Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( h \)

Angle \( h \) and the \( 62^\circ \) angle are supplementary (form a linear pair), so \( h + 62^\circ = 180^\circ \).
\( h = 180^\circ - 62^\circ = 118^\circ \)

Step2: Find angle \( g \)

Angle \( g \) and the \( 62^\circ \) angle are vertical angles (or \( g \) and \( h \) are supplementary, but vertical angles are equal to the opposite angle). Wait, actually, angle \( g \) and the \( 62^\circ \) angle: since \( h \) and \( g \) are supplementary? No, wait, the straight line: \( h + g = 180^\circ \), but also, the angle opposite to \( 62^\circ \) (vertical angle) would be equal. Wait, no, let's correct. The angle adjacent to \( 62^\circ \) on the straight line is \( h \), then \( g \) is vertical to \( 62^\circ \)? Wait, no, the two lines: one vertical, one transversal. So the angle \( g \) and the \( 62^\circ \) angle: actually, \( h \) and \( 62^\circ \) are supplementary, so \( h = 118^\circ \), then \( g \) and \( 62^\circ \) are vertical angles? Wait, no, the transversal crosses the vertical line. So the angle \( g \) and the \( 62^\circ \) angle: let's see, the vertical line is straight, so \( h + 62^\circ = 180^\circ \) (linear pair), so \( h = 118^\circ \). Then \( g \) and \( 62^\circ \) are vertical angles? Wait, no, \( g \) and the \( 62^\circ \) angle: actually, \( g \) is equal to \( 62^\circ \) because they are vertical angles? Wait, no, the transversal creates vertical angles. Wait, the angle above the transversal and \( 62^\circ \) is \( h \), then below the transversal, the angle \( g \) and the angle opposite to \( 62^\circ \): maybe I made a mistake. Wait, let's start over.

Wait, the vertical line is a straight line, so the sum of angles on a straight line is \( 180^\circ \). So for the upper intersection: the angle \( h \) and \( 62^\circ \) are adjacent, so \( h + 62^\circ = 180^\circ \), so \( h = 180 - 62 = 118^\circ \). Then angle \( g \) is vertical to the \( 62^\circ \) angle? Wait, no, \( g \) is adjacent to \( h \), so \( g + h = 180^\circ \), so \( g = 180 - 118 = 62^\circ \). Alternatively, \( g \) and \( 62^\circ \) are vertical angles, so they are equal. So \( g = 62^\circ \).

Step3: Find angle \( m \)

Angle \( m \) and \( 97^\circ \) are supplementary (linear pair), so \( m + 97^\circ = 180^\circ \).
\( m = 180^\circ - 97^\circ = 83^\circ \)

Step4: Find angle \( k \)

Angle \( k \) and \( 97^\circ \) are vertical angles, so \( k = 97^\circ \). Alternatively, \( k \) and \( m \) are supplementary, so \( k = 180 - 83 = 97^\circ \).

Answer:

• \( h = 118^\circ \)
• \( g = 62^\circ \)
• \( m = 83^\circ \)
• \( k = 97^\circ \)