8-2€¦ · EXAMPLE 4 Solve Problems Involving Isosceles Trapezoids All horizontal beams of the high-voltage transmission tower are parallel to the ground. 14 ft 14 ft 10 ft 1 2 10

Activity Assess

Kites and Trapezoids

I CAN… use triangle congruence to understand kites and trapezoids.

VOCABULARY• midsegment of a trapezoid

8-2 CRITIQUE & EXPLAIN

Manuel draws a diagram of kite PQRS with

⟷ QS as the line of symmetry over

a design of a kite-shaped key fob. He makes a list of conclusions based on the diagram.

• PR QS• QP =~ QR• SP =~ SR• PR bisects QS.• ∆PQR is an equilateral triangle.• ∆PSR is an isosceles triangle.

A. Which of Manuel’s conclusions do you agree with? Which do you disagree with? Explain.

B. Use Structure What other conclusions are supported by the diagram?

STUDY TIPRemember that you must show that both B and D are on the perpendicular bisector in order to show that one diagonal is the perpendicular bisector of the other. It is not sufficient to show that only one is on the perpendicular bisector.

ESSENTIAL QUESTION How are diagonals and angle measures related in kites and trapezoids?

EXAMPLE 1 Investigate the Diagonals of a Kite

How are the diagonals of a kite related?

D B

A

C

X

The diagonals of a kite are perpendicular to each other. Exactly one diagonal bisects the other.

Try It! 1. a. What is the measure of ∠AXB?

b. If AX = 3.8 , what is AC?

c. If BD = 10 , does BX = 5 ? Explain.

A kite has two pairs of congruent adjacent sides.

CONCEPTUAL UNDERSTANDING

Point B is equidistant from the endpoints of ̄ AC , as is D, so they lie on the perpendicular bisector of ̄ AC .

R

Q

P

S

LESSON 8-2 Kites and Trapezoids 365

PearsonRealize.com

Activity Assess

EXAMPLE 2 Use the Diagonals of a Kite

Quadrilateral PQRS is a kite with diagonals ‾ QS and ‾ PR .

QT

R

P

S

35°2

3

1

A. What is m∠1 ?

The diagonals of a kite are perpendicular, so m∠1 = 90 .

B. What is m∠2 ?

The sum of the angles of △PQT is 180.

m∠2 + 35 + 90 = 180

m∠2 = 55

C. What is m∠3?

Since △PQR is an isosceles triangle, m∠3 ≅ m∠QPT.

So, m∠3 = 35.

Try It! 2. Quadrilateral WXYZ is a kite.

a. What is m∠1 ?

b. What is m∠2 ?

The diagonals of a kite are perpendicular.

PROOF: SEE EXERCISE 12.

If...

Then... ‾ WY ⟂ ‾ XZ

W YE

X

Z

THEOREM 8-3

W Y

X

Z

32°2

1

COMMON ERRORYou may incorrectly assume angles are congruent just from their appearance. Always check that you can prove congruence first.

366 TOPIC 8 Quadrilaterals and Other Polygons Go Online | PearsonRealize.com

Activity Assess

EXAMPLE 3 Explore Parts of an Isosceles Trapezoid

Kiyo is designing a trapezoid-shaped roof. In order for the roof to be symmetric, the overlapping triangles △DAB and △ADC must be congruent. Will the roof be symmetric?

Step 1 Show △ABE ≅ △DCF .

A E F

B C

D

Step 2 Show △DAB ≅ △ADC .

By CPCTC, ∠DAB ≅ ∠ADC . By the Reflexive Property of Congruency, ‾ AD ≅ ‾ DA . So, △DAB ≅ △ADC by SAS.

The overlapping triangles are congruent, so the roof is symmetric.

_

BE ≅ _

CF , because _

BE and

_ CF are altitudes.

△ABE and △DCF are right triangles, so they are congruent by the HL Theorem.

The roof is an isosceles trapezoid, since ‾ DC ≅ ‾ AB .

GENERALIZEHow do the lengths of the diagonals and the way they intersect relate to the sides of a quadrilateral?

APPLICATION

Try It! 3. a. Given isosceles trapezoid PQRS, what are m∠P , m∠Q, and m∠S ?

b. Given ‾ ST ∥ ‾ RU , what is the measure of ∠TUR ?

P

Q R

S

135°

B

A D

C

47°

R

S T

U


Activity Assess

In an isosceles trapezoid, each pair of base angles is congruent.


The diagonals of an isosceles trapezoid are congruent.


If...

Then... ∠BAD ≅ ∠CDA, ∠ABC ≅ ∠DCB

If...

Then... ‾ AC ≅ ‾ DB

A D

B C

A D

B C

THEOREM 8-4

THEOREM 8-5

EXAMPLE 4 Solve Problems Involving Isosceles Trapezoids

All horizontal beams of the high-voltage transmission tower are parallel to the ground.

14 ft14 ft

10 ft 1210 ft

A. If m∠1 = 138 , what is m∠2 ?

The sum of the interior angle measures of a quadrilateral is 360.

m∠1 + m∠1 + m∠2 + m∠2 = 360

138 + 138 + 2(m∠2) = 360

276 + 2(m∠2) = 360

2(m∠2) = 84

m∠2 = 42

The measure of ∠2 is 42 .

The base angles are congruent.

MAKE SENSE AND PERSEVEREWhat other strategy might you use to solve this problem?

The top section is an isosceles trapezoid.The center section

is an isosceles trapezoid.

CONTINUED ON THE NEXT PAGE


14 ft 14 ft

4c + 3 6c – 5

Activity Assess

B. One cross support in the center of the tower measures 4c + 3, and the other measures 6c − 5 . What is the length of each cross support?

The cross supports are diagonals of an isosceles trapezoid, so they are congruent.

Step 1 Find the value of c.

4c + 3 = 6c − 5

8 = 2c

4 = c

Step 2 Find the lengths of the diagonals.

4c + 3 = 4(4) + 3

= 19

6c − 5 = 6(4) − 5

= 19

Each cross support measures 19 ft in length.

Try It! 4. Given isosceles trapezoid MNOP where the given expressions represent the measures of the diagonals, what is the value of a?

N

M

P

O

a + 13

2a – 1

In a trapezoid, the midsegment is parallel to the bases, and the length of the midsegment is half the sum of the lengths of the bases.

PROOF: SEE LESSON 11-2.

If...

Then... ‾ XY ∥ ‾ AD , ‾ XY ∥ ‾ BC ,

and XY = 1 __ 2 (AD + BC)

B C

X Y

A D

THEOREM 8-6 Trapezoid Midsegment Theorem

EXAMPLE 4 CONTINUED

GENERALIZEWhy might this strategy work for isosceles trapezoids but not for trapezoids with noncongruent legs?


Activity Assess

EXAMPLE 5 Apply the Trapezoid Midsegment Theorem

Paxton makes trapezoidal handbags for her friends. She stiches decorative trim along the top, middle, and bottom on both sides of the handbags. How much trim does she need for three handbags? Explain.

9 in.

6 in.

2 in. 2 in.

2 in. 2 in.

The top and bottom sides of the handbag are the bases of a trapezoid. The left and right sides are the legs. Since the middle segment divides both legs in half, it is the midsegment of the trapezoid. The midsegment of a trapezoid is the segment that connects the midpoints of the legs.

Let x represent the length of the midsegment in inches.

Step 1 Find the value of x.

x = 1 __ 2 (6 + 9)

x = 7.5

The length of the midsegment is 7.5 in.

Step 2 Find the amount of trim that she needs.

First, find the amount for one side.

6 + 9 + 7.5 = 22.5

Then, multiply by 2 for the number of sides per handbag and by 3 for the number of handbags.

22.5 ∙ 2 ∙ 3 = 135

Paxton needs 135 inches of trim.

Formulate

Compute

Interpret

Apply the Trapezoid Midsegment Theorem with the base lengths 6 and 9.

Try It! 5. Given trapezoid JKLM, what is KL?

25 m

J M

K

X Y

L

XY45

APPLICATION


W Q Y

X

Z

56° 12 mm

Concept Summary Assess

CONCEPT SUMMARY Kites and Trapezoids

Kites

A kite is a quadrilateral with two pairs of adjacent sides congruent and no pairs of opposite sides congruent. Exactly one diagonal is a perpendicular bisector of the other.

WORDS

Quadrilateral ABCD is a kite.

A E C

B

D

‾ AC ⟂ ‾ BD

BE = ED

DIAGRAMS

Do You UNDERSTAND?

1. ESSENTIAL QUESTION How are

diagonals and angle measures related in kites and trapezoids?

2. Error Analysis What is Reagan’s error?

RQ

P S

✗By Theorem 8-5, PR =~ QS

3. Vocabulary If ‾ XY is the midsegment of a trapezoid, what must be true about point X and point Y?

4. Construct Arguments Emaan says every kite is composed of 4 right triangles. Is he correct? Explain.

Do You KNOW HOW?

For Exercises 5–7, use kite WXYZ to find the measures.

5. m∠XQY

6. m∠YZQ

7. WY

For Exercises 8–10, use trapezoid DEFG with EG = 21 ft and m∠DGF = 77 to find each measure.

8. ED

9. DF

10. m∠DEF

11. What is the length of ‾ PQ ?

33 cm

54 cm CB

A D

QP

Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The length of the midsegment is the average of the lengths of the two bases. A trapezoid with congruent legs is an isosceles trapezoid that has congruent base angles and congruent diagonals.

Quadrilateral RSTU is an isosceles trapezoid.

R

A B

S T

U

8 ft

77°

FE

D G

‾ SU ≅ ‾ TR

AB = 1 __ 2 (ST + RU)

‾ AB ∥ ‾ ST ∥ ‾ RU

m∠S = m∠T

m∠R = m∠U


PRACTICE & PROBLEM SOLVING

UNDERSTAND PRACTICE

Additional Exercises Available Online

Practice Tutorial

16. Given kite ABCD, in which AN = 4.6 m , what is AC ? SEE EXAMPLE 1

17. Given kite RSTU, what is m∠RUS ? SEE EXAMPLE 2

R T

S

U

27°

18. Write a two-column proof to show that the diagonals of an isosceles trapezoid are congruent. SEE EXAMPLES 3 AND 4

A D

B C

19. Given trapezoid MNPQ, what is m∠MNP ? SEE EXAMPLE 4

N

M

P

Q78°

20. Given trapezoid WXYZ, what is XY ? SEE EXAMPLE 5

W

D E

Z

X Y

35 ft

XY43

12. Construct Arguments Write a two-column proof to show that the diagonals of a kite are perpendicular.

W Y

X

Z

13. Mathematical Connections Write a paragraph proof to show that each pair of base angles in an isosceles trapezoid is congruent.

P S

Q R

14. Error Analysis What is Emery’s error?

B

H

D

A C

✗

BD is the perpendicularbisector of AC,so HC = 8 in. becauseAC = 16 in.

16 in.

15. Higher Order Thinking Given kite JKLM with diagonal ‾ KM , JK < JM, and KL < LM , prove that ∠JMK is congruent to ∠LMK.

J L

K

M

N4.6 m

A C

B

D


Scan for Multimedia


APPLY ASSESSMENT PRACTICE

Mixed Review Available Online

Practice Tutorial


21. Model With Mathematics Gregory plans to make a kite like the one shown. He has 1,700 square inches of plastic sheeting. Does Gregory have enough plastic to make the kite? Explain.

17 in.

30 in.

17 in.

39 in

.

22. Reason Coach Murphy uses the map to plan a 2-mile run for the track team. How many times will the team run the route shown?

Mu

lberry St.

46 yd92 yd

Elm St.

48 yd

Start

80 yd

80 yd

48 yd

Pecan St.

Lamar A

ve.

Brazos Ave.

23. Use Structure Abby builds a bench with the seat parallel to the ground. She bends pipe to make the leg and seat supports. At what angles should she bend the pipe? Explain.

102°

24. The of a kite are always .

25. SAT/ACT Given trapezoid ABCD, what is the length of ‾ XY ?

A D

CB

X Y

6s + 1

4s − 2

s

Ⓐ 3 3 __ 5 Ⓑ 4 2 __

3 Ⓒ 5 Ⓓ 11 Ⓔ 18

26. Performance Task Cindy is a member of a volunteer group that built the play structure shown.

4 ft

12 ft

Part A Cindy wants to add three more trapezoid boards evenly spaced between the bottom and top boards of the triangular frame. Based on the average lengths of the top and bottom boards shown, what will be the average lengths of each of the three additional boards? Explain.

Part B The three boards will be trapezoids that have the same height as the top and bottom boards. How can Cindy use the lengths of the bases of the bottom board to determine the lengths of the bases of the three new boards?

Part C What other measurements should Cindy find to be certain that the boards will fit exactly onto the triangular frame?

? ?


8-2€¦ · EXAMPLE 4 Solve Problems Involving Isosceles Trapezoids All horizontal beams of the high-voltage transmission tower are parallel to the ground. 14 ft 14 ft 10 ft 1 2 10

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