7.1 Practice A - · PDF file... of a heptagon. 2. The sum of the measures of the interior angles of a convex polygon is 3060 . ... number of sides. 3. Find the measure of each

Geometry Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 232

7.1 Practice A

Name _________________________________________________________ Date _________

1. Find the sum of the measures of the interior angles of a heptagon.

2. The sum of the measures of the interior angles of a convex polygon is 3060 .°Classify the polygon by the number of sides.

3. Find the measure of each interior and exterior angle of a regular 30-gon.

In Exercises 4 and 5, find the value of x.

4. 5.

In Exercises 6 and 7, find the measures of ∠X and ∠Y.

6. 7.


8. 9.

10. A pentagon has three angles that are congruent and two other angles that are supplementary to each other. Find the measure of each of the three congruent angles in the pentagon.

11. You are designing an amusement park ride with cars that will spin in a circle around a center axis, and the cars are located at the vertices of a regular polygon. The sum of the measures of the angles’ vertices is 6120 .° If each car holds a maximum of four people, what is the maximum number of people who can be on the ride at one time?

120°W

X

Y

Z103°149°

V

W

X

Y

Z

U

108°

144°

122°

x°

100°

110°

160°

105°

105°

115°x°

60°

64°

36°

48°

x°x°

x°73°

109°

Copyright © Big Ideas Learning, LLC Geometry All rights reserved. Resources by Chapter

233

7.1 Practice B

Name _________________________________________________________ Date __________


1. 2.

In Exercises 3 and 4, find the measures of ∠X and ∠Y.

3. 4.


5. 6.

7. Find the measure of each interior angle and each exterior angle of a regular 24-gon.

8. Each exterior angle of a regular polygon has a measure of 18°. Find the number of sides of the regular polygon.

9. A polygon has two pairs of complementary interior angles and three sets of supplementary interior angles. The sum of the remaining interior angles is 1440 .°How many sides does the polygon have? Explain.

10. The figure shows interior angle measures of the kite.

a. Find the sum of the measures of the interior angles of the convex polygon.

b. Find the value of x.

127°

40°

x° 150° 102°

84°2x°

x°

130°

V

W Z

X Y

88° 100°

V U

W Z

YX

100°

92°

86°70°

x°

x°

(x + 30)°

x°x°

x°

x°

x°

110°x°

110°x°


237

7.2 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–4, find the value of each variable in the parallelogram.

1. 2.

3. 4.

5. Find the coordinates of the intersection of the diagonals of the parallelogram with vertices ( ) ( ) ( ) ( )2, 1 , 1, 3 , 6, 3 , and 3, 1 .− − −

In Exercises 6 and 7, three vertices of parallelogram ABCD are given. Find the remaining vertex.

6. ( ) ( ) ( )2, 0 , 2, 2 , 2, 2A B D− − − 7. ( ) ( ) ( )1, 3 , 1, 2 , 1, 2A C D− − − −

8. The measure of one interior angle of a parallelogram is 30° more than two times the measure of another angle. Find the measure of each angle of the parallelogram.

9. Your friend claims that you can prove that two parallelograms are congruent by proving that they have two pairs of congruent opposite angles. Is your friend correct? Explain your reasoning.

10. Use the diagram to write a two-column proof.

Given: PQRS is a parallelogram.

Prove: PQT RST≅

38

17 x + 3

y − 2

2u°

124°

(v − 3)°

27

3s

19

t + 5

P S

RQ

T

111°

3b°

15

a + 5


7.2 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–4, find the value of each variable in the parallelogram.

1. 2.

3. 4.

5. Find the coordinates of the intersection of the diagonals of the parallelogram with vertices ( ) ( ) ( ) ( )2, 4 , 4, 4 , 2, 12 , and 4, 4 .− − −

6. Three vertices of parallelogram ABCD are ( ) ( ) ( )1, 5 , 1, 1 , and 2, 2 .A B D Find the coordinates of the remaining vertex.

7. Use the diagram to write a two-column proof.

Given: CEHF is a parallelogram. D bisects CE and G bisects .FH

Prove: CDF HGE≅

8. State whether each statement is always, sometimes, or never true for a parallelogram. Explain your reasoning.

a. The opposite sides are congruent.

b. All four sides are congruent.

c. The diagonals are congruent.

d. The opposite angles are congruent.

e. The adjacent angles are congruent.

f. The adjacent angles are complementary.

43

124 4(4y − 1)

3x + 10

66°3v°

u°

3b°3a + 5

(b + 84)°5a − 9

C D E

F G H

3c + 7

4c − 8

d12

d − 823


7.3 Practice A

Name _________________________________________________________ Date _________

In Exercises 1 and 2, state which theorem you can use to show that the quadrilateral is a parallelogram.

1. 2.

In Exercises 3 and 4, find the value of x that makes the quadrilateral a parallelogram.

3. 4.

In Exercises 5 and 6, graph the quadrilateral with the given vertices in a coordinate plane. Then show that the quadrilateral is a parallelogram.

5. ( ) ( ) ( ) ( )4, 2 , 2, 1 , 4, 1 , 2, 2A B C D− − − − 6. ( ) ( ) ( ) ( )4, 1 , 1, 5 , 11, 0 , 8, 4E F G H− − −

7. Use the diagram to write a two-column proof. Given: A ABE∠ ≅ ∠

, AE CD BC DE≅ ≅

Prove: BCDE is a parallelogram.

8. In the diagram of the handrail for a staircase shown, 145 and .m A AB CD∠ = ° ≅

a. Explain how to show that ABDC is a parallelogram.

b. Describe how to prove that CDFE is a parallelogram.

c. Can you prove that EFHG is a parallelogram? Explain.

d. Find , , , and . m ACD m DCE m CEF m EFD∠ ∠ ∠ ∠

115°

115°

41

3x + 5

2x

2x

3x + 2 5x − 6

A

E D

B C

A

B

D

F

H

G

C

E


243

7.3 Practice B

Name _________________________________________________________ Date __________

In Exercises 1 and 2, state which theorem you can use to show that the quadrilateral is a parallelogram.

1. 2.

In Exercises 3 and 4, find the value of x that makes the quadrilateral a parallelogram.

3. 4.

In Exercises 5 and 6, graph the quadrilateral with the given vertices in a coordinate plane. Then show that the quadrilateral is a parallelogram.

5. ( ) ( ) ( ) ( )3, 1 , 3, 4 , 3, 2 , 3, 3W X Y Z− − − − 6. ( ) ( ) ( ) ( )4, 0 , 2, 2 , 5, 1 , 1, 3A B C D− − − −

7. Use the diagram to write a two-column proof. Given: A FDE∠ ≅ ∠

F is the midpoint of .AD

D is the midpoint of .CE

Prove: ABCD is a parallelogram.

8. A quadrilateral has two pairs of congruent angles. Can you determine whether the quadrilateral is a parallelogram? Explain your reasoning.

9. An octagon star is shown in the figure on the right.

a. Find , , and . m FCG m BCF m D∠ ∠ ∠

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

c. The length of AB is three times the length of .AD Write an expression for the perimeter of parallelogram ABCD in terms of the variable x.

(2x + 5)°

3x°

5.8

5.8

7.27.2

7x + 1

8x − 10

D

F

A B

CE

45°45°

135°

E

C G

F

B

A D


247

7.4 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–5, the diagonals of rhombus ABCD intersect at E. Given that m EAD CE DE67 , 5, and 12,∠ = ° = = find the indicated measure.

1. m AED∠

2. m ADE∠

3. m BAE∠

4. AE

5. BE

In Exercises 6 and 7, find the lengths of the diagonals of rectangle JKLM.

6. 3 44 1

JL xKM x

= += −

7. 32

2 6

1

JL x

KM x

= −

= +

In Exercises 8 and 9, decide whether quadrilateral WXYZ is a rectangle, a rhombus, or a square. Give all names that apply. Explain your reasoning.

8. ( ) ( ) ( ) ( )3, 1 , 3, 2 , 5, 2 , 5, 1W X Y Z− − − − 9. ( ) ( ) ( ) ( )4, 1 , 1, 4 , 2, 1 , 1, 2W X Y Z− −

10. Use the figure to write a two-column proof. Given: PSUR is a rectangle.

PQ TU≅

Prove: QS RT≅

11. In the figure, all sides are congruent and all angles are right angles.

a. Determine whether the quadrilateral is a rectangle. Explain your reasoning.

b. Determine whether the quadrilateral is a rhombus. Explain your reasoning.

c. Determine whether the quadrilateral is a square. Explain your reasoning.

d. Find .m AEB∠

e. Find .m EAD∠

67°

12

5

E

A

CB

D

S

P Q R

T U

A

E

D

B C


7.4 Practice B

Name _________________________________________________________ Date _________

In Exercises 1 and 2, decide whether quadrilateral JKLM is a rectangle, a rhombus, or a square. Give all names that apply. Explain your reasoning.

1. ( ) ( ) ( ) ( )3, 5 , 7, 6 , 6, 2 , 2, 1J K L M 2. ( ) ( ) ( ) ( )4, 1 , 1, 5 , 5, 2 , 2, 4J K L M− − − −

In Exercises 3–7, the diagonals of rhombus ABCD intersect at M. Given that m MAB MB AM53 , 16, and 12,∠ = ° = = find the indicated measure.

3. m AMD∠

4. m ADM∠

5. m ACD∠

6. DM

7. AC

8. Find the point of intersection of the diagonals of the rhombus with vertices ( ) ( ) ( ) ( )1, 2 , 3, 4 , 5, 8 , and 1, 6 .−

9. Use the figure to write a two-column proof. Given: WXYZ is a parallelogram. XWY XYW∠ ≅ ∠

Prove: WXYZ is a rhombus.

10. Your friend claims that you can transform every rhombus into a square using a similarity transformation. Is your friend correct? Explain your reasoning.

11. A quadrilateral has four congruent angles. Is the quadrilateral a parallelogram? Explain your reasoning.

12. A quadrilateral has two consecutive right angles. If the quadrilateral is not a rectangle, can it still be a parallelogram? Explain your reasoning.

13. Will a diagonal of a rectangle ever divide the rectangle into two isosceles triangles? Explain your reasoning.

X

W Z

Y

53°16

12

M

D C

BA


7.5 Practice A

Name _________________________________________________________ Date _________


1. 2.

In Exercises 3 and 4, find the length of the midsegment of the trapezoid with the given vertices.

3. ( ) ( ) ( ) ( )0, 3 , 4, 5 , 4, 2 , 0, 2A B C D− − 4. ( ) ( ) ( ) ( )3, 3 , 1, 3 , 3, 3 , 5, 3E F G H− − − −

In Exercises 5 and 6, give the most specific name for the quadrilateral. Explain your reasoning.

5. 6.

7. Describe and correct the error in finding the most specfice name for the quadrialteral.

8. Use the diagram to write a two-column proof. Given: ABCD is a parallelogram.

AE AD≅

Prove: ABCE is an isosceles trapezoid.

9. The figure shows a window in the shape of a kite.

a. Find .m XVW∠

b. Find .XY

c. Which angle is congruent to ?XYZ∠

The quadrilateral has two pairs of consecutive congruent sides and the diagonals are perpendicular. So, the quadrilateral is a kite.

94°x° 100°

x°

W Z

X Y X Y

ZW

A B

E D C

X

V

Z

W Y

22 in.

26 in.


253

7.5 Practice B

Name _________________________________________________________ Date __________

In Exercises 1 and 2, show that the quadrilateral with the given vertices is a trapezoid. Then decide whether it is isosceles.

1. ( ) ( ) ( ) ( )1, 2 , 1, 3 , 3, 4 , 3, 3T U V W− − − − 2. ( ) ( ) ( ) ( )0, 0 , 2, 4 , 5, 4 , 5, 0P Q R S


3. 4.

In Exercises 5 and 6, give the most specific name for the quadrilateral. Explain your reasoning.

5. 6.

7. Use the diagram to write a two-column proof. Given: VXYZ is a kite.

, XY YZ WX UZ≅ ≅ Prove: WXV UZV≅

8. Three vertices of a trapezoid are given by ( ) ( ) ( )3, 6 , 3, 2 , and 6, 8 .− − − Find the fourth vertex such that the trapezoid is an isosceles trapezoid.

9. Is it possible to have a concave kite? Explain your reasoning.

10. The diagram shows isosceles trapezoid JKLP with base lengths a and b, and height c.

a. Explain how you know JKMN is a rectangle. Write the area of JKMN.

b. Write the formula for the area of .JNP

c. Write and simplify the formula for the area of trapezoid JKLP.

12 cm

X cm

22 cm

VW U

ZX

Y

P N M

KJ a

c

bL

55°

2x°

x°

B C

A D

D E

G F

Documents

7.1 Practice A - · PDF file... of a heptagon. 2. The sum of the measures of the interior angles of a convex polygon is 3060 . ... number of sides. 3. Find the measure of each