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