In isosceles trapezoid ABCD, side AB is parallel to side CD. If the measure of \angle A is… — ACT

In isosceles trapezoid $ABCD$ , side $AB$ is parallel to side $CD$ . If the measure of $\angle A$ is $(3x + 10)^\circ$ and the measure of $\angle C$ is $(5x - 30)^\circ$ , what is the measure of $\angle B$ ?

A
$70^\circ$
B
$75^\circ$
$85^\circ$ Answer
D
$95^\circ$
E
$110^\circ$

Answer

The measure of

\angle B

85^\circ

The answer of

85^\circ

is correct because parallel sides

AB

and

CD

imply that consecutive interior angles

\angle A

and

\angle C

add up to

180^\circ

. Solving

(3x + 10) + (5x - 30) = 180

yields

x = 25

. Substituting

x = 25

into the expression for

\angle A

gives

3(25) + 10 = 85^\circ

. Because

ABCD

is an isosceles trapezoid, the base angles

\angle A

and

\angle B

adjacent to base

AB

are congruent, so

\angle B = 85^\circ

Step-by-Step Solution

Identify the relationship between

\angle A

and

\angle C

Since

AB \parallel CD

, angles

\angle A

and

\angle C

are consecutive interior angles along transversal

AC

(or leg

AD

), which means they are supplementary:

\angle A + \angle C = 180^\circ

Parallel lines cut by a transversal form supplementary consecutive interior angles.

Set up and solve the algebraic equation for

x

(3x + 10) + (5x - 30) = 180 \implies 8x - 20 = 180 \implies 8x = 200 \implies x = 25

Combine like terms and solve for

x

Calculate the measure of

\angle A

\angle A = 3(25) + 10 = 75 + 10 = 85^\circ

Substitute

x = 25

back into the expression for

\angle A

Determine the measure of

\angle B

using isosceles trapezoid properties

\angle B = \angle A = 85^\circ

In an isosceles trapezoid with

AB \parallel CD

, base angles along the same parallel base are congruent.

Key Concept

Properties of Isosceles Trapezoids and Consecutive Interior Angles

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