Question

find the measure of the missing angles.
answer attempt 1 out of 2
g = ° h = °
k = ° m = °

Explanation:

Step1: Find \( h \)

\( h \) and \( 130^\circ \) are supplementary (form a linear pair), so \( h + 130^\circ = 180^\circ \).
\( h = 180^\circ - 130^\circ = 50^\circ \)

Step2: Find \( g \)

\( g \) and \( 130^\circ \) are vertical angles? No, \( g \) and \( h \) are adjacent? Wait, \( g \) and \( 130^\circ \)? Wait, \( g \) and \( h \) are adjacent? Wait, actually, \( g \) and \( 130^\circ \): Wait, no, \( h \) and \( 130^\circ \) are supplementary, and \( g \) and \( 130^\circ \)? Wait, no, \( g \) and \( h \) are adjacent? Wait, the vertical line and the slanted line intersect, so \( g \) and \( 130^\circ \)? Wait, no, \( h \) and \( 130^\circ \) are supplementary, so \( h = 50^\circ \), then \( g \) and \( h \) are supplementary? Wait, no, the vertical line is straight, so \( g + h = 180^\circ \)? Wait, no, the vertical line is a straight line, so the angle adjacent to \( 130^\circ \) is \( h \), then \( g \) is vertical to \( 130^\circ \)? Wait, no, let's re-examine. The slanted line intersects the vertical line, so \( h \) and \( 130^\circ \) are supplementary (linear pair), so \( h = 50^\circ \). Then \( g \) and \( 130^\circ \) are vertical angles? Wait, no, vertical angles are equal. Wait, no, the angle opposite to \( 130^\circ \) would be \( g \)? Wait, no, the vertical line and slanted line intersect, so the angle \( h \) and \( 130^\circ \) are linear pair, so \( h = 50^\circ \). Then \( g \) is equal to \( 130^\circ \)? Wait, no, that can't be. Wait, no, the vertical line is straight, so the sum of \( h \) and \( g \) is \( 180^\circ \)? Wait, no, the vertical line is a straight angle (180 degrees). So the slanted line cuts it into \( h \) and \( g \). Wait, but there's also a \( 130^\circ \) angle. Wait, maybe \( 130^\circ \) and \( g \) are vertical angles? No, vertical angles are opposite each other. Wait, maybe I made a mistake. Let's look at the horizontal and vertical lines. The horizontal and vertical lines are perpendicular? No, because there's an \( 88^\circ \) angle. Wait, the horizontal line and vertical line intersect at \( k \), so \( m \) and \( 88^\circ \) are supplementary? Wait, no, \( m \) and \( 88^\circ \) are adjacent, forming a linear pair? Wait, the horizontal line is straight, so \( m + 88^\circ = 180^\circ \)? No, wait, the vertical line and horizontal line intersect, so the angles around \( k \) are right angles? No, because \( m \) is \( 88^\circ \), so \( k \) is \( 90^\circ \)? Wait, no, let's start over.

First, the slanted line and vertical line intersect: \( h \) and \( 130^\circ \) are linear pair (supplementary), so \( h = 180 - 130 = 50^\circ \). Then \( g \) and \( 130^\circ \) are vertical angles? Wait, no, vertical angles are equal. Wait, no, the angle opposite to \( 130^\circ \) is \( g \)? Wait, no, when two lines intersect, vertical angles are equal. So the slanted line and vertical line intersect, so the angle \( 130^\circ \) and \( g \) are vertical angles? No, that would mean \( g = 130^\circ \), but then \( h \) and \( g \) would be supplementary (since they are adjacent on a straight line), so \( h + g = 180 \), so \( h = 50 \), which matches. So \( g = 130^\circ \)? Wait, no, that can't be. Wait, no, the vertical line is straight, so the sum of \( h \) and \( g \) is \( 180^\circ \). So if \( h = 50^\circ \), then \( g = 180 - 50 = 130^\circ \). Yes, that makes sense. So \( g = 130^\circ \), \( h = 50^\circ \).

Now, the horizontal and vertical lines intersect at \( k \). The angle \( m \) and \( 88^\circ \) are linear pair (supplementary), so \( m + 88^\circ = 180^\circ \), so \( m = 180 - 88 = 92^\circ \). Then \( k \) is a right angle? No, wait, the vertical and horizontal lines: if they are perpendicular, then \( k = 90^\circ \), but \( m = 88^\circ \), so that's not perpendicular. Wait, no, the vertical line and horizontal line intersect, so the angles around \( k \) are: \( m = 88^\circ \), the angle opposite to \( m \) is also \( 88^\circ \), and \( k \) is adjacent to \( m \), so \( k + m = 180^\circ \)? No, wait, the horizontal line is straight, so the angle \( m \) and the angle adjacent to it (let's say \( k \)): wait, \( k \) is the angle between vertical and horizontal lines? Wait, the vertical line is up and down, horizontal is left and right. So at point \( k \), the vertical line and horizontal line intersect, so the angles are: \( 88^\circ \), \( m \), \( k \), and the angle opposite to \( 88^\circ \). Wait, \( m \) and \( 88^\circ \) are adjacent, forming a linear pair, so \( m = 180 - 88 = 92^\circ \). Then \( k \) is the angle between vertical and horizontal lines, so if we consider the vertical line, the angle \( k \) and the angle opposite to \( 88^\circ \) (which is \( 88^\circ \)): wait, no, the vertical line is straight, so the sum of \( k \) and \( 88^\circ \) is \( 180^\circ \)? No, that's not right. Wait, the horizontal line is straight, so the angle \( m \) and \( 88^\circ \) are supplementary, so \( m = 92^\circ \). Then \( k \) is the angle between vertical and horizontal lines, so since the vertical and horizontal lines intersect, the angle \( k \) is \( 90^\circ \)? No, that's only if they are perpendicular. But here, \( m = 88^\circ \), so maybe the vertical and horizontal lines are not perpendicular. Wait, no, the vertical line is straight, so the angle between vertical and horizontal lines: if the horizontal line is horizontal, then the vertical line is vertical, so they should be perpendicular, meaning \( k = 90^\circ \), but then \( m + 88^\circ = 90^\circ \)? No, that can't be. Wait, I think I messed up the diagram. Let's re-express:

- The vertical line (up-down) and slanted line intersect: angle \( 130^\circ \) and \( h \) are linear pair, so \( h = 50^\circ \). Then \( g \) is vertical to \( 130^\circ \), so \( g = 130^\circ \) (vertical angles are equal).

- The vertical line (up-down) and horizontal line (left-right) intersect at \( k \). The angle between horizontal line (right) and vertical line (up) is \( 88^\circ \), so \( m \) is the angle between horizontal line (left) and vertical line (up), so \( m \) and \( 88^\circ \) are supplementary (linear pair), so \( m = 180 - 88 = 92^\circ \). Then \( k \) is the angle between vertical line (down) and horizontal line (left), which is vertical to \( 88^\circ \), so \( k = 90^\circ \)? No, wait, no. Wait, the vertical and horizontal lines intersect, so the angles are: \( 88^\circ \) (right-up), \( m \) (left-up), \( k \) (left-down), and \( 90^\circ - 88^\circ = 2^\circ \)? No, this is confusing. Wait, maybe \( k \) is a right angle? No, the problem says \( 88^\circ \), so maybe the vertical and horizontal lines are not perpendicular. Wait, no, the vertical line is straight, so the sum of \( k \) and \( m \) is \( 180^\circ \), and \( m \) and \( 88^\circ \) are supplementary, so \( m = 92^\circ \), then \( k = 180 - 92 = 88^\circ \)? No, that's not right. Wait, I think I made a mistake. Let's use the fact that vertical angles are equal and linear pairs are supplementary.

For \( h \) and \( 130^\circ \): linear pair, so \( h = 180 - 130 = 50^\circ \).

For \( g \) and \( 130^\circ \): vertical angles? No, \( g \) and \( h \) are linear pair? Wait, no, the vertical line is straight, so \( h + g = 180^\circ \), so \( g = 180 - 50 = 130^\circ \).

For \( m \) and \( 88^\circ \): linear pair, so \( m = 180 - 88 = 92^\circ \).

For \( k \): the vertical line and horizontal line intersect, so \( k \) is vertical to \( 88^\circ \)? No, \( k \) is the angle between vertical and horizontal lines, so if we consider the horizontal line is straight, then \( k + m = 180^\circ \), but \( m = 92^\circ \), so \( k = 88^\circ \)? No, that's not. Wait, no, the vertical line is straight, so the angle \( k \) and the angle opposite to \( 88^\circ \) (which is \( 88^\circ \)): wait, no, the vertical and horizontal lines intersect, so the angles are: \( 88^\circ \) (right-up), \( m = 92^\circ \) (left-up), \( k = 88^\circ \) (left-down), and \( 92^\circ \) (right-down). Wait, that makes sense because vertical angles are equal: \( 88^\circ \) and \( k \) are vertical angles, so \( k = 88^\circ \)? No, vertical angles are opposite each other. So \( 88^\circ \) (right-up) and \( k \) (left-down) are vertical angles? No, vertical angles are formed by two intersecting lines, so the opposite angles. So the two lines are vertical (up-down) and horizontal (left-right), intersecting at \( k \). So the angles are:

- Right-up: \( 88^\circ \)

- Left-up: \( m \)

- Left-down: \( k \)

- Right-down: \( 90^\circ - 88^\circ = 2^\circ \)? No, this is wrong. Wait, maybe the vertical and horizontal lines are perpendicular, so all angles are \( 90^\circ \), but there's an \( 88^\circ \) angle, so that means the horizontal line is not horizontal? No, the diagram shows a horizontal line (left-right) and vertical line (up-down) intersecting at \( k \), with an \( 88^\circ \) angle between horizontal (right) and vertical (up). So the angle between horizontal (right) and vertical (up) is \( 88^\circ \), so the angle between horizontal (left) and vertical (up) is \( 180 - 88 = 92^\circ \) (linear pair), so \( m = 92^\circ \). Then the angle between vertical (down) and horizontal (left) is \( 88^\circ \) (vertical to \( 88^\circ \)), so \( k = 88^\circ \)? No, vertical angles are equal, so \( 88^\circ \) (right-up) and \( k \) (left-down) are vertical angles, so \( k = 88^\circ \). Wait, no, vertical angles are opposite, so right-up and left-down are vertical angles, so they are equal. So \( k = 88^\circ \). But then the angle between vertical (down) and horizontal (right) would be \( 92^\circ \), which is vertical to \( m = 92^\circ \). That makes sense.

So summarizing:

- \( h \): linear pair with \( 130^\circ \), so \( h = 180 - 130 = 50^\circ \)

- \( g \): vertical to \( 130^\circ \), so \( g = 130^\circ \) (wait, no, \( h \) and \( g \) are linear pair, so \( g = 180 - h = 180 - 50 = 130^\circ \))

- \( m \): linear pair with \( 88^\circ \), so \( m = 180 - 88 = 92^\circ \)

- \( k \): vertical to \( 88^\circ \)? No, \( k \) is adjacent to \( m \), so \( k + m = 180^\circ \)? No, the vertical line is straight, so \( k + m = 180^\circ \)? Wait, no, the vertical line is up-down, so the angle \( k \) is at the bottom, between vertical (down) and horizontal (left). The angle at the top, between vertical (up) and horizontal (left) is \( m = 92^\circ \), so the angle at the bottom (left-down) is \( k = 90^\circ \)? No, I'm getting confused. Let's use the fact that when two lines intersect, vertical angles are equal, and linear pairs are supplementary.

For the slanted line and vertical line:

- \( 130^\circ \) and \( h \): linear pair, so \( h = 50^\circ \)

- \( 130^\circ \) and \( g \): vertical angles? No, \( h \) and \( g \) are linear pair, so \( g = 130^\circ \)

For the horizontal and vertical line:

- \( 88^\circ \) and \( m \): linear pair, so \( m = 92^\circ \)

- \( 88^\circ \) and \( k \): vertical angles? No, \( m \) and \( k \) are linear pair, so \( k = 88^\circ \)? Wait, no, the vertical line is straight, so the angle at the top (left-up) is \( m = 92^\circ \), and the angle at the bottom (left-down) is \( k \), so \( m + k = 180^\circ \) (since they are on a straight line, the vertical line), so \( k = 180 - 92 = 88^\circ \). Yes, that makes sense. So \( k = 88^\circ \), \( m = 92^\circ \), \( g = 130^\circ \), \( h = 50^\circ \).

Step3: Verify \( k \)

Since \( m = 92^\circ \) and \( k \) are on the vertical line (straight line), \( m + k = 180^\circ \), so \( k = 180 - 92 = 88^\circ \). Correct.

Step4: Verify \( g \)

\( h = 50^\circ \) and \( g \) are on the vertical line (straight line), so \( h + g = 180^\circ \), so \( g = 180 - 50 = 130^\circ \). Correct.

Answer:

\( g = 130^\circ \), \( h = 50^\circ \), \( k = 88^\circ \), \( m = 92^\circ \)

So filling in the boxes:

\( g = \boxed{130}^\circ \)

\( h = \boxed{50}^\circ \)

\( k = \boxed{88}^\circ \)

\( m = \boxed{92}^\circ \)