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Geometry Homework Practice Workbook 1st Edition Chapter 2 Inductive Reasoning and Conjecture

Page 27 Problem 1 Answer

We have been given QA=QA in the question.

Here, we are asked to find the property of the given comparison.

Here, we have seen that both side parameter is same along with equal sign.

Therefore, we can say that it follows the reflexive property of equality which defines that a number or variable is equal to itself.

Hence, according to the given comparison QA=QA we have found that it follows the reflexive property of equality.

Page 27 Problem 2 Answer

We have been given AB≅BC, BC≅CE,AB≅CE in the question.

Here, we are asked to find the property among these comparisons.

Now, we have seen that all three sides are congruent to each other.

Finally, we will conclude that the given statement follows the transitive property of congruency which tells about similar shape and size.

Hence, using the given comparison AB≅BC,BC≅CE,AB≅CE we got that it followed the transitive property of congruency.

Page 27 Problem 3 Answer

We have been given PR=PQ+QR in the question.

We are asked to determine the property of the given comparison.

Here, we have seen that there are three points P,Q,R and Q is the midpoint.

Now, after evaluating the given comparison we can say that it follows the segment addition postulate which tells about the point on the line.

Hence, according to the given statement PR=PQ+QR we have found that it follows the segment addition postulate.

Page 27 Problem 4 Answer

We have been given AB+BC=EF+FG

AB+BC=AC

EF+FG=AC

​Here, we are asked to find the property of the given statement in the question.

Now, we have seen that there is a comparison between the sum of two points with other one or two points.

Now, we can say that it follows substitution and transitive property which tells about replacing the values and similarity.

Hence, according to the given statements

AB+BC=EF+FG

AB+BC=AC

EF+FG=AC

We have got that this statement follows substitution and transitive property.

Page 27 Problem 5 Answer

We have been given two statements SU≅LR

TU≅LN  in the question.

Here, we will prove the congruency of the given statement ST≅LR

Finally, using the given points and mathematical concepts we will conclude the final result.

We have SU≅LR

TU≅LN

Here, we have seen that T, and N are the midpoints of the lines SU,LR

Now, use each mathematical concept to prove ST≅LR

Using the given information

Hence, using the given statement

SU≅LR

TU≅LN

We prove that ST≅NR

Page 27 Problem 6 Answer

We are given a partially filled proof for the congruence of CD≅AB.

And it is given that, AB≅CD.

We are required to complete the proof.

Here, we will use properties of congruence to fill this.

We will compare the reasons and the statements to each other and then fill the given proof as,

We can fill the given proof as

Page 27 Problem 7 Answer

We are given that,AB≅DE

B is the midpoint of AC

E is the midpoint of DF

We are required to prove that, BC≅EF.

Here, we will use properties of congruence and equality to complete the given proof.

We will compare the reasons and the statements to each other and then fill the given proof as,

The given proof can be completed as,

Page 28 Problem 8 Answer

We are given a figure, in which the distance from Grays on to Apex is the same as the distance from Redding to Pine Bluff.

That is, GA≅RP.

We are required to prove that the distance from Grays on to Redding is equal to the distance from Apex to Pine Bluff. GR≅AP.

Here, we will use properties of equality to prove this.

The given statement can be proved as,

The proof of the distance from Grayson to Redding is equal to the distance from Apex to Pine Bluff is given by,

 

Geometry Homework Practice Workbook 1st Edition Chapter 2 Inductive Reasoning and Conjecture

Page 25 Problem 1 Answer

In this question, we have been given the value: 80=m∠A

We need to state the property that justifies the statement.

By using Inductive Reasoning and Conjecture, we will calculate the result.

As we see this statement uses the symmetric property.

The symmetric property of equality states that regardless of which side of an equal sign they are on, both sides are equal.

Hence, the given statement 80=m∠A, then 80=m∠A uses the symmetric property.

It states that regardless of which side of an equal sign they are on, both sides are equal.

Page 25 Problem 2 Answer

In this question, we have been given the value:

RS=TU

TU=YP

RS=YP

We need to state the property that justifies the statement.

By using Inductive Reasoning and Conjecture, we will calculate the result.

As we see this statement uses the transitive property of equality.

The transitive property of equality states that if a is related to b and b is related to c, then a is related to c.

Hence, the given statement RS=TU and TU=YP, then RS=YP uses the transitive property of equality. It states that if a is related to b and b is related to c, then a is related to c.

Page 25 Problem 3 Answer

In this question, we have been given the value:

7x=28

x=4

We need to state the property that justifies the statement.

By using Inductive Reasoning and Conjecture, we will calculate the result.

As we see this statement uses the division property of equality.

It states that when both sides of an equation are divided by the same non-zero number, the two sides remain equal, according to the division property of equality.

Hence, the given statement 7x=28,  then x=4 uses the division property of equality.

It states that when both sides of an equation are divided by the same non-zero number, the two sides remain equal, according to the division property of equality.

Page 25 Problem 4 Answer

In this question, we have been given the value:

VR+TY=EN+TY

VR=EN​

We need to state the property that justifies the statement.

By using Inductive Reasoning and Conjecture, we will calculate the result.

As we see this statement uses the subtraction property of equality.

The subtraction property of equality states that if one side of an equation is subtracted, the other side must likewise be subtracted to maintain the equation the same.

Hence, the given statement VR+TY=EN+TY,  then VR=EN uses the subtraction property of equality.

It states that if one side of an equation is subtracted, the other side must likewise be subtracted to maintain the equation the same.

Page 25 Problem 5 Answer

We are given m∠1=30 & m∠1=m∠2

​We have to justify the statement that if the above two equations are true then m∠2=30

We will be using some algebraic properties to justify the required statement.

We are given m∠1=30&

m∠1=m∠2​

By using transitive property we can say that m∠2=30

Thus, the transitive property is shown by the statement if ​m∠1=30&

m∠1=m∠2 then m∠2=30

Page 25 Problem 6 Answer

We are given a table as shown

Also, we are given an equation 8x−5=2x+1

We have to complete the table and prove that x=1

We will be using some algebraic properties to complete the table and prove the required result

We are given a table and an equation8x−5=2x+1

After completing the table we get,

Here, the highlighted portion in the table represent the blanks given in the table of question

Thus, we have completed the table and proved x=1 for the given equation 8x−5=2x+1

Page 25 Problem 7 Answer

We are given a figure and PQˉ≅QSˉ&QSˉ=STˉ

We have to prove PQ=ST

We will be using some algebraic properties and congruent lines concept to prove the same.

We are given a figure

Two-column proof to verify the conjecture is

Hence, we have written a two-column proof to verify the conjecture “if PQˉ≅QSˉ&QSˉ≅STˉ then

PQ=ST”, for the given figure

The two-column proof is

Page 26 Problem 8 Answer

In this question, we have been given the value:

m∠ABC+m∠CBD=90

m∠ABC=3x−5

m∠CBD=x+1/2

We need to write a two-column proof to verify each conjecture.

By using Inductive Reasoning and Conjecture, we will calculate the result.

We analyze the diagram

Since the ∠ABC and m∠CBD are complementary angles.

We substitute the values

m∠ABC+m∠CBD=90

3x−5+x+1/2=90

6x−10+x+1=90×2

7x−9=180

Further solving, we get

7x=189

x=189/7

x=27

​Hence, the value of x=27.

Since the ∠ABC and ∠CBD are complementary angles.

We substitute the values and analyze the diagram

Page 26 Problem 9 Answer

In this question, we have been given the value: I=prt

We need to solve the formula for r and justify each step.

By using Inductive Reasoning and Conjecture, we will calculate the result.

We divide the whole formula by pt to solve for r

I/p×t=p×r×t/p×t

I/p×t=r

​Hence, the formula for r is r=I/p×t. Since we divide the whole formula by pt and using the formula of simple interest.

Geometry Homework Practice Workbook 1st Edition Chapter 2 Inductive Reasoning and Conjecture

Page 23 Problem 1 Answer

We have been given a figure where lines and planes intersecting each other or just passing one another.

We have to show line p lies in a plane N

First, we will identify plane N and locate line p, we see that the points D and A are on line p which lies in plane N So, according to postulate 2.5  which states that if two points lie in a plane then the entire line containing those points lies in that plane, we get that line p lies in plane N.

Hence by postulate 2.5, which states that if two points lie in a plane then the entire line containing those points lies in that plane, we get that line p lies in plane N.

Page 23 Problem 2 Answer

We have been given a statement that three collinear points determine a plane.

We have to show three collinear points to determine a plane is true or not.

If three collinear points intersect then their intersection of joining may be a line or plane Postulate 2.7 states that two or more points intersect then their join of intersection may be line or plane, so the statement is sometimes true.

Hence two or more points intersect then their join of intersection may be line or plane.

So the statement “Three collinear points determine a plane” is sometimes true.

Page 23 Problem 3 Answer

We have been given a statement two points A and B determine a line.

We have to show two points A and B determine a line.

Postulate 2.1 states that through any two points there is exactly one line possible in any direction.

So the statement is always true.

Hence the statement is always true as by postulate 2.1 that that through any two points there is exactly one line.

Page 23 Problem 4 Answer

We have been given a statement “a plane contains at least three lines”.

We have to whether a plane contains at least three lines or not.

A plane contains at least three non-collinear points. If two points lie in a plane, then the line containing them lies in the plane.

If two planes intersect, then their intersection is a line.

As the plane has three non-collinear points so we can say the plane can have three lines but by two parallel lines also we can make a plane.

So the statement is not always true.

Hence as the plane has three non-collinear points so we can say the plane can have three lines but by two parallel lines also we can make a plane.

So the statement “A plane contains at least three lines.” is not always true.

Page 23 Problem 5 Answer

We have been given figure as,

We have to show that G,P are collinear.

We know that points G,P lies in the same plane.

We also know that any two points in the same plane are always collinear.

So, we can say G,P are collinear.

Hence, for the given figure, where, DG, and DP is in plane J, and H lines on DG, we stated the postulate that, two points in same plane are always collinear, to show that the given statement, G, P are collinear, is true.

Page 23 Problem 6 Answer

We have been given figure as,

We have to show that G,P are collinear.We know that points G,P lies in the same plane.

We also know that any two points in the same plane are always collinear.

So, we can say G,P are collinear.

Hence, for the given figure,where, DG,DP is in plane J, and H lines on DG, we stated the postulate that, two points in same plane are always collinear, to show that the given statement, G,P are collinear, is true.

Page 23 Problem 7 Answer

We have been given figure as,We also have been given that B,C are the midpoints of AC,BD respectively.

We have to prove that AB=CD.

We can prove it using midpoint properties.

We have been given figure as,

As,B is midpoint of AC, so we get, AB=BC ……(1)

As, C is the midpoint of BD, so we get, BC=CD……(2)

By equating equations (1) and (2), we get,

AB=CD

Hence, for the given figure,where, point B is the midpoint of AC and C is the midpoint of BD, we wrote a paragraph proof and using midpoint properties we showed that AB=CD.

Page 24 Problem 8 Answer

We have been given a figure as,

We have to show that the lines l and m intersect at point Q.We know that intersection of two lines is always a point.

We know that planes J,K intersect at line m.We can see that line l lies on plane K.So, we can say that line l will also intersect line m.

So, we can say that the intersection point of lines l,m is point Q.

Hence, for the given figure,graphics 2 for exercise 2 page 24 geometry we explained that the given statement, the lines intersect at point, is true, by stating the postulate that, intersection of two lines is always a point.

Page 24 Problem 9 Answer

Here we have given a statement about the intersection of planes and we have to prove that whether the statement is true for always, never, or for sometimes.

So the given statement is the ”Intersection of two planes contains at least two points”.

So as we already know that in geometry two mutually intersecting planes are always formed a straight line.

In other words, two planes intersect each other at a straight line, and we also know a straight line is made up of two points.

Hence, it is obvious that as the two intersecting planes contain a straight line, they must contain at least two points on their surface always.

Here we have given a statement about the intersection of planes and we have to prove that whether the statement is true for always, never, or for sometimes.

So the given statement is the ”Intersection of two planes contains at least two points”.

So as we already know that in geometry two mutually intersecting planes are always formed a straight line.

In other words, two planes intersect each other at a straight line, and we also know a straight line is made up of two points.

Hence, it is obvious that as the two intersecting planes contain a straight line, they must contain at least two points on their surface always.

Page 24 Problem 10 Answer

In the question, here we have given that two lines i.e.

line m and the line TQ lie on the plane A as shown in Fig.1 below.

We have to tell that the points L,T and line m lies in the same plane.

So we already know that line m lies on the plane A and it clearly visible from Fig.1 that points L and T are one line m.

Thus ultimately we can say that the points L,T will lie on the plane A.

Hence we can say the points L,T and line m lies in the same plane i.e. plane A.

In the question, here we have given that two lines i.e. line m and the line TQ lie on the plane as shown in Fig.1 below.

We have to tell that the points L,T, and line m lies in the same plane.

So we already know that line m lies on plane A and it clearly visible from Fig.1 that points L and T are one line m.

Thus ultimately we can say that  the points L,T will lie on the plane A.

Hence we can say the points L,T and line m lies in the same plane i.e. plane A

Page 24 Problem 11 Answer

In the question, here we have given that two lines i.e. line m and the line TQ lie on the plane as shown in Fig.1 below.

And we just have to prove that the lines m,ST intersect each other at a point T.

So it is very clear that from Fig.1 we can say that the lines i.e. line m and line TQ on plane A intersect at point T.

And as the line TS is also passing through that point T, hence we confidently can say that both the line m and TS are intersecting each other at point T.

In the question, here we have given that two lines i.e. line m and the line TQ lie on the plane as shown in Fig.1 below.

And we just have to prove that the lines m,ST intersect each other at a point T.

So it is very clear that from Fig.1 we can say that the lines i.e. line m and line TQ on plane A intersect at point T.

And as the line TS is also passing through that point T, hence we confidently can say that both the line m and TS are intersecting each other at point T.

Page 24 Problem 12 Answer

Here in the question,  we have given that AB=CD and E is the midpoint of the lines AB,CD  as shown in Fig.1 below.

We have to write a paragraph proof of AE≅ED.So as we know That E  is the midpoint of the line AB and CD, which means E is dividing the two lines i.e AB and CD into two equal parts.Hence implies that AE≅ED.

So as we know that That E is the midpoint of the line AB and CD, which means E is dividing the two lines i.e AB and CD into two equal parts.

Hence implies that AE≅ED.

Page 24 Problem 13 Answer

In this question, we have been given Points A,B and C are noncollinear.

Points B,C and D are noncollinear. Points A,B,C and D are noncoplanar.

We need to describe two planes that intersect inline BC.

By using Inductive Reasoning and Conjecture, we will calculate the result.

Since two points define a line, three-point define a plane.

Therefore, one plane passing through points B,C and A.

Point D in not included in this plane.

The other plane passing through the point B,C and D.

Point A in not included in this plane.

Thus, those two planes intersect inline BC.

Hence, those two planes intersect inline BC.

Since two points define a line, three-point define a plane.

The other plane passing through the point B,C and D.

Geometry Homework Practice Workbook 1st Edition Chapter 2 Inductive Reasoning and Conjecture

Page 21 Problem 1 Answer

We are given that if the sum of the measures of two angles is 180, then the angles are supplementary. m∠A+m∠B is 180.

We are given the conclusion that ∠A and∠B are supplementary.

We have to determine whether the stated conclusion is valid based on the given information.

We will use law of detachment and determine the result.

We have been given that if the sum of the measures of two angles is 180, then the angles are supplementary.

We are also given that m∠A+m∠B=1800

The statement​m∠A+m∠B=1800

⇒∠A+∠B=1800​

Thus, the sum of angles A and B is 180.

Hence the conclusion that ∠A and ∠B are supplementary is valid.

If the sum of the measures of two angles is 180, then the angles are supplementary and m∠A+m∠B=1800, then the conclusion that ∠A and∠B are supplementary is valid as the conclusion holds true using law of detachment.

Page 21 Problem 2 Answer

We are given that if the sum of the measures of two angles is 90, then the angles are complementary and m∠ABC is 45 and m∠DEF is 48

We are given the conclusion that ∠ABC and ∠DEF are complementary.

We have to determine whether the stated conclusion is valid based on the given information.

We will use law of detachment and calculate the result.

We have been given that  If the sum of the measures of two angles is 90, then the angles are complementary.

And m∠ABC=450

m∠DEF=480​

Calculating the sum we get,

m∠ABC+m∠DEF=45+48

m∠ABC+m∠DEF=930

Thus, the sum of ∠ABC,∠DEF is not 90.

Hence the conclusion that ∠ABC and ∠DEF are complimentary is not valid.

If the sum of the measures of two angles is 90, then the angles are complementary.

m∠ABC is 45 and m∠DEF is 48, then the conclusion that ∠ABC and ∠DEF  are complementary is invalid, as the sum of angles is not 90 and makes the conclusion false.

Page 21 Problem 3 Answer

We are given that if the sum of the measures of two angles is, then the angles are supplementary, and ∠1,∠and 2 are a linear pair.

We are given the conclusion that ∠1 and∠2 are supplementary.

We have to determine whether the stated conclusion is valid based on the given information.

We will use law of detachment and calculate the result.

We have been given that if the sum of the measures of two angles is 180, then the angles are supplementary.

And ∠1 and∠2 are a linear pair.

We know that in a linear pair sum of all the angles is 180.

Thus, ∠1+∠2=180.

And the conclusion ∠1 and∠2 are supplementary is valid.

If the sum of the measures of two angles is 180, then the angles are supplementary. ∠1,∠2 forms a linear pair.

The conclusion that ∠1 and∠2 are supplementary is valid as the conclusion holds true using the law of detachment.

Page 21 Problem 4 Answer

We are given that  If two angles are complementary, then the sum of their measures is 90.

If the sum of the measures of two angles is 90, then both of the angles are acute.

We have to draw a valid conclusion from each set of statements, We will use the law of syllogism and calculate the result

The statement if two angles are complementary, then the sum of their measures is 90 is true.

The statement If the sum of the measures of two angles is 90, then both of the angles are acute.

Then the valid conclusion that can be drawn from statements is, if two angles are complementary, then both of the angles are acute.

The valid conclusion that can be drawn from statements if two angles are complementary, then the sum of their measures is 90.

If the sum of the measures of two angles is 90, then both of the angles are acute. is, if two angles are complementary, then both of the angles are acute.

Page 21 Problem 5 Answer

We are given that if the heat wave continues, then air conditioning will be used more frequently.

If air conditioning is used more frequently, then energy costs will be higher.

We have to draw a valid conclusion from each set of statements, We will use the law of syllogism and calculate the result

The statement if the heat wave continues, then air conditioning will be used more frequently is true.

The statement if air conditioning is used more frequently, then energy costs will be higher is true.

Then the valid conclusion that can be drawn from statements is, if the heat wave continues, then energy costs will be higher.

The valid conclusion that can be drawn from statements if the heat wave continues, then air conditioning will be used more frequently is true and if air conditioning is used more frequently, then energy costs will be higher is true is, if the heat wave continues, then energy costs will be higher.

Page 21 Problem 6 Answer

We are given that if it is Tuesday, then Marla tutors chemistry.

If Marla tutors chemistry, then she arrives home at 4 P.M.We have to draw a valid conclusion from each set of statements, We will use the law of syllogism and calculate the result

The statement if it is Tuesday, then Marla tutors chemistry is true.

The statement if Marla tutors chemistry, then she arrives home at 4 P.M is true.

Then the valid conclusion that can be drawn from statements is if it is Tuesday, then Maria arrives home at 4 P.M

The valid conclusion that can be drawn from statements if it is Tuesday, then Marla tutors chemistry and if Marla tutors chemistry, then she arrives home at 4 P.M. is,  if it is Tuesday, then Maria arrives home at 4 P.M

Page 22 Problem 7 Answer

We are given that If a point is the midpoint of a segment, then it divides the segment into two congruent segments. R is the midpoint of QS

We are given the conclusion that QR≅RS

We have to determine whether the stated conclusion is valid based on the given information.

We will use law of detachment and determine the result.

We have been given that If a point is the midpoint of a segment, then it divides the segment into two congruent segments.

And R is the midpoint of QS

Then R, will divide QS into two congruent segments.

That is, QR≅RS

Thus, our conclusion is valid.

If a point is the midpoint of a segment, then it divides the segment into two congruent segments and R is the midpoint of QS then the conclusion QR≅RS is valid, as our conclusion holds true using law of detachment.

Page 22 Problem 8 Answer

We have a given statement.

We will conclude the statement.

We will be using mathematical operations to find the result.

We draw a segment AB≅BC that is

We can see in figure B is not a mid-point.

So, the conclusion which is given is invalid.

In the figure, we can see AB≅BC,  but B is not a midpoint of the line AC.

Hence, we can say the conclusion which is given is invalid.

Page 22 Problem 9 Answer

We have a given question.

We will give a conclusion which is right.

We know the hurricane category5

wind speed more than 157mph.

Hence, we can say a hurricane is a category 5, then winds are greater than 155mph and it can be blown down by trees, shrubs, and signs.

The category 5 hurricane speed is more than 157mph.

So, the conclusion is, we know a hurricane is a category 5, wind speed is greater than 155mph, and it can be blown down by trees, shrubs, and signs.

Page 22 Problem 10 Answer

We have a given statement.

We will find the conclusion of the statement is satisfy the Law of Detachment or the Law of Syllogism.

We know that the Law of Detachment is the order to allow what we desire to materialize in the physical universe.

Hence, we can say the whole even number, that is I am thinking, it can be divided by 4 that is a square.

The conclusion is, the square of the number I am thinking it can divisible by 4.

And it obeys the Law of Detachment.

Page 22 Problem 11 Answer

We have a given statement We will find the conclusion of the statement is satisfy the Law of Detachment or the Law of Syllogism.

We know the law of syllogism, also called reasoning by transitivity, is a valid argument form of deductive reasoning that follows a set pattern.

Hence, we can say if the virus is a parasite, then it can harms its host.

We can conclude the statement that is, if the virus is a parasite, then it can harm its host.

And it obeys the Law of Syllogism.

Geometry Homework Practice Workbook 1st Edition Chapter 2 Inductive Reasoning and Conjecture

Page 19 Problem 1 Answer

We are given a conditional statement ‘If you purchase a computer and do not like it, then you can return it within 30 days’.

We are required to identify the hypothesis and the conclusion.

Since, the given statement is in the ”if-then” form, we can say that the statement after ‘if’ will be the hypothesis and the statement after ‘then’ will be the conclusion.

That is, we have the hypothesis-you purchase a computer and do not like it and the conclusion-you can return it within 30 days.

In the given conditional statement, If you purchase a computer and do not like it, then you can return it within 30 days, we have the hypothesis-you purchase a computer and do not like it and the conclusion-you can return it within 30 days.

Page 19 Problem 2 Answer

We are given a conditional statement ‘If x+8=4, then x=−4’.We are required to identify the hypothesis and the conclusion.

Since, the given statement is in the ”if-then” form, we can say that the statement after ‘if’ will be the hypothesis and the statement after ‘then’ will be the conclusion.

That is, we have, hypothesis-x+8=4 and the conclusion-x=−4

For the given conditional statement ‘If x+8=4,then x=−4’, we have,the hypothes is -x+8=4 and the conclusion-x=−4

Page 19 Problem 3 Answer

We are given a conditional statement ‘If the drama class raises $2000, then they will go on tour’.We are required to identify the hypothesis and the conclusion.

Since, the given statement is in the ”if-then” form, we can say that the statement after ‘if’ will be the hypothesis and the statement after ‘then’ will be the conclusion.

That is, we have,the hypothesis- the drama class raises $2000 and the conclusion- they will go on tour

For the given conditional statement, ‘If the drama class raises $2000, then they will go on tour’, we have the hypothesis – the drama class raises $2000 and the conclusion – they will go on tour.

Page 19 Problem 4 Answer

We are given a statement ‘A polygon with four sides is a quadrilateral’.We are required to write it in the if-then form.

Here, we can write it as,If a polygon has four sides, then it is a quadrilateral.

The given statement ‘A polygon with four sides is a quadrilateral’ can be written in the if-then form as ‘If a polygon has four sides, then it is a quadrilateral’.

Page 19 Problem 5 Answer

We are given a statement ‘An acute angle has a measure less than 90’.We are required to write this in the if-then form.

We can formulate it as, If an angle measures less than 90, then it is an acute angle.

The statement ‘An acute angle has a measure less than 90’, can be written in the if-then form as, ‘If an angle measures less than 90, then it is an acute angle’.

Page 19 Problem 6 Answer

We are given a conditional statement ‘If you have five dollars, then you have five one-dollar bills’.We are required to it’s truth value.

The given conditional statement is False. As the conditional hypothesis is correct, but the conclusion is incorrect.

As a counter example, we know that five-dollars equals one five-dollar bill only.

Hence, we cannot have five-one dollar bills when you have five dollars.

The given a conditional statement ‘If you have five dollars, then you have five one-dollar bills’ if false.

As a counter example, we know that five-dollars equals one five-dollar bill only.

Hence, we cannot have five-one dollar bills when you have five dollars.

Page 19 Problem 7 Answer

In this question, a given statement is “If I roll two six-sided dice and the sum of the numbers is 11, then one die must be a five”.

Here, we have to tell the truth value of the given conditional statement.

We have to tell If the given statement is true then we have to explain reasoning but if the statement is false then we have to give a counterexample.

When this hypothesis is true, the conclusion is also true.

The sum can only be 11 if one die is a6 and the other is a 5.

The conditional statement is true.

Hence, the given conditional statement “If I roll two six-sided dice and the sum of the numbers is 11, then one die must be a five.” is true.

Page 19 Problem 8 Answer

In this question, a given statement is “If two angles are supplementary, then one of the angles is acute.”Here, we have to tell the truth value of the given conditional statement.

We have to tell If the given statement is true then we have to explain reasoning but if the statement is false then we have to give a counterexample.

False; both angles could be right angles. The hypothesis of the conditional is true, but the conclusion is false.

This counterexample shows that the conditional is false.

Hence, the given conditional statement “If two angles are supplementary, then one of the angles is acute.” is false.

Page 19 Problem 9 Answer

In this question, a given statement is “If 89 is divisible by 2, then 89 is an even number.”

Here, we have to write converse, inverse, and contra positive of the given conditional statement.

We have to tell If the given statement is true or false but if the statement is false then we have to give a counterexample.

So, Converse Statement is If 89 is an even number, then 89 is divisible by 2. which is true.

Inverse Statement is If 89 is not divisible by 2, then 89 is not an even number; which is true.

Contra positive Statement is If 89 is not an even number, then 89 is not divisible by 2. which is true.

Hence, the given statement “If 89 is divisible by 2, then 89 is an even number” is true for converse, inverse, and contra positive statements.

Page 20 Problem 10 Answer

Here, the given conditional statement is “If 3x+4=−5, then x=−3”.

As we know, The hypothesis of a conditional statement is the phrase immediately following the word if.

And The conclusion of a conditional statement is the phrase immediately following the word then.

So, the hypothesis is “3x+4=−5”.And the conclusion is “x=−3”

Hence, from the given statement “If3x+4=−5, then x=−3”

The hypothesis is “3x+4=−5” and the conclusion is “x=−3”.

Page 20 Problem 11 Answer

Here, the given conditional statement is “If you take a class in television broadcasting, then you will film a sporting event”.

As we know, The hypothesis of a conditional statement is the phrase immediately following the word if.

And The conclusion of a conditional statement is the phrase immediately following the word then.

So, the hypothesis is “you take a class in television broadcasting”.

And the conclusion is “you will film a sporting event”

Hence, from the given statement “If you take a class in television broadcasting, then you will film a sporting event” the hypothesis is “you take a class in television broadcasting” and the conclusion is “you will film a sporting event”.

Page 20 Problem 12 Answer

In this question, the given statement is “Those who do not remember the past are condemned to repeat it.”

We have to write the given statement in if-then form.

So, To write these statements in if-then form, identify the hypothesis and conclusion.

The word if is not part of the hypothesis.

The word then is not part of the conclusion.

So, the statement is “If you do not remember the past, then you are condemned to repeat it.”

Hence, the statement in if-then form is “If you do not remember the past, then you are condemned to repeat it”.

Page 20 Problem 13 Answer

In this question, the given statement is ” Adjacent angles share a common vertex and a common side.”

We have to write the given statement in if-then form.

So, To write these statements in if-then form, identify the hypothesis and conclusion.

The word if is not part of the hypothesis. The word then is not part of the conclusion.

So, the statement is “If two angles are adjacent, then they share a common vertex and a common side.”

Hence, the statement in if-then form is “If two angles are adjacent, then they share a common vertex and a common side.”

Page 20 Problem 14 Answer

In this question, a given statement is “If a and b  are negative, then a+b is also negative.”

Here, we have to tell the truth value of the given conditional statement.

We have to tell If the given statement is true then we have to explain reasoning but if the statement is false then we have to give a counterexample.

When this hypothesis is true, the conclusion is also true.

A negative added to another negative will just go further negative.

The conditional statement is true.

Hence, the given conditional statement “If a and b are negative, then a+b is also negative” is true.

Page 20 Problem 15 Answer

In this question, a given statement is “If two triangles have equivalent angle measures, then they are congruent.”Here, we have to tell the truth value of the given conditional statement.

We have to tell If the given statement is true then we have to explain reasoning but if the statement is false then we have to give a counterexample.

False, two triangles could have the same angle measures but different side lengths.

The hypothesis of the conditional is true, but the conclusion is false.

This counterexample shows that the conditional is false.

Hence, the given conditional statement “If two triangles have equivalent angle measures, then they are congruent” is false.

Page 20 Problem 16 Answer

In this question, a given statement is ” If the moon has purple spots, then it is June.”Here, we have to tell the truth value of the given conditional statement.

We have to tell If the given statement is true then we have to explain reasoning but if the statement is false then we have to give a counterexample.

When this hypothesis is false, the statement is true.

So, the given statement is true.

Hence, the given conditional statement “If the moon has purple spots, then it is June” is true.

Page 20 Problem 17 Answer

We are given, that older campers who attend Woodland Falls Camp are expected to work. Campers who are juniors wait on tables.

We have to write a conditional statement in if-then form.

The conditional statement will be: If you are an old camper who attends  Woodland Falls Camp then you are expected to work and if you are junior camper then you have to wait on table.

The conditional statement in if-then form for the given statement will be: If you are an old camper who attends  Woodland Falls Camp then you are expected to work and if you are junior camper then you have to wait on table.

Page 20 Problem 18 Answer

We are given, that older campers who attend Woodland Falls Camp are expected to work.

Campers who are juniors wait on tables. We have to write the converse of your conditional statement.

The conditional statement was: If you are an old camper who attends  Woodland Falls Camp then you are expected to work and if you are junior camper then you have to wait on table.

Its converse statement will be: If you are expected to work then you are an old camper who attends  Woodland Falls Camp and if you have to wait on table then you are junior camper.

Its converse statement for the conditional statement will be: If you are expected to work then you are an old camper who attends  Woodland Falls Camp and if you have to wait on table then you are junior camper.

Geometry Homework Practice Workbook 1st Edition Chapter 2 Inductive Reasoning and Conjecture

Page 17 Problem 1 Answer

In the question, we have given the following statements:

p:−3−2=−5

q: Vertical angles are congruent.r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

We have the task to write a compound statement for p and q conjunction or disjunction and find its truth value.

Here we can see that p and q:−3−2=−5 and vertical angles are congruent and we know that the vertical angles are always congruent.

So, the conjunction is true.

Hence our conjunction is true that p and q:−3−2=−5 and vertically opposite angles are congruent. from the given statements:

In the question, we have given the following statements:

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90.

Page 17 Problem 2 Answer

In the question, we have given the following statements:

p:−3−2=−5

q:Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

We have to write a compound statement for p∨r conjunction or disjunction and find its truth value.

Here we can see that, it is false that r:2+8>10 as the correct is r:2+8=10.

So, the conjunction is false.

Hence the compound statement is false that p∨r:−3−2=−5 and 2+8>10.

As the truth value will be for r:2+8=10 from the given statement.

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

Page 17 Problem 3 Answer

In the question, we have given the following statements:

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

We have the task to write a compound statement for p or s conjunction or disjunction and find its truth value.

Here we have p or s:−3−2=−5 or the sum of the measures of complementary angles is 90∘.

Since both of them are true and we have given ‘or’ in between.

So, the statement is true.

Hence our conjunction is true that p or s:−3−2=−5 or the sum of the complementary angles is 90∘from the given statements: p:−3−2=−5

q: Vertical angles are congruent.

r:2=8>10

s: The sum of the measures of complementary angles is 90∘.

Page 17 Problem 4 Answer

In the question, we have given the following statements:

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

We have the task to write a compound statement for r∨s conjunction or disjunction and find its truth value.

Here we have r or s:2+8>10 or the sum of the measures of complementary angles is 90∘.

Since one of them is true and we have given ‘or’ in between.

So, we conclude that the statement is true.

Hence we conclude that our conjunction is true that r or s:2+8>10 or the sum of the complementary angles is  from the given statements:

p:−3−2=−5

q:Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

Page 17 Problem 5 Answer

In the question, we have given the following statements:

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

We have to write a compound statement for p∧∼q conjunction or disjunction and find its truth value.

From the given ∼q: vertical angles are not congruent.

Now we have p∧∼q:−3−2=−5

and vertical angles are not congruent.

Since one of them is false and we have given ‘∧ implies and’ in between.

So, we conclude that the statement is false.

Hence we conclude that our conjunction is false that p∧∼q:−3−2=−5 and vertical angles are not congruent, from the given statements:

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

Page 17 Problem 6 Answer

In the question, we have given the following statements:

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

We have to write a compound statement for q∨∼r conjunction or disjunction and find its truth value.

From the given ∼r:2+8≤10.

Now Here we have q∨∼r: the sum of the measures of complementary angles is 90∘ or 2+8≤10 Since both of them are true and we have given ‘∧ implies or’ in between.

So, we conclude that the statement is true.

Hence we conclude that our conjunction is true that q∨∼r: the sum of the complementary angles is 90∘ or 2+8≤10 from the given statements:

p:−3−2=−5

q: Vertical angles are congruent.

r:2+8>10

s: The sum of the measures of complementary angles is 90∘.

Page 17 Problem 7 Answer

We have been given a truth table that is,

We have to complete it.

We will complete it by knowing the truth values of logical expressions.

Let us complete the truth table that is,

Hence, we have completed the given truth table that is,

Page 17 Problem 8 Answer

We have been given a truth table that is,

We have to complete it.

We will complete it by knowing the truth values of logical expressions.

Let us complete the truth table that is,

Hence, we have completed the given truth table that is,

Page 17 Problem 9 Answer

In this question, we have been given the value:∼q∧r

We need to construct a truth table for the compound statement.

By using Inductive Reasoning and Conjecture, we will calculate the result.

We construct a truth table for the given compound statement

Hence, a graph with rows and columns shows how the truth or falsity of a claim fluctuates depending on the truth or falsity of its constituents.

We construct a truth table for the given compound statement ∼q∧r is

Page 17 Problem 10 Answer

In this question, we have been given the value: ∼p∨∼r

We need to construct a truth table for the compound statement.

By using Inductive Reasoning and Conjecture, we will calculate the result.

We construct a truth table for the given compound statement

Hence, a graph with rows and columns shows how the truth or falsity of a claim fluctuates depending on the truth or falsity of its constituents.

We construct a truth table for the given compound statement ∼p∨∼r is

Page 18 Problem 11 Answer

We have been given the following statements that are,p: 60 seconds = 1-minute q: Congruent supplementary angles each have a measure of 90.

r:−12+11<−1

We have to write a disjunction statement for q∨r.

Therefore, the disjunction statement for q∨r is, ” congruent supplementary angles each have a measure of 90 or −12+11<−1″.

We see that the first statement is true and the second statement is false.

Since one of the statements in the disjunction is true, then the disjunction is also true.

Hence, we have written a disjunction statement for q∨r that is, ” congruent supplementary angles each have a measure of 90 or −12+11<−1″ and found that the truth value of the disjunction is true.

Page 18 Problem 12 Answer

We have been given the following statements that are,p: 60 seconds = 1-minute q: Congruent supplementary angles each have a measure of 90.

r.−12+11<−1 We have to write a conjunction statement for ∼p∨q.

First, we will find the negation of p that is,∼p”60 seconds≠1 minute”.

Therefore, the conjunction statement for ∼p∨q is,” 60 seconds≠1 minute and Congruent supplementary angles each have a measure of 90″.

We see that the first statement is false and the second statement is true.

Since one of the statements in the conjunction is false, then the truth value of the conjunction is false,

Hence, we have written a disjunction statement for ∼p∨q that is, “60 seconds≠1 minute and Congruent supplementary angles each have a measure of 90 “and found that the truth value of the conjunction statement is false.

Page 18 Problem 13 Answer

We have been given the following statements that are,p: 60 seconds = 1 minute q: Congruent supplementary angles each have a measure of 90.

r :−12+11<−1

We have to write a disjunction statement for ∼p∧∼r.

First, we will find the negation of p and r that is,∼p: 60 seconds≠1minute.

∼r:−12+11≮−1.

Therefore, the disjunction statement for ∼p∧∼r that is,” 60 seconds≠1 minute or −12+11≮−1.

We see that the first statement is false and the second statement is true.

Since one of the statements of the disjunction is true, then the truth value of the disjunction is true.

Hence, we have written a disjunction statement for ∼p∧∼r that is,” 60 seconds≠1 minute or −12+11≮−1″ and found that the truth value of the disjunction is true.

Page 18 Problem 14 Answer

We have been given a truth table that is,

We have to complete it.

We will complete it by knowing the truth values of logical expressions.

Let us complete the truth table that is,

Hence, we have completed the given truth table that is,

Page 18 Problem 15 Answer

We have been given a truth table that is,

We have to complete it.

We will complete it by knowing the truth values of logical expressions.

Let us complete the truth table that is,

Hence, we have completed the given truth table that is,

Page 18 Problem 16 Answer

We are given a compound statement q∨(p∧∼q).

We are required to construct a truth table for it.

Here, we will break the compound statement to its constituent simple statements and then find the truth or falsity of each to combine and get the table.

Breaking the given compound to simple statements and evaluating whether it is true or false, we get the truth table,

We get the truth table for q∨(p∧∼q) as,

Page 18 Problem 17 Answer

We are given a compound statement ∼q∧(∼p∨q).

We are required to construct a truth table for it.

Here, we will break the compound statement to its constituent simple statements and then find the truth or falsity of each to combine and get the table.

We know that,

∼q∧(∼p∨q)=∼[q∨(p∧∼q)]

Hence, we get,

We get the truth table for the compound statement ∼q∧(∼p∨q) as,

Page 18 Problem 18 Answer

We are given a Venn diagram that shows the number of students in the band who work after school or on the weekends.

We are required to find the number of students that work after school and on weekends.

We can see that from the Venn diagram, the intersection of the two given circles represent the number of students that work after school and on weekends and has the value 3.

Hence, the number of students that work after school and on weekends=3.

From the Venn diagram, we have, The number of students that work after school and on weekends=3

Page 18 Problem 19 Answer

We are given a Venn diagram that shows the number of students in the band who work after school or on the weekends.

We are required to find the number of students that work after school or on weekends.

Here, we can see that the Venn diagram shows the number of students in the band who work after school or on the weekends, and hence, the required problem asks us to find the union of both the circles in the diagram.

We get the union of the two circles given in the Venn diagram as, the number of students that work after school or weekends=5+3+17=25

​From the Venn diagram we have, the number of students that work after school or weekends=25

Geometry Homework Practice Workbook 1st Edition Chapter 2 Inductive Reasoning and Conjecture

Page 15 Problem 1 Answer

In this question, we have been given a figure.

We need to make a hypothesis that characterizes the sequence’s pattern.

Then use our hunch to figure out what comes next in the sequence.

By using Inductive Reasoning and Conjecture, we will calculate the result.

After following the pattern, We get

Two dark and one white block is added to the previous item to get the next item.

Hence, two dark and one white block is added to the previous item to get the next item. According to our conjecture, We get the below item

Page 15 Problem 2 Answer

In this question, we have been given a series: −4,−1,2,5,8

We need to make a hypothesis that characterizes the sequence’s pattern.

Then use our hunch to figure out what comes next in the sequence.

By using Inductive Reasoning and Conjecture, we will calculate the result.

After following the pattern, We get:

Since as we see+3 is added to the previous item to get the next item.

Hence, as we see +3 is added to the previous item to get the next item.

According to our conjecture, We get 11. After following the pattern

Page 15 Problem 3 Answer

In this question, we have been given a series: 6,11/2,5,9/2,4

We need to make a hypothesis that characterizes the sequence’s pattern.

Then use our hunch to figure out what comes next in the sequence.

By using Inductive Reasoning and Conjecture, we will calculate the result.

After following the pattern, We get:

According to the below pattern, we subtract from the previous item to get the next item.

Hence, According to the below pattern, we subtract from the previous item to get the next item.

According to our conjecture, we get 3.5.

Page 15 Problem 4 Answer

In this question, we have been given a series:−2,4,−8,16,−32.

We need to make a hypothesis that characterizes the sequence’s pattern.

Then use our hunch to figure out what comes next in the sequence.

By using Inductive Reasoning and Conjecture, we will calculate the result.

After following the pattern, We get:

According to the below pattern, we subtract from the previous item to get the next item.

Hence, According to the below pattern, we subtract from the previous item to get the next item.

According to our conjecture, we get 64.

Page 15 Problem 5 Answer

In this question, we have been given Points A, B, and C are collinear, and D is between B and C.

We need to make a conjecture about each value or geometric relationship.

By using Inductive Reasoning and Conjecture, we will calculate the result.

We draw figures according to the given statement.

Since all the given points are collinear.

Hence, all the given points are collinear. We draw figures according to the given statement.

According to our conjecture, We get

Page 15 Problem 6 Answer

In this question, we have been given a Point P is the midpoint of NQ.

We need to make a conjecture about each value or geometric relationship.

By using Inductive Reasoning and Conjecture, we will calculate the result.

Since all the given points are collinear.

We draw a figure according to the given statement.

Hence, all the given points are collinear.

We draw a figure according to the given statement.

According to our conjecture, We get

Page 15 Problem 7 Answer

In this question, we have been given ∠1,∠2,∠3 and ∠4 form four linear pairs.

We need to make a conjecture about each value or geometric relationship.

By using Inductive Reasoning and Conjecture, we will calculate the result.

We make figures according to the given statement.

 

∠1,∠2,∠3, and ∠4 are formed by two intersection lines.

Since all the given angles are linear pairs.

Hence, all the given angles are linear pairs. We make figures according to the given statement.

According to our conjecture, We get

Page 15 Problem 8 Answer

In this question, we have been given the value: ∠3≅∠4

We need to make a conjecture about each value or geometric relationship.

By using Inductive Reasoning and Conjecture, we will calculate the result.

m∠3=m∠4

We make figures according to the given statement.

∠3, and ∠4 are formed by two intersection lines.

Since both given angles are equal.

Hence, both given angles are equal m∠3=m∠4.

They are formed by two intersection lines.

According to our conjecture, We get

Page 15 Problem 9 Answer

In this question, we have been given AB, BC, and AC are congruent.

We need to determine whether each conjecture is true or false and give a counterexample for any false conjecture.

By using Inductive Reasoning and Conjecture, we will calculate the result.

We have been given that the lines are congruent, but we have not been given that the lines are in a straight line or not.

As we are not given any other information, so the values of the lines are maybe in different directions.

So, we can say that the given conjecture is false.

We can give a counterexample, that if all the three lines create a 60-degree angle.

Hence, we can say that the given conjecture is false. As we are not given any other information, the values of the lines are maybe in different directions.

We can give a counterexample, that if all three lines create a 60-degree angle.

Page 15 Problem 10 Answer

We have been given that,

If, AB+BC=AC, then, AB=BC.We have to determine whether the given conjecture is true or false, and give a counterexample if it is a false conjecture.

We have been given that AB+BC=AC, but we have not been given any other information about, AB, BC,AC.

So, AB=BC only if B is the midpoint of AC or ratio of AB, BC is 1:1.As we are not given any other information, so ratio of AB,BC can be any m:n ratio.

So, if m≠n, then AB is also not equal to BC So, we can say that the given conjecture is false.

We can give a counterexample, that if ratio of AB,BC is 2:3, then AB≠BC.

Hence, we determined that the given conjecture,

If, AB+BC=AC, then, AB=BC, is false as ratio of AB,BC can be any m:n ratio,

and gave a counter example that if ratio of AB,BC is 2:3 then AB≠BC.

Page 16 Problem 11 Answer

We have been given a figure as,

We have to make a conjecture about the next item in the given sequence.

We can see that the items in given sequence are vertically symmetrical at its center.

We can see that the dots increases by one on both sides of symmetry.

So, we can give the next item as,

Hence, for the given figure,

we made a conjecture about the next item as,

since dot increases by one on both sides of symmetry.

Page 16 Problem 12 Answer

We have been given a sequence as, 5,−10,15,−20.

We have to make a conjecture about the next item in the given sequence.

We can see that every item in sequence increases by five.

We can also see that every even item in sequence has a negative sign and odd item is positive .

So, we can write the next item in sequence as 25.

So, we get the whole sequence as, 5,−10,15,−20,25.

Hence, for the given sequence, 5,−10,15,−20, we made a conjecture about the next item as, 25, since every item is increased by five and odd item is positive.

Page 16 Problem 13 Answer

We have been given a sequence as, −2,1,−1/2,1/4,−1/8.

We have to make a conjecture about the next item in the given sequence.

We can see that every next item in sequence is divided by −2.

We can also call this type of sequence as geometric sequence where the common ratio is −1/2.

So, we get the next term in sequence as 1/16.

We get the whole sequence as, −2,1,−1/2,1/4,−1/8,1/16.

Hence, for the given sequence, −2,1,−1/2,1/4,−1/8, we made a conjecture about the next item as, 1/16 as it is geometric sequence with common ratio −1/2.

Page 16 Problem 14 Answer

We have been given a sequence as, 12,6,3,1.5,0.75.

We have to make a conjecture about the next item in the given sequence.

We can see that every next item in sequence is divided by 2.

We can also call this type of sequence as geometric sequence where the common ratio is 1/2.

So, we get the next term in sequence as 0.375.

We get the whole sequence as, 12,6,3,1.5,0.75,0.375.

Hence, for the given sequence, 12,6,3,1.5,0.75, we made a conjecture about the next item as, 0.375, as it is geometric sequence with common ratio 1/2.

Page 16 Problem 15 Answer

We have been given that, ∠ABC is right angle.

We have to make a conjecture about the given value or geometric relationship.

We will assume ABC as a triangle.

We know that in a triangle, sum of all angles is 180∘.

We are given ∠ABC as right angle.

So, we can make a conjecture that ∠A,∠C are complementary angles.

Hence, for the given geometric relationship, ∠ABC is right angle, we made a conjecture that the angles ∠A,∠C are complementary, if ABC is a triangle.

Page 16 Problem 16 Answer

We have been given that, point S is between R and T.

We have to make a conjecture about the given value or geometric relationship.

We can assume that all the given points are colinear.

We know that point S is between T, R.So, we can make a conjecture that RS+ST=RT.

Hence, for the given geometric relationship, point S is between R and T, we made a conjecture that RS+ST=RT, if points R,S,T are colinear.

Page 16 Problem 17 Answer

We have been given that, P,Q,R, and S are are non collinear and PQ,QR,RS,SP are congruent.

We have to make a conjecture about the given value or geometric relationship.

We will assume PQRS  to be a quadrilateral.We are given that PQ,QR,RS,SP are all congruent to each other.

We know that if all sides in a quadrilateral are equal, then it is a square or a rhombus.

Since we are not given anything about angles, we can make a conjecture that given quadrilateral is a rhombus.

Hence, for the given geometric relationship, where, P,Q,R,S are non collinear and PQ,QR,RS,SP are all congruent to each other, we made a conjecture that PQRS is a rhombus, if PQRS is a quadrilateral.

Page 16 Problem 18 Answer

We have been given that, ABCD is a parallelogram.

We have to make a conjecture about the given value or geometric relationship.

We will do it by the properties of parallelogram.

We know that opposite sides and angles in a parallelogram are congruent.

So, we can make a conjecture that,

AB≅CD,

AD≅BC,

∠A≅∠C,

∠B≅∠D.​

Hence, for the given geometric relationship, ABCD is a parallelogram, we made a conjecture that,

AB≅CD,

AD≅BC,

∠A≅∠C,

∠B≅∠D,

​as opposite sides and angles in a parallelogram are congruent.

Page 16 Problem 19 Answer

In the question, we have been given that points S,T and U are collinear and ST=TU, then T is the midpoint of SUˉ.

We have to tell that whether the above conjecture is true or false.

Here the above conjecture is true as in the figure 1, we can see that T is equidistance from both the points and it divides SUˉ in two equal parts that make ST=TU.

So, the above conjecture is true.

Hence the conjecture is true that if S,T and U are collinear and ST=TU, then T will be the midpoint of SUˉ[shown in figure 1].

Page 16 Problem 20 Answer

We have been given that if ∠1 and ∠2 are adjacent angles, then ∠1 and ∠2 form a linear pair.

We have been asked to tell whether the above conjecture is true or false.

Here we can see in the figure 1 that the ∠1 and ∠2 are adjacent angles but they are not making a linear pair.

So, we conclude that the given conjecture is false as they could make an angle of 60∘ or 90∘ .

Hence we conclude that the conjecture is false that if ∠1 and ∠2 are adjacent angle then ∠1 and ∠2 could an angle of 60∘ or 90∘ as shown in figure 1.

Page 16 Problem 21 Answer

In the following question, we have been given that if GHˉ and JKˉ form a right angle and intersect at P, then GHˉ⊥JKˉ.

We have the task to tell whether the conjecture is true or false and if it is a false conjecture then we have to give a counterexample.

Here we supposed the figure that is shown in the figure 1.

Now in the figure 1, we can see that GHˉ⊥JKˉ.So, the conjecture is true.

Hence we observe in the assumed figure 1, that the conjecture is true i.e., if GHˉ and JKˉ form a right angle and intersect at P, then GHˉ⊥JKˉ[shown in the figure 1].

Page 16 Problem 22 Answer

We have been given that each spring, Rachel starts sneezing when the pear trees on her street blossom.

She reasons that she is allergic to pear trees.

We have been asked to find a counterexample to Rachel’s conjecture.

Here the counterexample to Rachel’s conjecture will be “Rachel could be allergic to other flora that blooms at the same time as the pear trees”

Hence when  Each spring, Rachel starts sneezing when the pear trees on her street blossom.

She reasons that she is allergic to pear trees.

A counterexample to Rachel’s conjecture will be “Rachel could be allergic to other flora that blooms at the same time as the pear trees”