Need instant help while preparing the BIM Geometry  Chapter 7 topics? Then, here is the perfect guide for you all ie., Big Ideas Math Geometry Answers Chapter 7 Quadrilaterals and Other Polygons. Make use of this easy and helpful study resource at times of preparation and boost up your confidence to attempt the exam. Learn & Prepare by taking the help of BIM Book Geometry Answer Key and clarify all your doubts on the Ch 7 concepts. Big Ideas Math Geometry Chapter 7 Solution Key covers all 7.1 to 7.6 exercise questions, Chapter Tests, Review Tests, Assessments, Cumulative Practice, etc. which enhance your subject knowledge.

Big Ideas Math Book Geometry Answer Key Chapter 7 Quadrilaterals and Other Polygons

Learning the concepts of Ch 7 Quadrilaterals and Other Polygons with the help of our provided BIM Geometry Answers is the best way to answer all the questions asked in the annual exams. The fact that you can observe in this Big Ideas Math Book Geometry Solution Key is covering all exercise questions of Chapter 7 Quadrilaterals and Other Polygons. So, just click on the quick links mentioned below and score better marks in any final exams or competitive examinations.

Quadrilaterals and Other Polygons Maintaining Mathematical Proficiency

Solve the equation by interpreting the expression in parentheses as a single quantity.

Question 1.
4(7 – x) = 16

Question 2.
7(1 – x) + 2 = – 19

Question 3.
3(x – 5) + 8(x – 5) = 22

Determine which lines are parallel and which are perpendicular.

Question 4.

Question 5.

Question 6.

Question 7.
ABSTRACT REASONING
Explain why interpreting an expression as a single quantity does not contradict the order of operations.

Quadrilaterals and Other Polygons Mathematical Practices

Monitoring Progress

Use the Venn diagram above to decide whether each statement is true or false. Explain your reasoning.

Question 1.
Some trapezoids are kites.

Question 2.
No kites are parallelograms.

Question 3.
All parallelograms are rectangles.

Question 4.

Question 5.
Example 1 lists three true statements based on the Venn diagram above. Write six more true statements based on the Venn diagram.

Question 6.
A cyclic quadrilateral is a quadrilateral that can be circumscribed by a circle so that the circle touches each vertex. Redraw the Venn diagram so that it includes cyclic quadrilaterals.

7.1 Angles of Polygons

Exploration 1

The Sum of the Angle Measures of a Polygon

Work with a partner. Use dynamic geometry software.

a. Draw a quadrilateral and a pentagon. Find the sum of the measures of the interior angles of each polygon.
Sample

b. Draw other polygons and find the sums of the measures of their interior angles. Record your results in the table below.

c. Plot the data from your table in a coordinate plane.

d. Write a function that fits the data. Explain what the function represents.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to reason inductively about data.

Exploration 2

Measure of one Angle in a Regular Polygon

Work with a partner.

a. Use the function you found in Exploration 1 to write a new function that gives the measure of one interior angle in a regular polygon with n sides.

b. Use the function in part (a) to find the measure of one interior angle of a regular pentagon. Use dynamic geometry software are to check your result by constructing a regular pentagon and finding the measure of one of its interior angles.

c. Copy your table from Exploration 1 and add a row for the measure of one interior angle in a regular polygon with n sides. Complete the table. Use dynamic geometry software to check your results.

Question 3.
What is the sum of the measures of the interior angles of a polygon?

Question 4.
Find the measure of one interior angle in a regular dodecagon (a polygon with 12 sides).

Lesson 7.1 Angles of Polygons

Monitoring Progress

Question 1.
The Coin shown is in the shape of an 11-gon. Find the sum of the measures of the interior angles.

Question 2.
The sum of the measures of the interior angles of a convex polygon is 1440°. Classify the polygon by the number of sides.

Question 3.
The measures of the interior angles of a quadrilateral are x°, 3x°. 5x°. and 7x° Find the measures of all the interior angles.

Question 4.
Find m∠S and m∠T in the diagram.

Question 5.
Sketch a pentagon that is equilateral but not equiangular.

Question 6.
A convex hexagon has exterior angles with measures 34°, 49°, 58°, 67°, and 75°. What is the measure of an exterior angle at the sixth vertex?

Question 7.
An interior angle and an adjacent exterior angle of a polygon form a linear pair. How can you use this fact as another method to find the measure of each exterior angle in Example 6?

Exercise 7.1 Angles of Polygons

Question 1.
VOCABULARY
Why do vertices connected by a diagonal of a polygon have to be nonconsecutive?

Question 2.
WHICH ONE DOESNT BELONG?
Which sum does not belong with the other three? Explain your reasoning.

 the sum of the measures of the interior  angles of a quadrilateral the sum of the measures of the exterior angles of a quadrilateral the sum of the measures of the interior  angles of a pentagon the sum of the measures of the exterior angles of a pentagon

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, find the sum of the measures of the interior angles of the indicated convex po1gon.

Question 3.
nonagon

Question 4.
14-gon

Question 5.
16-gon

Question 6.
20-gon

In Exercises 7-10, the sum of the measures of the interior an1es of a convex polygon is given. Classify the polygon by the number of sides.

Question 7.
720°

Question 8.
1080°

Question 9.
2520°

Question 10.
3240°

In Exercises 11-14, find the value of x.

Question 11.

Question 12.

Question 13.

Question 14.

In Exercises 15-18, hod the value of x.

Question 15.

Question 16.

Question 17.

Question 18.

In Exercises 19 – 22, find the measures of ∠X and ∠Y.

Question 19.

Question 20.

Question 21.

Question 22.

In Exercises 23-26, find the value of x.

Question 23.

Question 24.

Question 25.

Question 26.

In Exercises 27-30, find the measure of each interior angle and each exterior angle of the indicated regular polygon.

Question 27.
pentagon

Question 28.
18-gon

Question 29.
45-gon

Question 30.
90-gon

ERROR ANALYSIS
In Exercises 31 and 32, describe and correct the error in finding the measure of one exterior angle of a regular pentagon.

Question 31.

Question 32.

Question 33.
MODELING WITH MATHEMATICS
The base of a jewelry box is shaped like a regular hexagon. What is the measure of each interior angle of the jewelry box base?

Question 34.
MODELING WITH MATHEMATICS
The floor of the gazebo shown is shaped like a regular decagon. Find the measure of each interior angle of the regular decagon. Then find the measure of each exterior angle.

Question 35.
WRITING A FORMULA
Write a formula to find the number of sides n in a regular polygon given that the measure of one interior angle is x°.

Question 36.
WRITING A FORMULA
Write a formula to find the number of sides n in a regular polygon given that the measure of one exterior angle is x°.

REASONING
In Exercises 37-40, find the number of sides for the regular polygon described.

Question 37.
Each interior angle has a measure of 156°.

Question 38.
Each interior angle has a measure of 165°.

Question 39.
Each exterior angle has a measure of 9°.

Question 40.
Each exterior angle has a measure of 6°.

Question 41.
DRAWING CONCLUSIONS
Which of the following angle measures are possible interior angle measures of a regular polygon? Explain your reasoning. Select all that apply.
(A) 162°
(B) 171°
(C) 75°
(D) 40°

Question 42.
PROVING A THEOREM
The Polygon Interior Angles Theorem (Theorem 7.1) states that the sum of the measures of the interior angles of a convex n-gon is (n – 2) • 180°. Write a paragraph proof of this theorem for the case when n = 5.

Question 43.
PROVING A COROLLARY
Write a paragraph proof of the Corollary to the Polygon Interior Angles Theorem (Corollary 7. 1).

Question 44.
MAKING AN ARGUMENT
Your friend claims that to find the interior angle measures of a regular polygon. you do not have to use the Polygon Interior Angles Theorem (Theorem 7. 1). You instead can use the Polygon Exterior Angles Theorem (Theorem 7.2) and then the Linear Pair Postulate (Postulate 2.8). Is your friend correct? Explain your reasoning.

Question 45.
MATHEMATICAL CONNECTIONS
In an equilateral hexagon. four of the exterior angles each have a measure of x°. The other two exterior angles each have a measure of twice the sum of x and 48. Find the measure of each exterior angle.

Question 46.
THOUGHT PROVOKING
For a concave polygon, is it true that at least one of the interior angle measures must be greater than 180°? If not, give an example. If so, explain your reasoning.

Question 47.
WRITING EXPRESSIONS
Write an expression to find the sum of the measures of the interior angles for a concave polygon. Explain your reasoning.

Question 48.
ANALYZING RELATIONSHIPS
Polygon ABCDEFGH is a regular octagon. Suppose sides $$\overline{A B}$$ and $$\overline{C D}$$ are extended to meet at a point P. Find m∠BPC. Explain your reasoning. Include a diagram with your answer.

Question 49.
MULTIPLE REPRESENTATIONS
The formula for the measure of each interior angle in a regular polygon can be written in function notation.

a. Write a junction h(n). where n is the number of sides in a regular polygon and h(n) is the measure of any interior angle in the regular polygon.
b. Use the function to find h(9).
c. Use the function to find n when h(n) = 150°.
d. Plot the points for n = 3, 4, 5, 6, 7, and 8. What happens to the value of h(n) as n gets larger?

Question 50.
HOW DO YOU SEE IT?
Is the hexagon a regular hexagon? Explain your reasoning.

Question 51.
PROVING A THEOREM
Write a paragraph proof of the Polygon Exterior Angles Theorem (Theorem 7.2). (Hint: In a convex n-gon. the sum of the measures of an interior angle and an adjacent exterior angle at any vertex is 180°.)

Question 52.
ABSTRACT REASONING
You are given a convex polygon. You are asked to draw a new polygon by increasing the sum of the interior angle measures by 540°. How many more sides does our new polygon have? Explain your reasoning.

Maintaining Mathematical Proficiency

Find the value of x.

Question 53.

Question 54.

Question 55.

Question 56.

7.2 Properties of Parallelograms

Exploration 1

Discovering Properties of Parallelograms

Work with a partner: Use dynamic geometry software.

a. Construct any parallelogram and label it ABCD. Explain your process.
Sample

b. Find the angle measures of the parallelogram. What do you observe?

c. Find the side lengths of the parallelogram. What do you observe?

d. Repeat parts (a)-(c) for several other parallelograms. Use your results to write conjectures about the angle measures and side lengths of a parallelogram.

Exploration 2

Discovering a Property of Parallelograms

Work with a partner: Use dynamic geometry software.

a. Construct any parallelogram and label it ABCD.

b. Draw the two diagonals of the parallelogram. Label the point of intersection E.

c. Find the segment lengths AE, BE, CE, and DE. What do you observe?

d. Repeat parts (a)-(c) for several other parallelograms. Use your results to write a conjecture about the diagonals of a parallelogram.
MAKING SENSE OF PROBLEMS
To be proficient in math, you need to analyze givens, constraints, relationships, and goals.

Question 3.
What are the properties of parallelograms?

Lesson 7.2 Properties of Parallelograms

Monitoring progress

Question 1.
Find FG and m∠G.

Question 2.
Find the values of x and y.

Question 3.
WHAT IF?
In Example 2, find in m∠BCD when m∠ADC is twice the measure of ∠BCD.

Question 4.
Using the figure and the given statement in Example 3, prove that ∠C and ∠F are supplementary angles.

Question 5.
Find the coordinates of the intersection of the diagonals of STUV with vertices S(- 2, 3), T(1, 5), U(6, 3), and V(3, 1).

Question 6.
Three vertices of ABCD are A(2, 4), B(5, 2), and ((3, – 1). Find the coordinates of vertex D.

Exercise 7.2 Properties of Parallelograms

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Why is a parallelogram always a quadrilateral, but a quadrilateral is only sometimes a parallelogram?

Question 2.
WRITING
You are given one angle measure of a parallelogram. Explain how you can find the other angle measures of the parallelogram.

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, find the value of each variable in the parallelogram.

Question 3.

Question 4.

Question 5.

Question 6.

In Exercises 7 and 8. find the measure of the indicated angle in the parallelogram.

Question 7.
Find m∠B.

Question 8.
Find m ∠ N.

In Exercises 9-16. find the indicated measure in LMNQ. Explain your reasoning.

Question 9.
LM

Question 10.
LP

Question 11.
LQ

Question 12.
MQ

Question 13.
m∠LMN

Question 14.
m∠NQL

Question 15.
m∠MNQ

Question 16.
m∠LMQ

In Exercises 17-20. find the value of each variable in the parallelogram.

Question 17.

Question 18.

Question 19.

Question 20.

ERROR ANALYSIS
In Exercises 21 and 22, describe and correct the error in using properties of parallelograms.

Question 21.

Question 22.

PROOF
In Exercises 23 and 24, write a two-column proof.

Question 23.
Given ABCD and CEFD are parallelograms.
Prove $$\overline{A B} \cong \overline{F E}$$

Question 24.
Given ABCD, EBGF, and HJKD arc parallelograms.
Prove ∠2 ≅∠3

In Exercises 25 and 26, find the coordinates of the intersection of the diagonals of the parallelogram with the given vertices.

Question 25.
W(- 2, 5), X(2, 5), Y(4, 0), Z(0, 0)

Question 26.
Q(- 1, 3), R(5, 2), S(1, – 2), T(- 5, – 1)

In Exercises 27-30, three vertices of DEFG are given. Find the coordinates of the remaining vertex.

Question 27.
D(0, 2), E(- 1, 5), G(4, 0)

Question 28.
D(- 2, – 4), F(0, 7), G(1, 0)

Question 29.
D(- 4, – 2), E(- 3, 1), F(3, 3)

Question 30.
E (1, 4), f(5, 6), G(8, 0)

MATHEMATICAL CONNECTIONS
In Exercises 31 and 32. find the measure of each angle.

Question 31.
The measure of one interior angle of a parallelogram is 0.25 times the measure of another angle.

Question 32.
The measure of one interior angle of a parallelogram is 50 degrees more than 4 times the measure of another angle.

Question 33.
MAKING AN ARGUMENT
m∠B = 124°, m∠A = 56°, and m∠C = 124°.

Question 34.
ATTENDING TO PRECISION
∠J and ∠K are Consecutive angles in a parallelogram. m∠J = (3t + 7)°. and m∠K = (5x – 11)°. Find the measure of each angle.

Question 35.
CONSTRUCTION
Construct any parallelogram and label it ABCD. Draw diagonals $$\overline{A C}$$ and $$\overline{B D}$$. Explain how to use paper folding to verify the Parallelogram Diagonals Theorem (Theorem 7.6) for ABCD.

Question 36.
MODELING WITH MATHEMATICS
The feathers on an arrow from two congruent parallelograms. The parallelograms are reflections of each other over the line that contains their shared side. Show that m ∠ 2 = 2m ∠ 1.

Question 37.
PROVING A THEOREM
Use the diagram to write a two-column proof of the Parallelogram Opposite Angles Theorem (Theorem 7.4).

Given ABCD is a parallelogram.
Prove ∠A ≅ ∠C, ∠B ≅ ∠D

Question 38.
PROVING A THEOREM
Use the diagram to write a two-column proof of the Parallelogram Consecutive Angles Theorem (Theorem 7.5).

Given PQRS is a parallelogram.
Prove x° + y° = 180°

Question 39.
PROBLEM SOLVING
The sides of MNPQ are represented by the expressions below. Sketch MNPQ and find its perimeter.
MQ = – 2x + 37 QP = y + 14
NP= x – 5 MN = 4y + 5

Question 40.
PROBLEM SOLVING
In LMNP, the ratio of LM to MN is 4 : 3. Find LM when the perimeter of LMNP is 28.

Question 41.
ABSTRACT REASONING
Can you prove that two parallelograms are congruent by proving that all their corresponding sides are congruent? Explain your reasoning.

Question 42.
HOW DO YOU SEE IT?
The mirror shown is attached to the wall by an arm that can extend away from the wall. In the figure. points P, Q, R, and S are the vertices of a parallelogram. This parallelogram is one of several that change shape as the mirror is extended.

a. What happens to m∠P as m∠Q increases? Explain.
b. What happens to QS as m∠Q decreases? Explain.
c. What happens to the overall distance between the mirror and the wall when m∠Q decreases? Explain.

Question 43.
MATHEMATICAL CONNECTIONS
In STUV m∠TSU = 32°, m∠USV = (x2)°, m∠TUV = 12x°, and ∠TUV is an acute angle. Find m∠USV.

Question 44.
THOUGHT PROVOKING
Is it possible that any triangle can be partitioned into four congruent triangles that can be rearranged to form a parallelogram? Explain your reasoning.

Question 45.
CRITICAL THINKING
Points W(1. 2), X(3, 6), and Y(6, 4) are three vertices of a parallelogram. How many parallelograms can be created using these three vertices? Find the coordinates of each point that could be the fourth vertex.

Question 46.
PROOF
In the diagram. $$\overline{E K}$$ bisects ∠FEH, and $$\overline{F J}$$ bisects ∠EFG. Prove that $$\overline{E K}$$ ⊥ $$\overline{F J}$$. (Hint: Write equations using the angle measures of the triangles and quadrilaterals formed.)

Question 47.
PROOF
Prove the congruent Parts of Parallel Lines Corollary: If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Given
Prove $$\overline{H K}$$ ≅ $$\overline{K M}$$
(Hint: Draw $$\overline{K P}$$ and $$\overline{M Q}$$ such that quadrilatcral GPKJ and quadrilateral JQML are parallelorams.)

Maintaining Mathematical Proficiency

Determine whether lines l and m are parallel. Explain your reasoning.

Question 48.

Question 49.

Question 50.

7.3 Proving That a Quadrilateral is a Parallelogram

Exploration 1

Proving That a Quadrilateral is a Parallelogram

Work with a partner: Use dynamic geometry software.

a. Construct an quadrilateral ABCD whose opposite sides are congruent.

c. Repeat pasts (a) and (b) for several other quadrilaterals. Then write a conjecture based on your results.

d. Write the converse of your conjecture. Is the Converse true? Explain.
REASONING ABSTRACTLY
To be proficient in math, you need to know and flexibly use different properties of objects.

Exploration 2

Proving That a Quadrilateral Is a Parallelogram

Work with a partner: Use dynamic geometry software.

a. Construct any quadrilateral ABCD whose opposite angles are congruent.

c. Repeat parts (a) and (b) for several other quadrilaterals. Then write a conjecture based on your results.

d. Write the converse of your conjecture. Is the converse true? Explain.

Question 3.
How can you prove that a quadrilateral is a parallelogram?

Question 4.

Lesson 7.3 Proving that a Quadrilateral is a Parallelogram

Monitoring Progress

Question 1.
In quadrilateral WXYZ, m∠W = 42°, m∠X = 138°, and m∠Y = 42°. Find m∠Z. Is WXYZ a parallelogram’? Explain your reasoning.

Question 2.
For what values of x and y is quadrilateral ABCD a parallelogram? Explain your reasoning.

State the theorem you can use to show that the quadrilateral is a parallelogram.

Question 3.

Question 4.

Question 5.

Question 6.
For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning.

Question 7.
Show that quadrilateral JKLM is a parallelogram.

Question 8.
Refer to the Concept Summary. Explain two other methods you can use to show that quadrilateral ABCD in Example 5 is a parallelogram.
Concept Summary
Ways to Prove a Quadrilateral is a Parallelogram

Exercise 7.3 Proving that a Quadrilateral is a Parallelogram

Vocabulary and Core Concept Check

Question 1.
WRITING

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different’? Find “both” answers.

 Construct a quadrilateral with opposite sides congruent. Construct a quadrilateral with one pair of parallel sides. Construct a quadrilateral with opposite angles congruent, Construct a quadrilateral with one pair of opposite sides congruent and parallel.

Monitoring Progress and Modeling with Mathematics

In Exercises 3-8, state which theorem you can use to show that the quadrilateral is a parallelogram.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

In Exercises 9-12, find the values of x and y that make the quadrilateral a para1leloram.

Question 9.

Question 10.

Question 11.

Question 12.

In Exercises 13-16, find the value of x that makes the quadrilateral a parallelogram.

Question 13.

Question 14.

Question 15.

Question 16.

In Exercises 17-20, graph the quadrilateral with the given vertices in a coordinate plane. Then show that the quadrilateral is a parallelogram.

Question 17.
A(0, 1), B(4, 4), C(12, 4), D(8, 1)

Question 18.
E(- 3, 0), F(- 3, 4), G(3, – 1), H(3, – 5)

Question 19.
J(- 2, 3), K(- 5, 7), L(3, 6), M(6, 2)

Question 20.
N(- 5, 0), P(0, 4), Q(3, 0), R(- 2, – 4)

ERROR ANALYSIS
In Exercises 21 and 22, describe and correct the error in identifying a parallelogram.

Question 21.

Question 22.

Question 23.
MATHEMATICAL CONNECTIONS

Question 24.
MAKING AN ARGUMENT
Your friend says you can show that quadrilateral WXYZ is a parallelogram by using the Consecutive Interior Angles Converse (Theorem 3.8) and the Opposite Sides Parallel and Congruent Theorem (Theorem 7.9). Is your friend correct? Explain your reasoning.

ANALYZING RELATIONSHIPS
In Exercises 25-27, write the indicated theorems as a biconditional statement.

Question 25.
Parallelogram Opposite Sides Theorem (Theorem 7.3) and Parallelogram Opposite Sides Converse (Theorem 7.7)

Question 26.
Parallelogram Opposite Angles Theorem (Theorem 7.4) and Parallelogram Opposite Angles Converse (Theorem 7.8)

Question 27.
Parallclorarn Diagonals Theorem (Theorem 7.6) and
Parallelogram Diagonals Converse (Theorem 7.10)

Question 28.
CONSTRUCTION
Describe a method that uses the Opposite Sides Parallel and Congruent Theorem (Theorem 7.9) to construct a parallelogram. Then construct a parallelogram using your method.

Question 29.
REASONING
Follow the steps below to construct a parallelogram. Explain why this method works. State a theorem to support your answer.

Step 1: Use a ruler to draw two segments that intersect at their midpoints.

Step 2: Connect the endpoints of the segments to form a parallelogram.

Question 30.
MAKING AN ARGUMENT
Your brother says to show that quadrilateral QRST is a parallelogram. you must show that $$\overline{Q R}$$ || $$\overline{T S}$$ and $$\overline{Q T}$$ || $$\overline{R S}$$. Your sister says that you must show that $$\overline{Q R} \cong \overline{T S}$$ and $$\overline{Q R} \cong \overline{T S}$$. Who is correct? Explain your reasoning.

REASONING
In Exercises 31 and 32, our classmate incorrectly claims that the marked information can be used to show thai the figure is a parallelogram. Draw a quadrilateral with the same marked properties that is clearly not a parallelogram.

Question 31.

Question 32.

Question 33.
MODELING WITH MATHEMATICS
You shoot a pool ball, and it rolls back to where it started, as shown in the diagram. The ball bounces off each wall at the same angle at which it hits the wall.

a. The ball hits the first wall at an angle of 63°. So m∠AEF = m∠BEH = 63°. What is m∠AFE? Explain your reasoning.
b. Explain why m∠FGD = 63°.
c. What is m∠GHC? m∠EHB?

Question 34.
MODELING WITH MATHEMATICS
In the diagram of the parking lot shown, m∠JKL = 60°, JA = LM = 21 feet, and KL = JM = 9 feet.

a. Explain how to show that parking space JKLM is a parallelogram.

b. Find m∠JML, m∠KJM, and m∠KLM.

c. $$\overline{L M}$$||$$\overline{N O}$$ and $$\overline{N O}$$ || $$\overline{P Q}$$ which theorem could you use to show that $$\overline{J K}$$ || $$\overline{P Q}$$?

REASONING
In Exercises 35-37. describe how to prove that ABCD is a parallelogram.

Question 35.

Question 36.

Question 37.

Question 38.
REASONING
Quadrilateral JKLM is a parallelogram. Describe how to prove that ∆MGJ ≅ ∆KHL.

Question 39.
PROVING A THEOREM
Prove the Parallelogram Opposite Angles Converse (Theorem 7.8). (Hint: Let x° represent m∠A and m∠C. Let y° represent m∠B and m∠D. Write and simplify an equation involving x and y)
Gien ∠A ≅ ∠C, ∠B ≅∠D
Prove ABCD is a parallelogram.

Question 40.
PROVING A THEOREM
Use the diagram of PQRS with the auxiliary line segment drawn to prove the Opposite Sides Parallel and Congruent Theorem (Theorem 7.9).
Given $$\overline{Q R}$$ || $$\overline{P S}$$. $$\overline{Q R} \cong \overline{P S}$$
Prove PQRS is a parallelogram.

Question 41.
PROVING A THEOREM
Prove the Parallelogram Diagonals Converse (Theorem 7.10).
Given Diagonals $$\overline{J L}$$ and $$\overline{K M}$$ bisect each other.
Prove JKLM is a parallelogram.

Question 42.
PROOF
Write a proof.
Given DEBF is a parallelogram.
AE = CF
Prove ABCD is a parallelogram.

Question 43.
REASONING
Three interior angle measures of a quadrilateral are 67°, 67°, and 113°, Is this enough information to conclude that the quadrilateral is a parallelogram? Explain your reasoning.

Question 44.
HOW DO YOU SEE IT?
A music stand can be folded up, as shown. In the diagrams. AEFD and EBCF are parallelograms. Which labeled segments remain parallel as the stand is folded?

Question 45.
CRITICAL THINKING
In the diagram, ABCD is a parallelogram, BF = DE = 12, and CF = 8. Find AE. Explain your reasoning.

Question 46.
THOUGHT PROVOKING
Create a regular hexagon using congruent parallelograms.

Question 47.
WRITING
The Parallelogram Consecutive Angles Theorem (Theorem 7.5) says that if a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Write the converse of this theorem. Then write a plan for proving the converse. Include a diagram.

Question 48.
PROOF
Write a proof.
Given ABCD is a parallelogram.
∠A is a right angle.
Prove ∠B, ∠C, and ∠D are right angles.

Question 49.
ABSTRACT REASONING
The midpoints of the sides of a quadrilateral have been joined to turn what looks like a parallelogram. Show that a quadrilateral formed by connecting the midpoints of the sides of any quadrilateral is always a parallelogram. (Hint: Draw a diagram. Include a diagonal of the larger quadrilateral. Show how two sides of the smaller quadrilateral relate to the diagonal.)

Question 50.
CRITICAL THINKING
Show that if ABCD is a parallelogram with its diagonals intersecting at E, then you can connect the midpoints F, G, H, and J of $$\overline{A E}$$, $$\overline{B E}$$, $$\overline{C E}$$, and $$\overline{D E}$$, respcetively, to form another parallelogram, FGHJ.

Maintaining Mathematical proficiency

Question 51.

Question 52.

Question 53.

Question 54.

7.1 – 7.3 Quiz

Find the value of x.

Question 1.

Question 2.

Question 3.

Find the measure of each interior angle and each exterior angle of the indicated regular polygon.

Question 4.
decagon

Question 5.
15-gon

Question 6.
24-gon

Question 7.
60-gon

Find the indicated measure in ABCD. Explain your reasoning.

Question 8.
CD

Question 9.

Question 10.
AE

Question 11.
BD

Question 12.
m∠BCD

Question 13.
m∠ABC

Question 14.

Question 15.
m∠DBC

State which theorem you can use to show that the quadrilateral is a paralle1oram.

Question 16.

Question 17.

Question 18.

Graph the quadrilateral with the given vertices in a coordinate plane. Then show that the quadrilateral is a parallelogram.

Question 19.
Q(- 5, – 2) R(3, – 2), S(1, – 6), T(- 7, – 6)

Question 20.
W(- 3, 7), X(3, 3), Y(1, – 3), Z(- 5, 1)

Question 21.
A stop sign is a regular polygon. (Section 7.1)

a. Classify the stop sign by its number of sides.

b. Find the measure of each interior angle and each exterior angle of the stop sign.

Question 22.
In the diagram of che staircase shown, JKLM is a parallelogram, $$\overline{Q T}$$ || $$\overline{R S}$$, QT = RS = 9 feet, QR = 3 feet, and m∠QRS = 123°.

a. List all congruent sides and angles in JKLM. Explain your reasoning.

b. Which theorem could you use to show that QRST is a parallelogram?

c. Find ST, m∠QTS, m∠TQR, and m∠TSR. Explain your reasoning.

7.4 Properties of Special Parallelograms

Exploration 1

Work with a partner: Use dynamic geometry software.

Sample

a. Draw a circle with center A.

b. Draw two diameters of the circle. Label the endpoints B, C, D, and E.

d. Is BDCE a parallelogram? rectangle? rhombus? square? Explain your reasoning.

e. Repeat parts (a)-(d) for several other circles. Write a conjecture based on your results.

Exploration 2

Work with a partner: Use dynamic geometry software.

Sample

a. Construct two segments that are perpendicular bisectors of each other. Label the endpoints A, B, D, and E. Label the intersection C.

c. Is AEBD a parallelogram? rectangle? rhombus? square? Explain your reasoning.

d. Repeat parts (a)-(c) for several other segments. Write a conjecture based on your results.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.

Question 3.
What are the properties of the diagonals of rectangles, rhombuses, and squares?

Question 4.
Is RSTU a parallelogram? rectangle? rhombus? square? Explain your reasoning.

Question 5.
What type of quadrilateral has congruent diagonals that bisect each other?

Lesson 7.4 Properties of Special Parallelograms

Monitoring Progress

Question 1.
For any square JKLM, is it always or sometimes true that $$\overline{J K}$$ ⊥ $$\overline{K L}$$? Explain your reasoning.

Question 2.
For any rectangle EFGH, is it always or sometimes true that $$\overline{F G} \cong \overline{G H}$$? Explain your reasoning.

Question 3.
A quadrilateral has four congruent sides and four congruent angles. Sketch the quadrilateral and classily it.

Question 4.
In Example 3, what is m∠ADC and m∠BCD?

Question 5.
Find the measures of the numbered angles in rhombus DEFG.

Question 6.
Suppose you measure only the diagonals of the window opening in Example 4 and they have the same measure. Can you conclude that the opening is a rectangle? Explain.

Question 7.
WHAT IF?
In Example 5. QS = 4x – 15 and RT = 3x + 8. Find the lengths of the diagonals of QRST.

Question 8.
Decide whether PQRS with vertices P(- 5, 2), Q(0, 4), R(2, – 1), and S(- 3, – 3) is a rectangle, a rhombus, or a square. Give all naines that apply.

Exercise 7.4 Properties of Special Parallelograms

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
What is another name for an equilateral rectangle?

Question 2.
WRITING
What should you look for in a parallelogram to know if the parallelogram is also a rhombus?

Monitoring Progress and Modeling with Mathematics

In Exercises 3-8, for any rhombus JKLM, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning.

Question 3.
∠L ≅∠M

Question 4.
∠K ≅∠M

Question 5.
$$\overline{J M} \cong \overline{K L}$$

Question 6.
$$\overline{J K} \cong \overline{K L}$$

Question 7.
$$\overline{J L} \cong \overline{K M}$$

Question 8.
∠JKM ≅ ∠LKM

Question 9.

Question 10.

Question 11.

Question 12.

In Exercises 13-16. find the measures of the numbered angles in rhombus DEFG.

Question 13.

Question 14.

Question 15.

Question 16.

In Exercises 17-22, for any rectangle WYYZ, decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning.

Question 17.
∠W ≅ ∠X

Question 18.
$$\overline{W X} \cong \overline{Y Z}$$

Question 19.
$$\overline{W X} \cong \overline{X Y}$$

Question 20.
$$\overline{W Y} \cong \overline{X Z}$$

Question 21.
$$\overline{W Y}$$ ⊥ $$\overline{X Z}$$

Question 22.
∠WXZ ≅∠YXZ

In Exercises 23 and 24, determine whether the quadrilateral is a rectangle.

Question 23.

Question 24.

In Exercises 25-28, find the lengths of the diagonals of rectangle WXYZ.

Question 25.
WY = 6x – 7
XZ = 3x + 2

Question 26.
WY = 14x + 10
XZ = 11x – 22

Question 27.
WY = 24x – 8
XZ = – 18x + 13

Question 28.
WY = 16x + 2
XZ = 36x – 6

In Exercises 29-34, name each quadrilateral – parallelogram, rectangle, rhombus, or square – for which the statement is always true.

Question 29.
It is equiangular.

Question 30.
It is equiangular and equilateral.

Question 31.
The diagonals are perpendicular.

Question 32.
Opposite sides are congruent.

Question 33.
The diagonals bisect each other.

Question 34.
The diagonals bisect opposite angles.

Question 35.
ERROR ANALYSIS
Quadrilateral PQRS is a rectangle. Describe and correct the error in finding the value of x.

Question 36.
ERROR ANALYSIS
Quadrilateral PQRS is a rhombus. Describe and correct the error in finding the value of x.

In Exercises 37 – 42, the diagonals of rhombus ABCD intersect at E. Ghen that m∠BAC = 53°, DE = 8, and EC = 6, find the indicated measure.

Question 37.
m∠DAC

Question 38.
m∠AED

Question 39.

Question 40.
DB

Question 41.
AE

Question 42.
AC

In Exercises 43-48. the diagonals of rectangle QRST intersect at P. Given that n∠PTS = 34° and QS = 10, find the indicated measure.

Question 43.
m∠QTR

Question 44.
m∠QRT

Question 45.
m∠SRT

Question 46.
QP

Question 47.
RT

Question 48.
RP

In Exercises 49-54. the diagonals of square LMNP intersect at K. Given that LK = 1. find the indicated measure.

Question 49.
m∠MKN

Question 50.
m∠LMK

Question 51.
m∠LPK

Question 52.
KN

Question 53.
LN

Question 54.
MP

In Exercises 55-69. decide whetherJKLM is a rectangle, a rhombus. or a square. Give all names that apply. Explain your reasoning.

Question 55.
J(- 4, 2), K(0, 3), L(1, – 1), M(- 3, – 2)

Question 56.
J(- 2, 7), K(7, 2), L(- 2, – 3), M(- 11, 2)

Question 57.
J(3, 1), K(3, – 3), L(- 2, – 3), M(- 2, 1)

Question 58.
J(- 1, 4), K(- 3, 2), L(2, – 3), M(4, – 1)

Question 59.
J(5, 2), K(1, 9), L(- 3, 2), M(1, – 5)

Question 60.
J(5, 2), K(2, 5), L(- 1, 2), M(2, – 1)

MATHEMATICAL CONNECTIONS
In Exercises 61 and 62, classify the quadrilateral. Explain your reasoning. Then find the values of x and y.

Question 61.

Question 62.

Question 63.
DRAWING CONCLUSIONS
In the window, $$\overline{B D}$$ ≅ $$\overline{D F}$$ ≅ $$\overline{B H}$$ ≅ $$\overline{H F}$$. Also, ∠HAB, ∠BCD, ∠DEF, and ∠FGH are right angles.

a. Classify HBDF and ACEG. Explain your reasoning.

b. What can you conclude about the lengths of the diagonals $$\overline{A E}$$ and $$\overline{G C}$$? Given that these diagonals intersect at J, what can you conclude about the lengths of $$\overline{A J}$$, $$\overline{J E}$$, $$\overline{C J}$$ and $$\overline{J G}$$? Explain.

Question 64.
ABSTRACT REASONING
Order the terms in a diagram so that each term builds off the previous term(s). Explain why each figure is in the location you chose.

square

rectangle

rhombus

parallelogram

CRITICAL THINKING
In Exercises 65-70, complete each statement with always, sometimes, or never. Explain your reasoning.
Question 65.
A square is ____________ a rhombus.

Question 66.
A rectangle is __________ a square.

Question 67.
A rectangle _____________ has congruent diagonals.

Question 68.
The diagonals of a square _____________ bisect its angles.

Question 69.
A rhombus __________ has four congruent angles.

Question 70.
A rectangle ____________ has perpendicular diagonals.

Question 71.
USING TOOLS
You want to mark off a square region for a garden at school. You use a tape measure to mark off a quadrilateral on the ground. Each side of the quadrilateral is 2.5 meters long. Explain how you can use the tape measure to make sure that the quadrilateral is a square.

Question 72.
PROVING A THEOREM
Use the plan for proof below to write a paragraph proof for one part of the Rhombus Diagonals Theorem (Theorem 7. 11).

Given ABCD is a parallelogram.
$$\overline{A C}$$ ⊥ $$\overline{B D}$$
Prove: ABCD is a rhombus.
Plan for Proof: Because ABCD is a parallelogram. its diagonals bisect each other at X. Use $$\overline{A C}$$ ⊥ $$\overline{B D}$$ to show that ∆BXC ≅ ∆DXC. Then show that $$\overline{B C}$$ ≅ $$\overline{D C}$$. Use the properties of a parallelogram to show that ABCD is a rhombus.

PROVING A THEOREM
In Exercises 73 and 74, write a proof for parts of the Rhombus Opposite Angles Theorem (Theorem 7.12).

Question 73.
Given: PQRS is a parallelogram.
$$\overline{P R}$$ bisects ∠SPQ and ∠QRS.
$$\overline{S Q}$$ bisects ∠PSR and ∠RQP.
Prove: PQRS is a rhombus.

Question 74.
Given: WXYZ is a rhombus
Prove: $$\overline{W Y}$$ bisects ∠ZWX and ∠XYZ.
$$\overline{Z X}$$ bisects ∠WZY and ∠YXW.

Question 75.
ABSTRACT REASONING
Will a diagonal of a square ever divide the square into two equilateral triangles? Explain your reasoning.

Question 76.
ABSTRACT REASONING
Will a diagonal of a rhombus ever divide the rhombus into two equilateral triangles? Explain your reasoning.

Question 77.
CRITICAL THINKING

Question 78.
HOW DO YOU SEE IT?
What other information do you need to determine whether the figure is a rectangle?

Question 79.
REASONING
Are all rhombuses similar? Are all squares similar? Explain your reasoning.

Question 80.
THOUGHT PROVOKING
Use the Rhombus Diagonals Theorem (Theorem 7. 1I) to explain why every rhombus has at least two lines of symmetry.

PROVING A COROLLARY
In Exercises 81-83, write the corollary as a conditional statement and its converse. Then explain why each statement is true.

Question 81.
Rhombus Corollary (Corollary 7.2)

Question 82.
Rectangle Corollary (Corollary 7.3)

Question 83.
Square Corollary (Corollary 7.4)

Question 84.
MAKING AN ARGUMENT
Your friend claims a rhombus will never have congruent diagonals because it would have to be a rectangle. Is your friend correct? Explain your reasoning.

Question 85.
PROOF
Write a proof in the style of your choice.
Gien ∆XYZ ≅ ∆XWZ, ∠XYW ≅ ∠ZWY
Prove WVYZ is a rhombus.

Question 86.
PROOF
Write a proof in the style of your choice.
Given
Prove ABCD is a rectangle.

PROVING A THEOREM
In Exercises 87 and 88. write a proof for part of the Rectangle Diagonals Theorem (Theorem 7.13).

Question 87.
Given PQRS is a retangle.
Prove $$\overline{P R} \cong \overline{S Q}$$

Question 88.
Given PQRS is a parallelogram.
$$\overline{P R} \cong \overline{S Q}$$
Prove PQRS is a rectangle.

Maintaining Mathematical Proficiency

$$\overline{D E}$$ is a midsegment of ∆ABC. Find the values of x and y.

Question 89.

Question 90.

Question 91.

7.5 Properties of Trapezoids and Kites

Exploration 1

Sample

Work with a partner. Use dynamic geometry software.

a. Construct a trapezoid whose base angles are congruent. Explain your process.

c. Repeat parts (a) and (b) for several other trapezoids. Write a conjecture based on your results.
PERSEVERE IN SOLVING PROBLEMS
To be proficient in math, you need to draw diagrams of important features and relationships, and search for regularity or trends.

Exploration 2

Discovering a Property of Kites

Work with a partner. Use dynamic geometry software.

Sample

a. Construct a kite. Explain your process.

b. Measure the angles of the kite. What do you observe?

c. Repeat parts (a) and (b) for several other kites. Write a conjecture based on your results.

Question 3.
What are some properties of trapezoids and kites?

Question 4.
Is the trapezoid at the left isosceles? Explain.

Question 5.
A quadrilateral has angle measures of 7o, 70°, 1100, and 110°, Is the quadrilateral a kite? Explain.

Lesson 7.5 Properties of Trapezoids and Kites

Monitoring progress

Question 1.
The points A(- 5, 6), B(4, 9) C(4, 4), and D(- 2, 2) form the vertices of a quadrilateral. Show that ABCD is a trapezoid. Then decide whether it is isosceles.

In Exercises 2 and 3, use trapezoid EFGH.

Question 2.
If EG = FH, is trapezoid EFGH isosceles? Explain.

Question 3.
If m∠HEF = 70° and ,m∠FGH = 110°, is trapezoid EFGH isosceles? Explain.

Question 4.
In trapezoid JKLM, ∠J and ∠M are right angles, and JK = 9 centimeters. The length of midsegment $$\overline{N P}$$ of trapezoid JKLW is 12 centimeters. Sketch trapezoid JKLM and its midsegment. Find ML. Explain your reasoning.

Question 5.
Explain another method you can use to find the length of $$\overline{Y Z}$$ in Example 4.

Question 6.
In a kite. the measures of the angles are 3x° 75°, 90°, and 120°. Find the value of x. What are the measures of the angles that are congruent?

Question 7.
Quadrilateral DEFG has at least one pair of opposite sides congruent. What types of quadrilaterals meet this condition?

Question 8.

Question 9.

Question 10.

Exercise 7.5 Properties of Trapezoids and Kites

Vocabulary and Core Concept Check

Question 1.
WRITING
Describe the differences between a trapezoid and a kite.

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Is there enough information to prove that trapezoid ABCD is isosceles?

Is there enough information to prove that $$\overline{A B}$$ ≅ $$\overline{D C}$$?

Is there enough information to prove that the non-parallel sides of trapezoid ABCD are congruent?

Is there enough information to prove that the legs of trapezoid ABCD are congruent?

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, show that the quadrilateral with the given vertices is a trapezoid. Then decide whether it is isosceles.

Question 3.
W(1, 4), X(1, 8), Y(- 3, 9), Z(- 3, 3)

Question 4.
D(- 3, 3), E(- 1, 1), F(1, – 4), G(- 3, 0)

Question 5.
M(- 2, 0), N(0, 4), P(5, 4), Q(8, 0)

Question 6.
H(1, 9), J(4, 2), K(5, 2), L(8, 9)

In Exercises 7 and 8, find the measure of each angle in the isosceles trapezoid.

Question 7.

Question 8.

In Exercises 9 and 10. find the length of the midsegment of the trapezoid.

Question 9.

Question 10.

In Exercises 11 and 12, find AB.

Question 11.

Question 12.

In Exercises 13 and 14, find the length of the midsegment of the trapezoid with the given vertices.

Question 13.
A(2, 0), B(8, – 4), C(12, 2), D(0, 10)

Question 14.
S(- 2, 4), T(- 2, – 4), U(3, – 2), V(13, 10)

In Exercises 15 – 18, Find m ∠ G.

Question 15.

Question 16.

Question 17.

Question 18.

Question 19.
ERROR ANALYSIS
Describe and correct the error in finding DC.

Question 20.
ERROR ANALYSIS
Describe and correct the error in finding m∠A.

In Exercises 21 – 24. given the most specific name for the quadrilateral. Explain your reasoning.

Question 21.

Question 22.

Question 23.

Question 24.

REASONING
In Exercises 25 and 26, tell whether enough information is given in the diagram to classify the quadrilateral by the indicated name. Explain.

Question 25.
rhombus

Question 26.
Square

MATHEMATICAL CONNECTIONS
In Exercises 27 and 28, find the value of x.

Question 27.

Question 28.

Question 29.
MODELING WITH MATHEMATICS
In the diagram, NP = 8 inches, and LR = 20 inches. What is the diameter of the bottom layer of the cake?

Question 30.
PROBLEM SOLVING
You and a friend arc building a kite. You need a stick to place from X to Wand a stick to place from W to Z to finish constructing the frame. You want the kite to have the geometric shape of a kite. How long does each stick need to be? Explain your reasoning.

REASONING
In Exercises 31 – 34, determine which pairs of segments or angles must be congruent so that you can prove that ABCD is the indicated quadrilateral. Explain our reasoning. (There may be more than one right answer.)

Question 31.
isosceles trapezoid

Question 32.
Kite

Question 33.
Parallelogram

Question 34.
square

Question 35.
PROOF
Write a proof
Given $$\overline{J L} \cong \overline{L N}$$, $$\overline{K M}$$ is a midsegment of ∆JLN
Prove Quadrilateral JKMN is an isosceles trapezoid.

Question 36.
PROOF
Write a proof
Given ABCD is a kite.
$$\overline{A B} \cong \overline{C B}$$, $$\overline{A D} \cong \overline{C D}$$
Prove $$\overline{C E} \cong \overline{A E}$$

Question 37.
ABSTRACT REASONING
Point U lies on the perpendicular bisector of $$\overline{R T}$$. Describe the set of points S for which RSTU is a kite.

Question 38.
REASONING
Determine whether the points A(4, 5), B(- 3, 3), C(- 6, – 13), and D(6, – 2) are the vertices of a kite. Explain your reasoning.

PROVING A THEOREM
In Exercises 39 and 40, use the diagram to prove the given theorem. In the diagram, $$\overline{E C}$$’ is drawn parallel to $$\overline{A B}$$.

Question 39.
Isosceles Trapezoid Base Angles Theorem (Theorem 7.14)
Given ABCD is an isosceles trapezoid.
$$\overline{B C}$$ || $$\overline{A D}$$
Prove ∠A ≅ ∠D, ∠B ≅ ∠BCD

Question 40.
Isosceles Trapezoid Base Angles Theorem (Theorem 7.15)
Given ABCD is a trapezoid
∠A ≅ ∠D, $$\overline{B C}$$ || $$\overline{A D}$$
Prove ABCD is an isosceles trapezoid.

Question 41.
MAKING AN ARGUMENT
Your cousin claims there is enough information to prove that JKLW is an isosceles trapezoid. Is your cousin correct? Explain.

Question 42.
MATHEMATICAL CONNECTIONS
The bases of a trapezoid lie on the lines y = 2x + 7 and y = 2x – 5. Write the equation of the line that contains the midsegment of the trapezoid.

Question 43.
CONSTRUCTION
$$\overline{A C}$$ and $$\overline{B D}$$ bisect each other.
a. Construct quadrilateral ABCD so that $$\overline{A C}$$ and $$\overline{B D}$$ are congruent. hut not perpendicular. Classify the quadrilateral. Justify your answer.

b. Construct quadrilateral ABCD so that $$\overline{A C}$$ and $$\overline{B D}$$ are perpendicular. hut not congruent. Classify the quadrilateral. Justify your answer.

Question 44.
PROOF Write a proof.
Given QRST is an isosceles trapezoid.
Prove ∠TQS ≅ ∠SRT

Question 45.
MODELING WITH MATHEMATICS
A plastic spiderweb is made in the shape of a regular dodecagon (12-sided polygon). $$\overline{A B}$$ || $$\overline{P Q}$$, and X is equidistant from the vertices of the dodecagon.

a. Are you given enough information to prove that ABPQ is an isosceles trapezoid?
b. What is the measure of each interior angle of ABPQ

Question 46.
ATTENDING TO PRECISION
In trapezoid PQRS, $$\overline{P Q}$$ || $$\overline{R S}$$ and $$\overline{M N}$$ is the midsegment of PQRS. If RS = 5 . PQ. what is the ratio of MN to RS?
(A) 3 : 5
(B) 5 : 3
(C) 1 : 2
(D) 3 : 1

Question 47.
PROVING A THEOREM
Use the plan for proof below to write a paragraph proof of the Kite Opposite Angles Theorem (Theorem 7.19).
Given EFGH is a Kite.
, 
Prove ∠E ≅ ∠G, ∠F ∠H

Plan for Proof: First show that ∠E ≅ ∠G. Then use an indirect argument to show that ∠F ∠H.

Question 48.
HOW DO YOU SEE IT?
One of the earliest shapes used for cut diamonds is called the table cut, as shown in the figure. Each face of a cut gem is called a facet.

a. $$\overline{B C}$$ || $$\overline{A D}$$, and $$\overline{A B}$$ and $$\overline{D C}$$ are not parallel. What shape is the facet labeled ABCD?
b. $$\overline{D E}$$ || $$\overline{G F}$$, and $$\overline{D G}$$ and $$\overline{E F}$$ are congruent but not parallel. What shape is the facet labeled DEFG?

Question 49.
PROVING A THEOREM
In the diagram below, $$\overline{B G}$$ is the midsegment of ∆ACD. and $$\overline{G E}$$ is the midsegment of ∆ADF Use the diagram to prove the Trapezoid Midsegment Theorem (Theorem 7.17).

Question 50.
THOUGHT PROVOKING

Question 51.
PROVING A THEOREM
To prove the biconditional statement in the Isosceles Trapezoid Diagonals Theorem (Theorem 7.16), you must prove both Parts separately.
a. Prove part of the Isosceles Trapezoid Diagonals Theorem (Theorem 7. 16).
Given JKLM is an isosecles trapezoid.
$$\overline{K L}$$ || $$\overline{J M}$$, $$\overline{J L} \cong \overline{K M}$$
Prove $$\overline{J L} \cong \overline{K M}$$

b. Write the other parts of the Isosceles Trapezoid Diagonals Theorem (Theorem 7. 16) as a conditional. Then prove the statement is true.

Question 52.
PROOF
What special type of quadrilateral is EFGH? Write a proof to show that your answer is Correct.
Given In the three-dimensional figure, $$\overline{J L} \cong \overline{K M}$$. E, F, G, and H arc the midpoints of $$\overline{J L}$$. $$\overline{K l}$$, $$\overline{K M}$$, and $$\overline{J M}$$. respectively.
Prove EFGH is a ____________ .

Maintaining Mathematical Proficiency

Describe a similarity transformation that maps the blue preimage to the green image.

Question 53.

Question 54.

7.1 Angles of Polygons

Question 1.
Find the sum of the measures of the interior angles of a regular 30-gon. Then find the measure of each interior angle and each exterior angle.

Find the va1ue of x.

Question 2.

Question 3.

Question 4.

7.2 Properties of Parallelograms

Find the value of each variable in the parallelogram.

Question 5.

Question 6.

Question 7.

Question 8.
Find the coordinates of the intersection of the diagonals of QRST with vertices Q(- 8, 1), R(2, 1). S(4, – 3), and T(- 6, – 3).

Question 9.
Three vertices of JKLM are J(1, 4), K(5, 3), and L(6, – 3). Find the coordinates of vertex M.

7.3 Proving that a Quadrilateral is a Parallelogram

State which theorem you can use to show that the quadrilateral is a parallelogram.

Question 10.

Question 11.

Question 12.

Question 13.
Find the values of x and y that make the quadrilateral a parallelogram.

Question 14.
Find the value of x that makes the quadrilateral a parallelogram.

Question 15.
Show that quadrilateral WXYZ with vertices W(- 1, 6), X(2, 8), Y(1, 0), and Z(- 2, – 2) is a parallelogram.

7.4 Properties of Special Parallelograms

Question 16.

Question 17.

Question 18.

Question 19.
Find the lengths of the diagonals of rectangle WXYZ where WY = – 2y + 34 and XZ = 3x – 26.

Question 20.
Decide whether JKLM with vertices J(5, 8), K(9, 6), L(7, 2), and M(3, 4) is a rectangle. a rhombus, or a square. Give all names that apply. Explain.

7.5 Properties of Trapezoids and Kites

Question 21.
Find the measure of each angle in the isosceles trapezoid WXYZ.

Question 22.
Find the length of the midsegment of trapezoid ABCD.

Question 23.
Find the length of the midsegment of trapezoid JKLM with vertices J(6, 10), K(10, 6), L(8, 2), and M(2, 2).

Question 24.
A kite has angle measures of 7x°, 65°, 85°, and 105°. Find the value of x. What are the measures of the angles that are congruent?

Question 25.
Quadrilateral WXYZ is a trapezoid with one pair of congruent base angles. Is WXYZ all isosceles trapezoid? Explain your reasoning.

Question 26.

Question 27.

Question 28.

Find the value of each variable in the parallelogram.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.
In a convex octagon. three of the exterior angles each have a measure of x°. The other five exterior angles each have a measure of (2x + 7)°. Find the measure of each exterior angle.

Question 8.
Quadrilateral PQRS has vertices P(5, 1), Q(9, 6), R(5, 11), and 5(1, 6), Classify quadrilateral PQRS using the most specific name.

Determine whether enough information is given to show that the quadrilateral is a parallelogram. Explain your reasoning.

Question 9.

Question 10.

Question 11.

Question 12.
Explain why a parallelogram with one right angle must be a rectangle.

Question 13.
Summarize the ways you can prove that a quadrilateral is a square.

Question 14.
Three vertices of JKLM are J(- 2, – 1), K(0, 2), and L(4, 3),
a. Find the coordinates of vertex M.

b. Find the coordinates of the intersection of the diagonals of JKLM.

Question 15.
You are building a plant stand with three equally-spaced circular shelves. The diagram shows a vertical cross section of the plant stand. What is the diameter of the middle shell?

Question 16.
The Pentagon in Washington. D.C., is shaped like a regular pentagon. Find the measure of each interior angle.

Question 17.
You are designing a binocular mount. If $$\overline{B C}$$ is always vertical, the binoculars will point in the same direction while they are raised and lowered for different viewers. How can you design the mount so $$\overline{B C}$$ is always vertical? Justify your answer.

Question 18.
The measure of one angle of a kite is 90°. The measure of another angle in the kite is 30°. Sketch a kite that matches this description.

Quadrilaterals and Other Polygons Cummulative Assessment

Question 1.
Copy and complete the flowchart proof of the Parallelogram Opposite Angles Theorem (Thm. 7.4).
Given ABCD is a parallelogram.
Prove ∠A ≅ ∠C, ∠B ≅ ∠D

Question 2.
Use the steps in the construction to explain how you know that the circle is inscribed within ∆ABC.

Question 3.
Your friend claims that he can prove the Parallelogram Opposite Sides Theorem (Thm. 7.3) using the SSS Congruence Theorem (Thm. 5.8) and the Parallelogram Opposite Sides Theorem (Thin. 7.3). Is your friend correct? Explain your reasoning.

Question 4.
Find the perimeter of polygon QRSTUV Is the polygon equilateral? equiangular? regular? Explain your reasoni ng.

Question 5.
Choose the correct symbols to complete the proof of the Converse of the Hinge Theorem (Theorem 6. 13).

Given
Prove m ∠ B > m ∠ E
Step 1 Assume temporarily that m ∠ B m ∠ E. Then it follows that either m∠B____ m∠E or m∠B ______ m ∠ E.

Step 2 If m ∠ B ______ m∠E. then AC _____ DF by the Hinge Theorem (Theorem 6. 12). If, m∠B _______ m ∠ E. then ∠B _____ ∠E. So. ∆ABC ______ ∆DEF by the SAS Congruence Theorem (Theorem 5.5) and AC _______ DF.

Step 3 Both conclusions contradict the given statement that AC _______ DF. So, the temporary assumption that m ∠ B > m ∠ E Cannot be true. This proves that m ∠ B ______ m ∠ E.
>     <    =    ≠    ≅

Question 6.
Use the Isosceles Trapctoid Base Angles Conersc (Thm. 7.15) to prove that ABCD is an isosceles trapezoid.
Given $$\overline{B C}$$ || $$\overline{A D}$$. ∠EBC ≅ ∠¿ECB, ∠ABE ≅ ∠DCE
Prove ABCD is an isoscelcs trapezoid.

$$\overline{Q S} \cong \overline{R T}$$
 Statements Reasons 1. $$\overline{Q S} \cong \overline{R T}$$ 1. Given 2. __________________________ 2. Parallelogram Opposite Sides Theorem (Thm. 7.3) 3. __________________________ 3. SSS Congruence Theorem (Thm. 5.8) 4. __________________________ 4. Corresponding parts of congruent triangles are congruent. 5. __________________________ 5. Parallelogram Consecutive Angles Theorem (Thm. 7.5 6. __________________________ 6. Congruent supplementary angles have the same measure. 7. __________________________ 7. Parallelogram Consecutive Angles Theorem (Thm. 7.5) 8. __________________________ 8. Subtraction Property of Equality 9. __________________________ 9. Definition of a right angle 10. __________________________ 10. Definition of a rectangle