Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( h \)

Angles on a straight line sum to \( 180^\circ \). So \( h + 71^\circ = 180^\circ \), thus \( h = 180^\circ - 71^\circ = 109^\circ \).

Step2: Find angle \( g \)

Vertical angles are equal, or \( g \) and \( 71^\circ \) are vertical? Wait, no, \( g \) and the \( 71^\circ \) angle? Wait, actually, \( g \) and the \( 71^\circ \) angle: Wait, no, \( h \) and \( g \) with the \( 71^\circ \). Wait, \( g \) is vertical to the \( 71^\circ \)? No, wait, \( g \) and the \( 71^\circ \) angle: Wait, the angle \( g \) and the \( 71^\circ \) angle are vertical? Wait, no, let's see. The horizontal line and the vertical line intersect, so \( g \) is equal to \( 71^\circ \)? Wait, no, \( h \) and \( g \) are adjacent, so \( h + g + 71^\circ \)? No, wait, the two lines intersect, so vertical angles. Wait, the horizontal line and the vertical line: the angle \( 71^\circ \) and angle \( g \) are vertical? Wait, no, the angle \( h \) and the angle opposite to \( 71^\circ \)? Wait, maybe I made a mistake. Wait, the angle \( h \) and the \( 71^\circ \) angle are supplementary (since they are on a straight line), so \( h = 180 - 71 = 109^\circ \). Then angle \( g \) is equal to \( 71^\circ \) because they are vertical angles? Wait, no, vertical angles: when two lines intersect, opposite angles are equal. So the angle \( g \) and the \( 71^\circ \) angle are vertical? Wait, the horizontal line and the vertical line: the angle between the vertical line and the horizontal line (the \( 71^\circ \)) and the angle \( g \) (between the vertical line and the other horizontal line segment) – yes, they are vertical angles, so \( g = 71^\circ \).

Step3: Find angle \( k \)

Angles on a straight line sum to \( 180^\circ \). So \( k + 151^\circ = 180^\circ \), thus \( k = 180^\circ - 151^\circ = 29^\circ \).

Step4: Find angle \( m \)

Angle \( m \) and the \( 151^\circ \) angle: are they supplementary? Wait, no, angle \( m \) and angle \( k \): Wait, angle \( m \) and the \( 151^\circ \) angle – no, angle \( m \) and angle \( k \): Wait, the two lines (the slanted one and the vertical one) intersect, so vertical angles. Wait, angle \( m \) and the angle adjacent to \( 151^\circ \): Wait, \( k = 29^\circ \), and angle \( m \) is equal to \( k \) because they are vertical angles? Wait, no, the slanted line and the vertical line intersect, so angle \( m \) and the angle \( k \) – wait, no, the angle \( m \) and the \( 151^\circ \) angle: Wait, \( m + 151^\circ = 180^\circ \)? No, \( k = 29^\circ \), and angle \( m \) is equal to \( k \) because they are vertical angles? Wait, no, let's see: the vertical line and the slanted line intersect, so the angle \( m \) and the angle \( k \) – wait, \( k \) is \( 29^\circ \), and angle \( m \) is equal to \( k \)? Wait, no, maybe angle \( m \) is equal to \( 29^\circ \) because they are vertical angles? Wait, no, let's re-examine. The angle \( 151^\circ \) and angle \( m \): are they supplementary? \( 151 + m = 180 \)? No, \( 151 + 29 = 180 \), so \( m = 29^\circ \)? Wait, no, angle \( m \) and angle \( k \) are vertical angles, so \( m = k = 29^\circ \). Alternatively, angle \( m \) and the \( 151^\circ \) angle: \( m + 151 = 180 \)? No, \( 151 + 29 = 180 \), so \( m = 29^\circ \). Wait, maybe I confused. Let's do step by step.

First, angle \( h \): adjacent to \( 71^\circ \) on a straight line, so \( h = 180 - 71 = 109^\circ \).

Angle \( g \): vertical to \( 71^\circ \), so \( g = 71^\circ \).

Angle \( k \): adjacent to \( 151^\circ \) on a straight line, so \( k = 180 - 151 = 29^\circ \).

Angle \( m \): vertical to \( k \), so \( m = 29^\circ \), or adjacent to \( 151^\circ \)? Wait, no, the slanted line and the vertical line: the angle \( m \) and the angle \( k \) are vertical angles, so \( m = k = 29^\circ \). Alternatively, angle \( m \) and the \( 151^\circ \) angle: \( m + 151 = 180 \)? No, \( 151 + 29 = 180 \), so \( m = 29^\circ \). So that's correct.

Answer:

• \( h = 109^\circ \)
• \( g = 71^\circ \)
• \( k = 29^\circ \)
• \( m = 29^\circ \)