Match each geometric angle relationship with the correct calculated value of x. — Armed Services Vocational Aptitude Battery (ASVAB)

Match each geometric angle relationship with the correct calculated value of $x$ .

Angles $A$ and $B$ are complementary. $m\angle A = (3x - 15)^\circ$ and $m\angle B = (2x + 10)^\circ$ . $x = 19$
Angles $C$ and $D$ are vertical angles. $m\angle C = (5x - 30)^\circ$ and $m\angle D = (2x + 15)^\circ$ . $x = 15$
Angles $E$ and $F$ form a linear pair. $m\angle E = (5x + 20)^\circ$ and $m\angle F = (3x + 16)^\circ$ . $x = 18$
Angles $G$ and $H$ are alternate interior angles. $m\angle G = (3x + 45)^\circ$ and $m\angle H = (6x - 15)^\circ$ . $x = 20$

Answer

Match complementary to x = 19, vertical to x = 15, linear pair to x = 18, and alternate interior to x = 20.

Each geometric relationship dictates a distinct algebraic setup. Complementary angles sum to 90, vertical angles are set equal to each other, a linear pair sums to 180, and alternate interior angles are set equal to each other. Solving these linear equations gives the matching

x

values.

Step-by-Step Solution

Set up the equation for the complementary angles.

5x - 5 = 90

Complementary angles sum to

90^\circ

Solve for

x

in the complementary equation.

x = 19

Add 5 to both sides, then divide by 5.

Set up the equation for the vertical angles.

5x - 30 = 2x + 15

Vertical angles are congruent (equal in measure).

Solve for

x

in the vertical angles equation.

x = 15

Subtract 2x from both sides and add 30 to both sides, yielding 3x = 45.

Set up the equation for the linear pair.

8x + 36 = 180

Angles forming a linear pair are supplementary and sum to

180^\circ

Solve for

x

in the linear pair equation.

x = 18

Subtract 36 from both sides, then divide by 8.

Set up the equation for the alternate interior angles.

3x + 45 = 6x - 15

Alternate interior angles on parallel lines are congruent.

Solve for

x

in the alternate interior equation.

x = 20

Subtract 3x from both sides and add 15 to both sides, yielding 3x = 60.

Key Concept

Translating geometric angle relationships into algebraic equations to solve for unknown variables.

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