a line segment AB.Using protractor from the point A measure 130∘ and mark it as C.Join AC.

Thus ∠BAC=1300  is the required 30-degree angle as shown below:

Hence the angle with measure 1300 is drawn below:

Carnegie Learning Geometry Chapter 1 Page 19 Problem 32 Answer

Question 32.

How do you draw an acute angle of 30° using a protractor?

Answer:

Acute angle can be drawn as follows, here ∠ABC=30.

Therefore, the diagram is as follows,

Tools Of Geometry Exercise 1.2 Carnegie Learning 2nd Edition Answers Page 20 Problem 33 Answer

Question 33.

How do you draw a right angle using a protractor?

Answer:

Right angle can be drawn as like;

Therefore, the figure below is a right angle diagram.

Carnegie Learning Geometry Chapter 1 Page 20 Problem 34 Answer

Question 34.

How do you draw an obtuse angle using a protractor?

Answer:

Obtuse angle can be drawn as like;

Therefore, the figure below is an obtuse angle diagram.

Page 20 Problem 35 Answer

Question 35.

How do you draw a straight angle using a protractor?

Answer:

Straight angle that measures 180 can be drawn as follows;

Therefore, the resultant diagram is

Page 21 Problem 36 Answer

Question 36.

How do you measure an angle using a protractor and complete the statement for the measured angle?

Answer:

In question number 5 , ∠A and ∠B are drawn . ∠A≅∠B

Measure the angle A  and complete the statement

Use a protractor to measure ∠A we got from question 5 Note down the measure of angle A.

Suppose measure of angle A is 30 degree .

then write the statement as m∠A=30° is read as ‘ measure of angle A is equal to 30 degrees”

For example  m∠A=30° is read as ‘ measure of angle A is equal to 30 degrees”

Carnegie Learning Geometry Chapter 1 Page 21 Problem 37 Answer

Question 37.

Given that ∠A and ∠B are drawn such that ∠A≅∠B, use a protractor to measure ∠B and complete the statement.

Answer:

Given

In question number 5 ,  ∠A and ∠B are drawn  ∠A≅∠B

Measure the angle B and complete the statement

Use a protractor to measure ∠A we got from question 5° is read as ‘ measure of angle A is equal to 30 degrees”

Note down the measure of angle A.

Suppose measure of angle B is 30 degree . then write the statement as m∠B=30°

For example m∠B=30 ° is read as ‘ measure of angle A is equal to 30 degrees”

Page 21 Problem 38 Answer

Question 38.

How do you read the notation m∠DEF = 110°?

Answer:

Given : m∠DEF=110° here ‘m’ means the measure and less than symbol represents angle° symbol represents degree °

m∠DEF=110°, read as measure of angle DEF is 110 degrees

m∠DEF=110°,  read as “measure of angle DEF is 110 degrees”

Carnegie Learning Geometry Chapter 1 Page 23 Problem 39 Answer

Question 39.

What are the steps to duplicate angle ∠A and construct an angle twice its measure?

Answer:

The steps to duplicate angle ∠A

First duplicate the exact copy that is angle A by following steps

Draw a straight line and label point C on one end

Draw and arc with center A and use the same radius to draw and arc with center C

Label the points as B, D on A  and label E on C

Draw an arc with  E as center by taking  radius BD . Label the intersect as F

Draw another  arc with F as center and same radius BD. Label the intersection as G

Now ∠GCE  is twice the measure of angle A∠GCE  is twice the measure of angle A

Geometry Student Text 2nd Edition Chapter 1 Tools of Geometry

Carnegie Learning Geometry Student Text 2nd Edition Chapter 1 Exercise 1.1 Solution Page 4 Problem 1 Answer

Question 1.

Given the graph where each point represents the age and height of different children:

1. Identify which axis represents the age of the children.
2. Determine the point that represents the oldest child based on the graph.
3. Explain how you identified the point that represents the oldest child.

Read and learn More Carnegie Learning Geometry Student Text 2nd Edition Solutions

Answer:

The given graph is :-

Each point in the graph represents the age and height of different children.

We have to find the point that represents the oldest child.

The given model is:-

We can see that the x-axis represents the age of children.

So the maximum value of x, gives the maximum age.

We can see that The point E has the maximum value of x.

So the the point E represents the oldest children.

The point E  represents the oldest children.

Page 4 Problem 2 Answer

Question 2.

Given the graph where each point represents the age and height of different children:

1. Identify which axis represents the age of the children.
2. Determine the points that represent children of the same age based on the graph.
3. Explain how you identified the points that represent children of the same age.

Answer:

The given graph is:-

Each point in the graph represents the age and height of different children.

We have to find the point which represents the children of same age.

Carnegie Learning Geometry Chapter 1

Page 4 Problem 3 Answer

Question 3.

Given the graph where each point represents the age and height of different children:

1. Identify which axis represents the height of the children.
2. Determine the points that represent children of the same height based on the graph.
3. Explain how you identified the points that represent children of the same height.

Answer:

The given graph is:-

Each point in the graph represents the age and height of different children.

We have to find the points that represents the children of same height.

The given graph is :-

We can see that the points C and A has the same value of y and y represents the height of children.

So that the We can see that the points C and A has the same value of y and y represents the height of children.

So that the points C and A represents the children of same height.

Similarly, the points B and D has the same value of y.

So the points B and D represents the children of same height.

The points A and C represents the children of same height.

The points B and D also represents the children of same height.

Carnegie Learning Geometry Chapter 1 Page 4 Problem 4 Answer

Question 4.

Given that a mathematical model of several points is shown, where each point represents the age and height of a different child:

1. Which point represents the oldest child?
2. Which points represent children of the same age?
3. Which points represent children of the same height?

Based on the answers provided:

• The point E represents the oldest child.
• The points A and B represent children of the same age.
• The points A and C represent children of the same height.
• The points B and D represent children of the same height.

Explain how the (x,y) coordinate system is used to represent the age and height of different children.

Answer:

Given that, A mathematical model of several points is shown. Each point represents the age and  height of a different child.

From question 1to 3, we have,

1. Which point represents the oldest child?
2. Which points represent children of the same age?
3. Which points represent children of the same height?

And their answers are,

1. The point E represents the oldest children.
2. The points A and B represents the children of same age.
3. The pointsA andC represents the children of same height.

and also The points B and D represents the children of same height.

From above solutions we can say that, each point represents the age on x−axis and  height on y−axis of a different child.

So, (x,y) coordinate system is used to represent height and age of different child.

(x,y) coordinate system is used to represent height and age of different child.

Page 4 Problem 5 Answer

Question 5.

Given a mathematical model of several lines:

1. Determine whether point C represents a specific line in the model.
2. Explain the characteristics of a line in geometry.
3. Justify your conclusion about whether C represents a specific line using the properties of lines and points.

Answer:

The given mathematical model of several lines is :-

We have to check that is C determine a specific line in the model.

The given mathematical model of several lines is :-

We know that a line is a straight continuous arrangement of several points.

Also, it has infinite length but no width and  arrowheads are indicate that a line extends infinitely in opposite directions.

But C is not a arrangement of several points. It is just a single point.

So C is not a specific line in the given mathematical model.

No, C  is not a specific line in the given mathematical model.

Carnegie Learning Geometry Chapter 1 Page 5 Problem 6 Answer

Question 6.

Given a mathematical model of several lines:

1. Determine whether CD represents a specific line in the model.
2. Explain the characteristics of a line and a line segment in geometry.
3. Justify your conclusion about whether CD determines a specific line using the properties of lines and line segments.

Answer:

The given mathematical model of several lines is :-

We have to check that is CD determine a specific line in the model.

The given mathematical model of several lines is :-

We can see that CD is a line segment of line m.

We know that a line is described as a straight continuous arrangement of an infinite number of points.

CD is also continuous arrangement of infinite points.

Also a line segment is also a line.

So CD determines a specific line in the model.

Yes, CD determines the specific line in the given model.

Solutions For Tools of Geometry Exercise 1.1 In Carnegie Learning Geometry Page 5 Problem 7 Answer

Question 7.

Given a mathematical model of several lines:

1. Determine whether m represents a specific line in the model.
2. Explain the characteristics of a line in geometry.
3. Justify your conclusion about whether m determines a specific line using the properties of lines.

Answer:

The given mathematical model of several lines is :-

We have to check that is m determines the specific line in the model.

The given mathematical model of several lines is :-

We know that a line is a straight continuous arrangement of several points.

Also it has infinite length but no width and arrowheads are indicate that a line extends infinitely in opposite directions.

In the above model, m satisfies all of these conditions.

So m is a specific line in the model.

Yes, m is a specific line in the given mathematical model.

Carnegie Learning Geometry Chapter 1 Page 5 Problem 8 Answer

Question 8.

Given a mathematical model of several lines:

1. How many points are needed to describe a specific line?
2. Explain the characteristics of a line in geometry and why it requires a certain number of points to be described.

Answer:

The given mathematical model of several lines is:-

We have find that how many points are needed to describe a specific line.

We know that:-

A line is described as a straight continues arrangement of infinite number of points.

So to describe a specific line we need a continuous infinite points that should be in straight arrangement.

To describe a specific line we need a continues infinite points that should be in straight arrangement.

Tools Of Geometry Solutions Chapter 1 Exercise 1.1 Carnegie Learning Geometry Page 5 Problem 9 Answer

Question 9.

Given a mathematical model of several lines:

1. Identify the line represented by AB in the model.
2. Determine an alternative name for the line AB.
3. Explain how you identified the alternative name for line AB.

Answer:

The given mathematical model of several lines is :-

We have to find alternative name for line AB.

The given mathematical model of several lines is :-

We can see that AB represents a line in the model.

N is also represents the same line.

So we can say that the alternative name for line AB is n.

In the given mathematical model, alternative name for line AB is n .

Carnegie Learning Geometry Chapter 1 Page 5 Problem 10 Answer

Question 10.

Given a mathematical model of several lines:

1. Identify three points that are collinear in the model.
2. Explain the definition of collinear points.
3. Choose a line from the model and list the three points on that line which are collinear.

Answer:

The given mathematical model of several lines is :

We have to give name of three points that are collinear.

The given mathematical model of several lines is :-

We know that the points that are located on same line are called collinear.

Choose the line l from the given model.

We can see that three points A,C,E are located on the same line l.

So that the points A,C,E are collinear.

The three points A,C,E are collinear.

Step-By-Step Solutions For Carnegie Learning Geometry Chapter 1 Exercise 1.1 Page 5 Problem 11 Answer

Question 11.

Given a mathematical model of several lines:

1. Define what it means for points to be collinear.
2. Identify three points in the model that are not collinear.
3. Provide three sets of points from the model that are not collinear.

Answer:

Given that, Line AB can be read as “line AB. and

We have to Name three points that are not collinear in the model.

We know that, Collinear points are points that are located on the same line. So the points which are not on same line are not a colinear points.

From given model the following points are not colinear,

3. A,B,D
4. A,C,D
5.  C,E,D

The following points are not colinear,

1. A,B,D

3. A,C,D
4. C,E,D

Carnegie Learning Geometry Chapter 1 Page 6 Problem 12 Answer

Question 12.

Given planes p, w, and z in the figure:

1. Determine whether there is a single point that is common to all three planes.
2. Explain what it means for planes to be parallel.
3. Based on the figure, conclude whether the given planes are parallel or intersecting.
4. Justify your conclusion about the relationship between the planes.

Answer:

Given planes p,w,and z are given in figure,

From given figure we can conclude that, there is no single point which is common to given three planes.

So given plane are not intersecting each other.

Therefore, given planes are parallel.

Given three planes are parallel.

Carnegie Learning Geometry Chapter 1 Exercise 1.1 Free Solutions Page 6 Problem 13 Answer

Question 13.

Given planes p, w, and z in the figure:

1. Determine the relationship between the planes p, w, and z.
2. Explain how the planes intersect with each other.
3. State the number of points where the planes intersect.
4. Conclude the type of intersection that occurs among the planes.

Answer:

Given planes p, w and z are given figure below,

From above figure of planes, we can say that all planes p, w and z intersect with each other at a single line.

So there are infinite number of points of intersection.

All planes p, w and z intersect with each other at a single line.

So there are infinite number of points of intersection.

i.e. Three Planes Intersecting in a Line

Carnegie Learning Geometry Chapter 1 Page 6 Problem 14 Answer

Question 14.

Given planes p, w, and z in the figure:

1. Describe the relationship between planes p and w.
2. Explain how planes p and w intersect with plane z.
3. Determine if planes p and w intersect with each other.
4. Conclude the type of intersection that occurs among the planes.

Answer:

Given planes p, w and z are given in figure below,

From above figure we can say that, the planes p and w are intersects the plane z at two different lines but the planes p and w are not intersectes to each other so the planes p and w are parallel planes.

The planes p and w are intersects the plane z at two different lines but the planes p and w are not intersects to each other so the planes p and w are parallel planes.

i.e. Two Parallel Planes and the Other Cuts Each in a Line.

Page 6 Problem 15 Answer

Question 15.

Given planes p, w, and z in the figure:

1. Describe how plane p intersects with plane w.
2. Describe how plane p intersects with plane z.
3. Describe how plane w intersects with plane z.
4. Determine the type of intersection that occurs among the planes p, w, and z.
5. Conclude how the planes p, w, and z intersect with each other in the figure.

Answer:

Given planes p, w and z are shown in figure below,

From the above figure we can say that, Plane p and w intersect the plane z.

Plane p and z intersect the plane w.

Plane w and z intersect the plane p That means all planes p, w and z intersect each other but not in single line, they are intersects each other in two different lines.

The planes p, w and z intersects each other in two different lines.

i.e. Each Plane Cuts the Other Two in a Line.

Carnegie Learning Geometry Chapter 1 Page 6 Problem 16 Answer

Question 16.

Given planes p, w, and z in the figure:

1. Describe the intersection of planes p and w.
2. Describe the intersection of planes p and z.
3. Describe the intersection of planes w and z.
4. Determine the type of intersection that occurs among the planes p, w, and z.
5. Conclude how the planes p, w, and z intersect with each other in the figure.

Answer:

Given planes p, w and z are shown in figure below,

From above figure we can conclude that, all three planes p, w and z are intersects each other at single point.

The intersection of the three planes is a point.

Carnegie Learning Geometry Chapter 1 Page 6 Problem 17 Answer

Question 17.

List and describe all possible intersections of three planes.

Provide the following scenarios in your answer:

1. Intersecting at a point.
2. Each plane cutting the other two in a line.
3. Two parallel planes with the third plane cutting each in a line.
4. Three planes intersecting in a line.
5. Two coincident planes with the third plane intersecting them in a line.
6. Three parallel planes.

Answer:

We have to list all of the possible intersections of three planes.

Following are the list all of the possible intersections of three planes.

Intersection of Planes.

Intersecting at a Point. Each Plane Cuts the Other Two in a Line.

Two Parallel Planes and the Other Cuts Each in a Line.

Three Planes Intersecting in a Line.

Two Coincident Planes and the Other Intersecting Them in a Line.

Three Parallel Planes.

Carnegie Learning Geometry Exercise 1.1 Student Solutions Page 6 Problem 18 Answer

Question 18.

List and describe all possible intersections of a line and a plane.

Provide the following scenarios in your answer:

1. The line intersects the plane at a point.
2. The line is parallel to the plane.
3. The line is on the plane.

Answer:

Following are the all possible intersections of a line and a plane.

Line intersects the plane at a point

line is parallel to the plane

line is on the plane

All possible intersections of a line and a plane,

1.   Line intersects the plane at a point
2.  line is parallel to the plane
3. line is on the plane

Carnegie Learning Geometry Chapter 1 Page 7 Problem 19 Answer

Question 19.

Given collinear points A, B, and C such that point B is located halfway between points A and C:

1. Sketch the figure with points A, B, and C.
2. Ensure that B is exactly halfway between A and C.

Answer:

Given that, a geometric figure using collinear points A, B, and C such that point B is located halfway between points A and C.

We have to Sketch the figure.

Required sketch is

Page 7 Problem 20 Answer

Question 20.

Given collinear points A, B, and C such that point B is located halfway between points A and C:

1. Draw the figure with points A, B, and C ensuring that they are collinear.
2. Place point B exactly halfway between points A and C.

Answer:

Given that, a geometric figure using collinear points A, B, and C such that point B is located halfway between points A and C.

We have to draw the figure.

Required sketch is

Carnegie Learning Geometry Chapter 1 Page 7 Problem 21 Answer

Question 21.

Consider the following:

1. Sketch a geometric figure using collinear points A, B, and C such that point B is located halfway between points A and C.
2. Draw a geometric figure using collinear points A, B, and C such that point B is located halfway between points A and C.
3. Explain the difference between sketching a figure and drawing a figure.
4. For part 1(a), sketch the figure without the use of any tools.
5. For part 1(b), draw the figure using a ruler and straightedge to ensure accuracy.

Answer:

In the question 1 (a), we sketch the figure, the figure is created without the use of tools and in the question 1 (b), we draw a geometric figure, the figure is created with the use of ruler and straightedge.

Therefore, in the question 1 (a), we sketch the figure and in the question 1 (b), we draw a figure.

Page 7 Problem 22 Answer

Question 22.

Draw and label three coplanar lines.

1. Begin by drawing a line AB.
2. Next, draw a line CD in the same plane as line AB.
3. Finally, draw a line EF in the same plane as lines AB and CD.

Answer:

We have to draw and label three coplanar lines.

First we draw a line AB then draw a line CD and line EF in the same plane.

The required coplanar lines are

The required three coplanar lines are

Carnegie Learning Geometry Chapter 1 Page 8 Problem 23 Answer

Question 23.

Consider the following:

Your classroom is in the shape of a square.

1. Define skew lines and explain their properties.
2. Identify two lines in your classroom that are skew lines.
3. Draw a figure to represent the square shape of your classroom and label the two skew lines L1 and L2.

Answer:

Skew lines are two or more lines that are located in the same plane.

My classroom is in square shape.

So the two skew line are L1 and L2 which represent the two lines of my classroom in figure given below.

Therefore,  the two skew line areL1 and L2 which represent the two lines of my classroom.

Page 8 Problem 24 Answer

Question 24.

1. Draw a line segment and mark point A at one endpoint of the line segment.
2. Mark point B anywhere on the line segment.
3. Label the line segment \(\overrightarrow{A B}\).

Answer:

First we draw a line and mark A on the end point of the line and B is anywhere on the line.

The sketch of A⃗B is

The sketch of A⃗B is

Carnegie Learning Geometry Chapter 1 Page 8 Problem 25 Answer

Question 25.

1. Draw a line and mark point B at one endpoint of the line.
2. Mark point A anywhere on the line.
3. Label the directed line segment \(\overrightarrow{B A}\)

Answer:

First we draw a line and mark B on the end point of the line and A is anywhere on the line.

The sketch of B⃗A is

The sketch of B⃗A is

Page 8 Problem 26 Answer

Question 26.

1. Define a ray and explain its properties.
2. Explain how to properly name a ray using two capital letters.
3. Given points A and B:
4. Draw ray \(\overrightarrow{A B}\)and explain how it is different from ray \(\overrightarrow{B A}\).
5. Clarify why \(\overrightarrow{A B}\) and \(\overrightarrow{B A}\) are not the same ray.

Answer:

A ray is portion of a line that begins with a single point and extends infinitely in one direction. The endpoint of a tray is the single point where the ray begins.

A ray is named using two capital letters, the first representing the endpoint and second representing any other point on the ray.

Ray AB can be written using symbols A⃗B as and is read as “ray AB”.

So, the A⃗B and B⃗A are not the same ray , the are two different ray.

Therefore, the A⃗B and B⃗A are not the same ray.

Tools Of Geometry Exercise 1.1 Carnegie Learning 2nd Edition Answers Page 8 Problem 27 Answer

Question 27.

Given a figure where G is the endpoint of a ray and F is any point on the ray:

1. Define what a ray is and describe its properties.
2. Identify the endpoint of the ray in the given figure.
3. Name the ray using the correct notation.
4. Explain why the given figure represents the ray \(\overrightarrow{G F}\) and not \(\overrightarrow{F G}\).

Answer:

A ray is portion of a line that begins with a single point and extends infinitely in one direction.

The endpoint of a tray is the single point where the ray begins.

In the given figure G is the endpoint of the ray and the point F is the any point on the ray.

So, the given figure represent the ray GF or G⃗F.

So, the given figure represent the ray GF or G⃗F.

Carnegie Learning Geometry Chapter 1 Page 9 Problem 28 Answer

Question 28.

1. Draw a line segment and mark point A on one endpoint and point B on the other endpoint.
2. Label the line segment as \(\overline{A B} \).

Answer:

First we draw a line and mark the point A on one endpoints and point B on the other endpoint of the line segment.

The line segment AB is

Therefore, the line segment AB is

Page 9 Problem 29 Answer

Question 29.

1. Draw a line segment and mark point A on one endpoint and point B on the other endpoint.
2. Label the line segment as \(\overline{B A} \).

Answer:

First we draw a line and mark the point A on one endpoints and point B on the other endpoint of the line segment.

The line segment BA is

Therefore, the line segment BA is

Carnegie Learning Geometry Chapter 1 Page 9 Problem 30 Answer

Question 30.

1. Define a line segment and describe its properties.
2. Explain what endpoints of a line segment are.
3. Given points A and B, discuss whether \(\overline{A B} \) and \(\overline{B A} \) represent the same line segment.
4. Justify why \(\overline{A B} \) are names for the same line segment.

Answer:

A line segment is portion of a line that includes two points and all the collinear points between the two points.

The endpoints of a line segment are the points where the line segment begins and ends.

So, line segment AB is the same as line segment BA. Both pass through the same two points A and B.

Therefore, ABˉ and BAˉ names for same line segment is true.

Page 9 Problem 31 Answer

Question 31.

How do you measure the length of a line segment using a ruler, and what is the length of the line segment AB if measured to be 5 inches?

Answer:

Given-A line segment AB is given.

To Find-Length of the line segment AB.

(1) Line segment AB can be measured by the ruler by placing the ruler over line segment AB.

(2) The length between the two ends of the ruler gives the measure of the line segment AB.

(3) The length is 5 inches.

The length of the line segment AB is 5 inches.

Page 9 Problem 32 Answer

Question 32.

Fill in the blanks for the following statement:

“AB= _______ inches” is read as “the distance from point A to point B is equal to ______ inches.”

Use the given measurement of 5 inches to complete the statement.

“AB = 5 inches” is read as “the distance from point A to point B is equal to 5 inches.”

Answer:

Given-

“AB=  _______inches” is read as “the distance from point A to point B is equal to ______inches.”

To Find-It is needed to fill in the blanks.

“AB=5 inches” is read as “the distance from point A to point B is equal to 5 inches.”

“AB=5 inches” is read as “the distance from point A to point B is equal to 5 inches.”

Carnegie Learning Geometry Chapter 1 Page 9 Problem 33 Answer

Question 33.

Fill in the blanks for the following statement:

“mAB = _______ inches” is read as “the measure of line segment AB is equal to ______ inches.”

Use the given measurement of 5 inches to complete the statement.

“mAB = 5 inches” is read as “the measure of line segment AB is equal to 5 inches.”.

Answer:

Given-

“mAB =_______ inches” is read as “the measure of line segment AB is equal to ____inches

To Find- It is needed to fill in the blanks.

“mAB = 5 inches” is read as “the measure of line segment AB is equal to 5 inches.

“mAB = 5 inches” is read as “the measure of line segment AB is equal to 5 inches.

Page 9 Problem 34 Answer

Question 34.

Given: “mCF = 3 inches”

To find: How to read this statement.

It is read as: “The measure of line segment CF is 3 inches.”

It is read as: “The measure of line segment CF is 3 inches.”

Answer:

Given-“mCF = 3 inches”

To Find- It is needed to find out how to read this?

It is read as-“measure of line segment CF is 3 inches.”

It is read as “measure of line segment CF is 3 inches.”

Page 9 Problem 35 Answer

Question 35.

Given “SP = 8 inches,” how should this be read?

It should be read as “the measure of line segment SP is 8 inches.

Answer:

Given- “SP =8 inches”

To Find- It is needed to find out how to read this?

It is read as-“measure of line segment SP is 8 inches.”

It is read as “measure of line segment SP is 8 inches.”

Carnegie Learning Geometry Chapter 1 Page 9 Problem 36 Answer

Question 36.

To Find: The name of the geometric figure using the symbols.

The given figure is a line segment. It can be named as FGˉ.

This represents the line segment FG.

What is the name of the geometric figure?

Answer:

Given-

To Find- The name of the geometric figure using the symbols.

The given figure is a line segment. It can be named as-FGˉ

This represents the line segment FG.

The name of the geometric figure is FGˉ.

Page 4 Problem 37 Answer

Question 37.

To Find: The name of the geometric figure using the symbols.

The given figure is a line and it can be named as FGˉ.

This means that it is a line FG and it extends in both directions.

What is the name of the figure?

Answer:

Given-

To Find- The name of the geometric figure using the symbols.

The given figure is a line and it can be named as-

This means that it is a line FG and it extends in both directions.

The name of the figure is

It is a line.

Carnegie Learning Geometry Chapter 1 Page 10 Problem 38 Answer

Question 38.

It is necessary to draw and label two congruent line segments.

Congruent line segments have equal lengths.

These are the two-line segments AB and CD.

Both have equal lengths and hence they are congruent line segments.

What are the two congruent line segments?

Answer:

It is needed to draw and label two congruent line segments.

The Congruent Line Segments have equal lengths.

These are the two-line segments AB and CD.

Both have equal lengths and hence they are Congruent Line Segments.

The two congruent line segments are-

Page 10 Problem 39 Answer

Question 39.

To Find: Using symbols, it is needed to write three valid conclusions based on the figure. Also, how is each conclusion read?

Three valid conclusions and their readings are:

1. \(\overline{F G} \| \overline{H I}\) (FG and HI are line segments)
2. \(m \overline{F G}=10 \mathrm{~cm} \text { and } m \overline{H I}=10 \mathrm{~cm}\) (The measures of line segments FG and HI are 10cm)
3. \(m \overline{F G}=m \overline{H I}=10 \mathrm{~cm}\) (Line segments FG and HI are equal)

What are the three valid conclusions and their readings?

Answer:

Given-

To Find- Using symbols, it is needed to write three valid conclusions based on the figure. Also, how each conclusion are read?

Three valid conclusions and their readings are-

(1) FGˉHIˉ (FG and HI are line segments)

(2) ​mFGˉ=10 cm mHIˉ=10 cm (measure of line segments FG and HI are 10 cm)

(3) mFGˉ=mHIˉ=10 cm

Line segment FG and HI are equal.

Three valid conclusions and their readings are-

(1) FGˉHIˉ  (FG and HI are line segments)

(2) mFGˉ=10cmmHI=10cm  (measure of the line segments FG and HI are 10 cm)

(3) mFGˉ=mHIˉ=10 cm (Line segments FG and HI are equal.)

Carnegie Learning Geometry Chapter 1 Page 10 Problem 40 Answer

Question 40.

To specify if there is a difference between GH and HG.

A ray is the portion of a line that begins with a single point and extends infinitely in one direction.

The endpoint of a ray is the single point where the ray begins.

A ray is named using two capital letters, the first representing the endpoint and the second representing any other point on the ray.

Ray GH can be written using symbols as \(\overrightarrow{G H}\) and is read as “ray GH”.

So GH and HG are not the same ray; they are two different rays.

Hence, GH and HG are not the same.

Is there a difference between GH and HG?

Answer:

Given : To specify is here is a difference between GH and HG

A ray is the portion of a line that begins with a single point and extends infinitely in one direction.

The endpoint of a tray is the single point where the ray begins.

A ray is named using two capital letters, the first representing the endpoint and the second representing any other point on the ray.

Ray GH can be written using symbols as GH and is read as “ray GH”.

So GH and HG are not the same ray, they are two different rays.

Hence, GH and HG are not same.

Page 10 Problem 41 Answer

Question 41.

A line segment is named using two capital letters representing the two endpoints of the line segment.

HG is a line segment that starts at H and ends at G, while GH is a line segment that starts at G and ends at H.

HG and GH are two different line segments starting at H and G, and ending at G and H respectively.

What is the difference between the line segments HG and GH?

Answer:

Given :

A line segment is named using two capital letters representing the two endpoints of the line segment.

HG is a line segment which starts at H and ends at G while GH is a line segment which starts at G and ends at H.

HG and GH are two different line segments starting at H and G , and ends at G and H respectively.

Carnegie Learning Geometry Chapter 1 Page 11 Problem 42 Answer

Question 42.

It is given that JK = MN and \(\overline{J K}=\overline{M N}\).

To explain the difference between the statements:

It is represented as: \(\overline{J K}=\overline{M N}\)

It shows that JK and MN have the same magnitude and direction.

Hence, the given statements are different.

How are the statements JK = MN and \(\overline{J K}=\overline{M N}\) different?

Answer:

It is given that JK=MN and JKˉ=MNˉ.

To explain the difference of the statements.

It is represented as,

∣JKˉ∣=∣MNˉ∣

It shows the JK and MN are same magnitude and same direction.

Hence, the given statements are different.