Basic Mathematics for College Students

In the figure above, we can view as a transversal cutting the parallel lines

and. Since and are interior angles on the same side of a transversal, they are

supplementary. Similarly, is a transversal cutting the parallel lines and. Since

and are interior angles on the same side of a transversal, they are also

supplementary. These observations lead us to the conclusion that there are always two

pairs of supplementary angles in any trapezoid.

B C

DC

·

AB

·

BC

·

DC A D

·

AB

·

AD

·

770 Chapter 9 An Introduction to Geometry

A B

D C

Upper base

Lower base

Lower base

angles

Upper base

angles

Leg

Leg

Self Check 3 EXAMPLE (^3) Refer to trapezoid KLMNbelow, with KLNM. Find and .x y
Refer to trapezoid below,
with. Find and .HIKJ x y

HIJK

K L

N M

x

82° y

121°

StrategyWe will use the interior angles property twice to write two equations

that mathematically model the situation.

WHYWe can then solve the equations to find and.

Solution and are interior angles on the same side of transversal that

cuts the parallel lines segments and. Similarly, and are interior

angles on the same side of transversal that cuts the parallel lines segments

and. Recall that if two parallel lines are cut by a transversal, interior angles on

the same side of the transversal are supplementary. We can use this fact twice—

once to find and a second time to find.

The sum of the measures of supplementary

angles is 180°.

Substitute xfor m(K) and 82° for m(N).

To isolate x,subtract 82° from both sides.

Thus, is 98°.

The sum of the measures of supplementary

angles is 180°.

Substitute 121° for m(L) and y for m(M).

To isolate y,subtract 121° from both sides.

Thus, is 59°.y

y59°

121 °y180°

m(L)m(M)180°

x

x98°

x 82 °180°

m(K)m(N)180°

x y

NM

LM KL

·

KL NM L M

KN

·

K N

x y

Now TryProblem 29

HI

K J

y

x

93°

79°

If the nonparallel sides of a trapezoid are the

same length, it is called an isosceles trapezoid.

The figure on the right shows isosceles trapezoid

with. In an isosceles trapezoid,

both pairs of base angles are congruent.In the

figure,D  Eand .G  F

DEFG DG  EF

GF

DE

Trapezoid

Isosceles trapezoid

17

18

7

0

10

 82

98

18

7

0

10

 121

59