In a geometric diagram, angle \alpha and angle \beta are complementary. Furthermore, angle… — Armed Services Vocational Aptitude Battery (ASVAB)

In a geometric diagram, angle $\alpha$ and angle $\beta$ are complementary. Furthermore, angle $\beta$ and angle $\gamma$ are supplementary. If the measure of angle $\alpha$ is represented by the expression $(2x - 5)^\circ$ and the measure of angle $\gamma$ is represented by $(5x - 2)^\circ$ , what is the measure of angle $\beta$ ?

$37^\circ$ Answer
B
$97^\circ$
C
$135^\circ$
D
$153^\circ$

Answer

37^\circ

The correct answer is found by recognizing that angle

\beta

can be expressed in two ways:

90^\circ - \alpha

and

180^\circ - \gamma

. Setting

90 - (2x - 5)

equal to

180 - (5x - 2)

allows us to solve for

x = 29

. Substituting

x

back into the first expression yields

95 - 2(29) = 37^\circ

Step-by-Step Solution

Express angle beta in terms of x using the complementary relationship.

\beta = 90 - (2x - 5) = 95 - 2x

Since alpha and beta are complementary, their sum is 90 degrees.

Express angle beta in terms of x using the supplementary relationship.

\beta = 180 - (5x - 2) = 182 - 5x

Since beta and gamma are supplementary, their sum is 180 degrees.

Set the two expressions for beta equal to each other and solve for x.

95 - 2x = 182 - 5x \Rightarrow 3x = 87 \Rightarrow x = 29

Both expressions represent the same angle, so they must be algebraically equivalent.

Substitute the value of x back into either expression to find the measure of angle beta.

\beta = 95 - 2(29) = 95 - 58 = 37^\circ

This provides the final numerical measure of the target angle.

Key Concept

Algebraic application of complementary and supplementary angle relationships

Estimated Time:2m 0s

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