Question

the measure of \\( \angle 3 \\) is \\( 42 ^ { \circ } \\).
what is the measure of \\( \angle 1 \\) in degrees?

Explanation:

Step1: Find the measure of ∠5

Since ∠3 and ∠5 are vertical angles, they are equal.
So, \( \angle5=\angle3 = 42^{\circ}\)

Step2: Find the measure of ∠1

Since ∠1 and ∠5 are complementary (they form a right - angle with ∠2 = 90°), we use the formula \( \angle1+\angle5 = 90^{\circ}\)
Substitute \( \angle5 = 42^{\circ}\) into the formula: \( \angle1=90^{\circ}-\angle5\)
\( \angle1=90^{\circ}- 42^{\circ}=48^{\circ}\)

Answer:

\(48^{\circ}\)