Translate the following geometric relationships into algebraic equations and perform the f… — Armed Services Vocational Aptitude Battery (ASVAB)

Translate the following geometric relationships into algebraic equations and perform the first step of simplification. Pair each geometric description on the left with its corresponding simplified equation on the right.

An angle measuring $(3x)^\circ$ and its complement measuring $30^\circ$ . $3x = 60$
An angle measuring $(3x)^\circ$ and its supplement measuring $30^\circ$ . $3x = 150$
Two intersecting lines forming vertical angles of $(3x)^\circ$ and $30^\circ$ . $x = 10$
Two adjacent angles on a straight line measuring $(3x)^\circ$ and $(2x + 30)^\circ$ . $5x = 150$

Answer

Match the complementary angles with

3x = 60

; the supplementary angles with

3x = 150

; the vertical angles with

x = 10

; and the adjacent angles on a straight line with

5x = 150

Each geometric term dictates a specific mathematical equality: 'complement' means the sum is

90^\circ

, 'supplement' and 'straight line' mean the sum is

180^\circ

, and 'vertical angles' means the measures are equal to each other. Applying these definitions and performing basic algebraic simplification yields the correct matches.

Step-by-Step Solution

Formulate and simplify the equation for the complementary angles.

The setup is

3x + 30 = 90

, which simplifies to

3x = 60

Complementary angles always add up to

90^\circ

Formulate and simplify the equation for the supplementary angles.

The setup is

3x + 30 = 180

, which simplifies to

3x = 150

Supplementary angles always add up to

180^\circ

Formulate and simplify the equation for the vertical angles.

The setup is

3x = 30

, which simplifies to

x = 10

Vertical angles (angles opposite each other at an intersection) are geometrically equal.

Formulate and simplify the equation for the adjacent angles on a straight line.

The setup is

3x + (2x + 30) = 180

. Combining the

x

terms gives

5x + 30 = 180

, and subtracting 30 gives

5x = 150

Angles that form a linear pair on a straight line must sum to

180^\circ

Key Concept

Translating Geometric Angle Relationships into Algebraic Equations

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