In a geometric plane, straight lines L_1 and L_2 intersect at point P. The measure of the … — GRE General Test

In a geometric plane, straight lines $L_1$ and $L_2$ intersect at point $P$ . The measure of the obtuse angle formed by the intersection of $L_1$ and $L_2$ is $(5x - 10)^\circ$ , and the measure of an adjacent acute angle is $(2x + 15)^\circ$ . Ray $PQ$ originates from point $P$ , is perpendicular to line $L_1$ , and lies entirely within the interior of the $(5x - 10)^\circ$ obtuse angle. What is the measure, in degrees, of the angle formed between ray $PQ$ and line $L_2$ ?

$25^\circ$ Answer
B
$35^\circ$
C
$50^\circ$
D
$65^\circ$
E
$155^\circ$

Answer

25^\circ

Two intersecting lines form adjacent angles that sum to

180^\circ

. Solving

(5x - 10) + (2x + 15) = 180

gives

7x + 5 = 180

, so

x = 25

. The obtuse angle measure is

5(25) - 10 = 115^\circ

. Since ray

PQ

is perpendicular to line

L_1

, it forms a

90^\circ

angle with line

L_1

. The remaining angle between ray

PQ

and line

L_2

within the obtuse angle region is

115^\circ - 90^\circ = 25^\circ

Step-by-Step Solution

Set up an algebraic equation using the supplementary angle relationship.

(5x - 10) + (2x + 15) = 180

Adjacent angles formed by two intersecting straight lines lie on a straight line and are supplementary, summing to

180^\circ

Solve for the variable

x

7x + 5 = 180 \implies 7x = 175 \implies x = 25

Combining like terms simplifies the linear equation.

Calculate the degree measure of the obtuse angle.

Obtuse angle =

5(25) - 10 = 125 - 10 = 115^\circ

Substitute

x = 25

back into the expression

(5x - 10)^\circ

Calculate the angle between ray

PQ

and line

L_2

Angle =

115^\circ - 90^\circ = 25^\circ

Ray

PQ

is perpendicular to line

L_1

(

90^\circ

) and lies inside the

115^\circ

angle, dividing the obtuse angle into a

90^\circ

portion and the remaining angle adjacent to line

L_2

Key Concept

Supplementary angles on a straight line and angle subtraction with perpendicular rays

Estimated Time:1m 30s

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