A trapezoid base legs Base of a trapezoid are two consecutive … · 2018-07-05 · Holt McDougal...

Holt McDougal Geometry

6-6 Properties of Kites and Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base.



If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.



Theorem 6-6-5 is a biconditional statement. So it is true both “forward” and “backward.”

Reading Math



Isos. trap. s base

Example 3A: Using Properties of Isosceles

Trapezoids

Find mA.

Same-Side Int. s Thm.

Substitute 100 for mC.

Subtract 100 from both sides.

Def. of s

Substitute 80 for mB

mC + mB = 180°

100 + mB = 180

mB = 80°

A B

mA = mB

mA = 80°



Example 3B: Using Properties of Isosceles

Trapezoids

KB = 21.9 and MF = 32.7.

Find FB.

Isos. trap. s base

Def. of segs.

Substitute 32.7 for FM.

Seg. Add. Post.

Substitute 21.9 for KB and 32.7 for KJ.

Subtract 21.9 from both sides.

KJ = FM

KJ = 32.7

KB + BJ = KJ

21.9 + BJ = 32.7

BJ = 10.8



Isos. trap. s base

Same-Side Int. s Thm.

Def. of s

Substitute 49 for mE.

mF + mE = 180°

E H

mE = mH

mF = 131°

mF + 49° = 180°

Simplify.

Check It Out! Example 3a

Find mF.



Check It Out! Example 3b

JN = 10.6, and NL = 14.8. Find KM.

Def. of segs.

Segment Add Postulate

Substitute.

Substitute and simplify.

Isos. trap. s base

KM = JL

JL = JN + NL

KM = JN + NL

KM = 10.6 + 14.8 = 25.4



Example 4A: Applying Conditions for Isosceles

Trapezoids

Find the value of a so that PQRS is isosceles.

a = 9 or a = –9

Trap. with pair base s isosc. trap.

Def. of s

Substitute 2a2 – 54 for mS and a2 + 27 for mP.

Subtract a2 from both sides and add 54 to both sides.

Find the square root of both sides.

S P

mS = mP

2a2 – 54 = a2 + 27

a2 = 81



Example 4B: Applying Conditions for Isosceles

Trapezoids

AD = 12x – 11, and BC = 9x – 2. Find the value of x so that ABCD is isosceles.

Diags. isosc. trap.

Def. of segs.

Substitute 12x – 11 for AD and 9x – 2 for BC.

Subtract 9x from both sides and add 11 to both sides.

Divide both sides by 3.

AD = BC

12x – 11 = 9x – 2

3x = 9

x = 3



Check It Out! Example 4

Find the value of x so that PQST is isosceles.

Subtract 2x2 and add 13 to both sides.

x = 4 or x = –4 Divide by 2 and simplify.

Trap. with pair base s isosc. trap.

Q S

Def. of s

Substitute 2x2 + 19 for mQ and 4x2 – 13 for mS.

mQ = mS

2x2 + 19 = 4x2 – 13

32 = 2x2



The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it.



Example 5: Finding Lengths Using Midsegments

Find EF.

Trap. Midsegment Thm.

Substitute the given values.

Solve. EF = 10.75



Check It Out! Example 5

Find EH.

Trap. Midsegment Thm.

Substitute the given values.

Simplify.

Multiply both sides by 2. 33 = 25 + EH

Subtract 25 from both sides. 8 = EH

1 16.5 = (25 + EH) 2



Lesson Quiz: Part I

In kite HJKL, mKLP = 72°, and mHJP = 49.5°. Find each measure. 1. mLHJ 2. mPKL

.

81° 18°

A trapezoid base legs Base of a trapezoid are two consecutive … · 2018-07-05 · Holt McDougal...

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