During the alignment of a satellite's solar panels, sensors record various angular positio… — Armed Services Vocational Aptitude Battery (ASVAB)

During the alignment of a satellite's solar panels, sensors record various angular positions as algebraic expressions. Pair each geometric relationship described below with the exact numerical value of its unknown variable.

A linear pair of angles measuring $(3x + 10)^\circ$ and $(2x - 5)^\circ$ . Value of $x$ . $35$
Two adjacent angles forming a right angle, measuring $(4y - 12)^\circ$ and $(y + 12)^\circ$ . Value of $y$ . $18$
Vertical angles formed by intersecting support struts, measuring $(6z - 18)^\circ$ and $(4z + 14)^\circ$ . Value of $z$ . $16$
Corresponding angles along parallel deployment tracks, measuring $(7w - 3)^\circ$ and $(4w + 24)^\circ$ . Value of $w$ . $9$

Answer

The linear pair matches to

x = 35

. The right angle matches to

y = 18

. The vertical angles match to

z = 16

. The corresponding angles match to

w = 9

Each relationship dictates a specific mathematical equation based on geometric definitions: linear pairs sum to 180, right angle components sum to 90, vertical angles are equal, and corresponding angles are equal. Solving these multi-term algebraic equations yields the correct numerical values.

Step-by-Step Solution

Set up and solve an equation for the linear pair.

3x + 10 + 2x - 5 = 180 \Rightarrow 5x = 175 \Rightarrow x = 35

Angles in a linear pair are supplementary and their measures sum to 180 degrees.

Set up and solve an equation for the adjacent angles forming a right angle.

4y - 12 + y + 12 = 90 \Rightarrow 5y = 90 \Rightarrow y = 18

Angles that form a right angle are complementary and their measures sum to 90 degrees.

Set up and solve an equation for the vertical angles.

6z - 18 = 4z + 14 \Rightarrow 2z = 32 \Rightarrow z = 16

Vertical angles are congruent, meaning their measures are equal.

Set up and solve an equation for the corresponding angles.

7w - 3 = 4w + 24 \Rightarrow 3w = 27 \Rightarrow w = 9

Corresponding angles formed by parallel lines and a transversal are congruent.

Key Concept

Using geometric angle relationships (supplementary, complementary, vertical, and corresponding) to write and solve algebraic equations.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic