CK-12 Geometry Concepts

6.6. Trapezoids http://www.ck12.org

Investigation: Midsegment Property

Tools Needed: graph paper, pencil, ruler

1. Draw a trapezoid on your graph paper with verticesA(− 1 , 5 ),B( 2 , 5 ),C( 6 , 1 )andD(− 3 , 1 ). Notice this is
   NOT an isosceles trapezoid.


2. Find the midpoint of the non-parallel sides either by using slopes or the midpoint formula. Label themEand
   F. Connect the midpoints to create the midsegment.


3. Find the lengths ofAB,EF, andCD. Can you write a formula to find the midsegment?

Midsegment Theorem:The length of the midsegment of a trapezoid is the average of the lengths of the bases, or
EF=AB+ 2 CD.

Example A

Look at trapezoidT RAPbelow. What ism^6 A?

T RAPis an isosceles trapezoid. So,m^6 R= 115 ◦. To findm^6 A, set up an equation.

115 ◦+ 115 ◦+m^6 A+m^6 P= 360 ◦

230 ◦+ 2 m^6 A= 360 ◦→m^6 A=m^6 P

2 m^6 A= 130 ◦

m^6 A= 65 ◦

Notice thatm^6 R+m^6 A= 115 ◦+ 65 ◦= 180 ◦. These angles will always be supplementary because of the Consecutive
Interior Angles Theorem. Therefore, the two angles along the same leg (or non-parallel side) are always going to be
supplementary. Only in isosceles trapezoids will opposite angles also be supplementary.

Example B

Write a two-column proof.

Given: TrapezoidZOIDand parallelogramZOIM

(^6) D∼= (^6) I
Prove:ZD∼=OI
TABLE6.8:
Statement Reason

1. TrapezoidZOIDand parallelogramZOIM,^6 D∼=^6 I Given
   2.ZM∼=OI Opposite Sides Theorem
   3.^6 I∼=^6 ZMD Corresponding Angles Postulate
   4.^6 D∼=^6 ZMD Transitive PoC
   5.ZM∼=ZD Base Angles Converse
   6.ZD∼=OI Transitive PoC