Question

use the figure. what is the value of x?
a. 38
b. 59
c. 83
d. 96

Explanation:

Step1: Find the measure of angle \( y \)

In a triangle, the sum of interior angles is \( 180^\circ \). For the left triangle, we have angles \( 46^\circ \), \( 51^\circ \), and \( y^\circ \). So, \( 46 + 51 + y = 180 \).
\[
46 + 51 + y = 180 \\
97 + y = 180 \\
y = 180 - 97 \\
y = 83
\]

Step2: Use vertical angles and triangle angle sum for the right triangle

Angle \( y \) and the vertical angle to it in the right triangle are equal (vertical angles are congruent). So the vertical angle is also \( 83^\circ \). Now, in the right triangle, we have angles \( 38^\circ \), \( x^\circ \), and \( 83^\circ \). The sum of angles in a triangle is \( 180^\circ \), so \( 38 + x + 83 = 180 \).
\[
38 + x + 83 = 180 \\
121 + x = 180 \\
x = 180 - 121 \\
x = 59
\]

Answer:

B. 59