Question

use the figure shown. if ( mangle 2 = 42 ) and ( mangle 6 = 126 ), what is ( mangle 4 )?
a. 12
b. 54
c. 96
d. 138

Explanation:

Step1: Find \( m\angle3 \)

Since \( \angle3 \) and \( \angle6 \) are supplementary (they form a linear pair), we use the supplementary angles property: \( m\angle3 + m\angle6 = 180^\circ \).
Given \( m\angle6 = 126^\circ \), then \( m\angle3 = 180^\circ - 126^\circ = 54^\circ \).

Step2: Find \( m\angle1 \)

In a triangle, the sum of interior angles is \( 180^\circ \). We know \( m\angle2 = 42^\circ \) and \( m\angle3 = 54^\circ \), so \( m\angle1 + m\angle2 + m\angle3 = 180^\circ \).
Substituting the known values: \( m\angle1 + 42^\circ + 54^\circ = 180^\circ \).
Simplify: \( m\angle1 + 96^\circ = 180^\circ \), so \( m\angle1 = 180^\circ - 96^\circ = 84^\circ \). Wait, no, wait—wait, actually, \( \angle1 \), \( \angle2 \), and \( \angle3 \) are the interior angles? Wait, no, looking at the diagram, \( \angle1 \) and \( \angle4 \) are supplementary, \( \angle2 \) and \( \angle5 \) are supplementary, and \( \angle3 \) and \( \angle6 \) are supplementary. Also, the sum of interior angles of a triangle: \( \angle1 + \angle2 + \angle3 = 180^\circ \). Wait, but maybe another approach: \( \angle6 \) is an exterior angle? Wait, no, \( \angle6 \) and \( \angle3 \) are adjacent, forming a linear pair. Then, in the triangle, angles \( \angle1 \), \( \angle2 \), \( \angle3 \) sum to 180. Then \( \angle1 = 180 - \angle2 - \angle3 = 180 - 42 - 54 = 84 \). Then \( \angle4 \) and \( \angle1 \) are supplementary (linear pair), so \( m\angle4 = 180 - m\angle1 = 180 - 84 = 96 \)? Wait, no, that contradicts. Wait, maybe I made a mistake. Wait, let's re-examine.

Wait, \( \angle6 = 126^\circ \), so \( \angle3 = 180 - 126 = 54^\circ \) (linear pair). Then, in the triangle, \( \angle1 + \angle2 + \angle3 = 180 \), so \( \angle1 = 180 - 42 - 54 = 84 \). Then \( \angle4 \) and \( \angle1 \) are supplementary, so \( \angle4 = 180 - 84 = 96 \)? But option C is 96. Wait, but let's check again.

Wait, maybe \( \angle6 \) is an exterior angle. Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non-adjacent interior angles. Wait, \( \angle6 \) is an exterior angle, so \( \angle6 = \angle1 + \angle2 \)? Wait, no, \( \angle6 \) is adjacent to \( \angle3 \), so \( \angle6 = 180 - \angle3 \), and \( \angle3 = 180 - (\angle1 + \angle2) \), so \( \angle6 = \angle1 + \angle2 \). Wait, that's the exterior angle theorem. So \( m\angle6 = m\angle1 + m\angle2 \). Then \( 126 = m\angle1 + 42 \), so \( m\angle1 = 126 - 42 = 84 \). Then \( \angle4 \) and \( \angle1 \) are supplementary, so \( m\angle4 = 180 - 84 = 96 \). So that's correct. So \( m\angle4 = 96^\circ \), which is option C.

Answer:

C. 96