Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( h \)

Angles on a straight line sum to \( 180^\circ \). So, \( h + 131^\circ = 180^\circ \).
\( h = 180^\circ - 131^\circ = 49^\circ \)

Step2: Find angle \( g \)

Vertical angles are equal, or angles on a straight line. \( g \) and \( 131^\circ \) are vertical angles? Wait, no, \( h \) and \( g \): Wait, \( h + 131^\circ = 180^\circ \), and \( g \) is vertical to \( 131^\circ \)? Wait, no, let's see: The straight line, so \( h + 131^\circ = 180^\circ \), and \( g \) is adjacent to \( h \)? Wait, no, actually, \( g \) and \( 131^\circ \) are vertical angles? Wait, no, the two lines intersect, so vertical angles. Wait, the straight line (the vertical line) and the slanted line intersect, so \( h \) and \( g \) are adjacent, and \( h + 131^\circ = 180^\circ \), so \( g = 131^\circ \) (vertical angle with \( 131^\circ \))? Wait, no, let's correct. When two lines intersect, adjacent angles are supplementary (sum to \( 180^\circ \)), and vertical angles are equal. So the slanted line and the vertical line intersect, creating angle \( h \), \( 131^\circ \), \( g \), and another angle. So \( h + 131^\circ = 180^\circ \) (supplementary), so \( h = 49^\circ \). Then \( g \) is vertical to \( 131^\circ \)? No, \( g \) is adjacent to \( h \), so \( g + h = 180^\circ \)? Wait, no, the vertical line is straight, so the sum of angles on a straight line is \( 180^\circ \). So the angles around the intersection: \( h \), \( 131^\circ \), and the other two angles. Wait, maybe better: \( h \) and \( 131^\circ \) are supplementary (they form a linear pair), so \( h = 180 - 131 = 49^\circ \). Then \( g \) is equal to \( 131^\circ \) because they are vertical angles? Wait, no, \( g \) is adjacent to \( h \), so \( g + h = 180^\circ \)? Wait, no, the vertical line is a straight line, so the angles on one side of the slanted line: \( h \) and \( 131^\circ \) sum to \( 180^\circ \), so \( h = 49^\circ \). Then the angle opposite to \( 131^\circ \) (which is \( g \))? Wait, no, when two lines intersect, vertical angles are equal. So the angle opposite to \( 131^\circ \) is \( g \)? Wait, no, the slanted line and vertical line intersect, so the four angles: \( h \), \( 131^\circ \), \( g \), and the angle opposite to \( h \). So \( h \) and \( 131^\circ \) are supplementary, \( g \) and \( 131^\circ \) are vertical? No, \( g \) is adjacent to \( h \), so \( g = 131^\circ \) (because \( h + g = 180^\circ \)? No, \( h = 49^\circ \), so \( g = 180 - 49 = 131^\circ \), which is equal to the given \( 131^\circ \) (vertical angle). Okay, that makes sense.

Step3: Find angle \( m \)

Angles on a straight line sum to \( 180^\circ \). So \( m + 54^\circ = 180^\circ \).
\( m = 180^\circ - 54^\circ = 126^\circ \)

Step4: Find angle \( k \)

Vertical angles are equal, so \( k = 54^\circ \) (vertical angle with \( 54^\circ \)), or using supplementary: \( k + m = 180^\circ \), so \( k = 180 - 126 = 54^\circ \).

Answer:

\( h = 49^\circ \), \( g = 131^\circ \), \( m = 126^\circ \), \( k = 54^\circ \)