Take note Theorem 6-20

Theorem

If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

If …

ABCD is an isosceles trapezoid

Then …

A C ¯ ≅ B D ¯ eh c bar , approximately equal to , b d bar

You will prove Theorem 6-20 in Exercise 54.

Take note Theorem 6-21: Trapezoid Midsegment Theorem

Theorem

If a quadrilateral is a trapezoid, then

1. the midsegment is parallel to the bases, and
2. the length of the midsegment is half the sum of the lengths of the bases.

If …

TRAP is a trapezoid with midsegment M N ¯ m n bar

Then …

1. M N ¯ ∥ T P ¯ , M N ¯ ∥ R A ¯ , m n bar , parallel to , t p bar , comma , m n bar , parallel to , r eh bar , comma and
2. M N = 1 2 ( T P + R A ) m n equals , 1 half , open t p plus r eh close

You will prove Theorem 6-21 in Lesson 6-9.

Problem 3 Using the Midsegment of a Trapezoid

Algebra Q R ¯ q r bar is the midsegment of trapezoid LMNP. What is x?

Q R = 1 2 ( L M + P N ) Trapezoid Midsegment Theorem x + 2 = 1 2 [ ( 4 x − 10 ) + 8 ] Substitute. x + 2 = 1 2 ( 4 x − 2 ) Simplify. x + 2 = 2 x − 1 Distributive Property 3 = x Subtract  x  and add  1  to each side. table with 5 rows and 3 columns , row1 column 1 , q r , column 2 equals , 1 half , open l m plus p n close , column 3 cap trapezoid cap midsegment cap theorem , row2 column 1 , x plus 2 , column 2 equals , 1 half , left bracket open 4 x minus 10 close plus 8 right bracket , column 3 cap substitute. , row3 column 1 , x plus 2 , column 2 equals , 1 half , open 4 x minus 2 close , column 3 cap simplify. , row4 column 1 , x plus 2 , column 2 equals 2 x minus 1 , column 3 cap distributive cap property , row5 column 1 , 3 , column 2 equals x , column 3 cap subtract . x . and add . 1 . to each side. , end table