In the plane, line L_1 is parallel to line L_2. A transversal line T intersects L_1 and L_… — GRE General Test

In the plane, line $L_1$ is parallel to line $L_2$ . A transversal line $T$ intersects $L_1$ and $L_2$ . One of the acute angles formed at the intersection of $L_1$ and $T$ measures $(4x - 10)^\circ$ , and an alternate interior angle on $L_2$ measures $(2x + 30)^\circ$ . What is the degree measure of one of the obtuse angles formed at the intersection of $L_1$ and $T$ ?

A
$20^\circ$
B
$50^\circ$
C
$70^\circ$
$110^\circ$ Answer
E
$140^\circ$

Answer

The degree measure of the obtuse angle is

110^\circ

Since lines

L_1

and

L_2

are parallel, alternate interior angles are equal in measure. Setting

(4x - 10)^\circ = (2x + 30)^\circ

gives

2x = 40

, so

x = 20

. Substituting

x = 20

into

4x - 10

gives an acute angle of

70^\circ

. Because angles on a straight line are supplementary, the obtuse angle measures

180^\circ - 70^\circ = 110^\circ

, which corresponds to the correct choice.

Step-by-Step Solution

Set alternate interior angles equal to solve for

x

4x - 10 = 2x + 30 \implies 2x = 40 \implies x = 20

When two parallel lines are cut by a transversal, alternate interior angles are congruent.

Substitute

x = 20

into the expression for the acute angle.

Acute angle

= 4(20) - 10 = 70^\circ

This yields the degree measure of the acute angle formed by the intersection.

Calculate the measure of the supplementary obtuse angle.

Obtuse angle

= 180^\circ - 70^\circ = 110^\circ

Adjacent angles along a straight line are supplementary and sum to

180^\circ

Key Concept

Alternate Interior Angles and Supplementary Angles

Estimated Time:50s

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