find the measure of ∠gkh.
find the measure of ∠mkh.

Question

Accepted Answer

### First Sub - question: Find the measure of $\angle GKH$
# Explanation:
## Step 1: Recall the property of supplementary angles
Angles on a straight line (supplementary angles) add up to $180^{\circ}$. We know that $\angle JK G = 130^{\circ}$ and $\angle JK G$ and $\angle GKH$ are supplementary (they form a linear pair).
## Step 2: Calculate the measure of $\angle GKH$
Let the measure of $\angle GKH$ be $x$. Then we have the equation $130^{\circ}+x = 180^{\circ}$. Solving for $x$, we subtract $130^{\circ}$ from both sides of the equation: $x=180^{\circ}- 130^{\circ}=50^{\circ}$.
# Answer:
The measure of $\angle GKH$ is $50^{\circ}$.

### Second Sub - question: Find the measure of $\angle MKH$
# Explanation:
## Step 1: Recall the property of vertical angles or supplementary angles (also, note the linear pair and vertical angle relationships)
First, we know that $\angle JK G = 155^{\circ}$, and $\angle JK M$ and $\angle JK G$ are supplementary? Wait, no. Actually, $\angle MK H$ and $\angle JK G$ are vertical angles? Wait, no. Let's look at the lines. $JH$ is a straight line and $MG$ is a straight line intersecting at $K$. So $\angle MK H$ and $\angle JK G$ are supplementary? Wait, no. Wait, $\angle JK M$ and $\angle GKH$? Wait, no. Let's use the fact that $\angle MK H$ and $\angle JK M$ are supplementary, and $\angle JK M$ and $\angle JK G$ (which is $155^{\circ}$) are vertical angles? Wait, no. Wait, $\angle JK G$ and $\angle MK H$: since $JH$ and $MG$ are straight lines, $\angle MK H$ and $\angle JK G$ are supplementary? Wait, no. Wait, the sum of angles on a straight line is $180^{\circ}$. Let's see, $\angle JK M$ and $\angle MK H$ form a linear pair. And $\angle JK M$ is equal to $\angle GKH$? No, wait, $\angle JK G$ is $155^{\circ}$, so $\angle GKH$ would be $180 - 155=25^{\circ}$? No, wait, no. Wait, for the second diagram, $JH$ is a straight line, $MG$ is a straight line intersecting at $K$. So $\angle MK H$ and $\angle JK G$ are vertical angles? No, $\angle MK H$ and $\angle JK M$ are supplementary. And $\angle JK M$ is equal to $\angle GKH$? Wait, no. Wait, let's start over. The angle $\angle JK G = 155^{\circ}$, and $\angle MK H$ and $\angle JK G$: since $JH$ is a straight line, and $MG$ is a straight line, the angle $\angle MK H$ and $\angle JK M$ are supplementary. And $\angle JK M$ is vertical to $\angle GKH$? No, $\angle JK M$ and $\angle GKH$ are vertical angles? Wait, no. Wait, $\angle JK G$ and $\angle MK H$: actually, $\angle MK H$ is equal to $\angle JK G$? No, that can't be. Wait, no, let's use the linear pair. The angle adjacent to $\angle JK G$ (which is $155^{\circ}$) on the straight line $JH$ is $180 - 155 = 25^{\circ}$, but that's $\angle GKH$. Wait, no, for the second diagram, we need to find $\angle MK H$. Wait, $\angle MK H$: since $MG$ and $JH$ intersect at $K$, $\angle MK H$ and $\angle JK M$ are supplementary, and $\angle JK M$ is equal to $\angle GKH$? No, wait, $\angle JK G$ is $155^{\circ}$, so $\angle MK H$ is equal to $\angle JK G$? No, that's not right. Wait, no, vertical angles are equal. $\angle JK G$ and $\angle MK H$: are they vertical angles? Let's see, the intersection of $JH$ and $MG$ at $K$. So the vertical angle of $\angle JK G$ is $\angle MK H$? Yes! Because when two lines intersect, vertical angles are equal. Wait, no, if $JH$ and $MG$ are two intersecting lines, then $\angle JK G$ and $\angle MK H$ are vertical angles? Wait, no, $\angle JK G$ and $\angle MK H$: let's label the angles. Let's say line $JH$ is horizontal, line $MG$ is slanting. So at point $K$, the angles: $\angle JK G$ (top - left), $\angle GKH$ (top - right), $\angle MK H$ (bottom - right), $\angle JK M$ (bottom - left). So $\angle JK G$ and $\angle MK H$ are vertical angles? No, $\angle JK G$ and $\angle MK H$: $\angle JK G$ is top - left, $\angle MK H$ is bottom - right. So they are vertical angles? Yes! Because vertical angles are opposite each other when two lines intersect. So if two lines intersect, vertical angles are equal. Wait, no, vertical angles are opposite angles formed by two intersecting lines. So $\angle JK G$ and $\angle MK H$ are vertical angles? Wait, no, $\angle JK G$ and $\angle MK H$: let's check the vertices. The lines are $JH$ (from $J$ to $H$) and $MG$ (from $M$ to $G$). The intersection is at $K$. So the four angles are: $\angle JKM$ (between $J$ and $M$), $\angle MKH$ (between $M$ and $H$), $\angle HKG$ (between $H$ and $G$), and $\angle GKJ$ (between $G$ and $J$). So $\angle GKJ$ (which is $155^{\circ}$) and $\angle MKH$: are they vertical angles? No, $\angle GKJ$ and $\angle MKH$ are not vertical angles. $\angle GKJ$ and $\angle MKH$: $\angle GKJ$ is adjacent to $\angle HKG$, and $\angle MKH$ is adjacent to $\angle JKM$. Wait, I think I made a mistake earlier. Let's use the fact that $\angle MKH$ and $\angle JKM$ are supplementary (they form a linear pair along $JH$). And $\angle JKM$ is equal to $\angle HKG$ (vertical angles). Wait, no, $\angle JKM$ and $\angle HKG$ are vertical angles. And $\angle HKG$ and $\angle GKJ$ are supplementary (since $JH$ is a straight line). So $\angle HKG=180^{\circ}-\angle GKJ = 180 - 155 = 25^{\circ}$. Then $\angle JKM=\angle HKG = 25^{\circ}$ (vertical angles). Then $\angle MKH = 180^{\circ}-\angle JKM=180 - 25 = 155^{\circ}$? No, that can't be. Wait, no, I'm getting confused. Wait, let's look at the diagram again. In the second diagram, the angle given is $155^{\circ}$ at $K$ between $J$ and $G$. So the angle between $J$ and $M$ is vertical to the angle between $H$ and $G$. Wait, maybe a simpler way: $\angle MKH$ and the $155^{\circ}$ angle are supplementary? No, wait, no. Wait, the sum of angles on a straight line is $180^{\circ}$. So if one angle is $155^{\circ}$, the adjacent angle (forming a linear pair) is $180 - 155 = 25^{\circ}$, but that's not $\angle MKH$. Wait, no, in the second diagram, $\angle MKH$: let's see, the line $MG$ is a straight line, and $JH$ is a straight line. So $\angle MKH$ and $\angle JKM$ are supplementary. And $\angle JKM$ is equal to the angle opposite to the $155^{\circ}$ angle? Wait, no, the $155^{\circ}$ angle and $\angle JKM$ are vertical angles? No, the $155^{\circ}$ angle is $\angle JKG$, and $\angle JKM$ is vertical to $\angle HKG$. Wait, I think I need to use the property that vertical angles are equal and linear pairs sum to $180^{\circ}$. Let's start over for the second sub - question:

## Step 1: Identify the relationship between angles
We know that $JH$ is a straight line, so the sum of angles on $JH$ at point $K$ is $180^{\circ}$. Also, $MG$ is a straight line, so the sum of angles on $MG$ at point $K$ is $180^{\circ}$. The angle $\angle JKG = 155^{\circ}$, and $\angle MKH$ and $\angle JKG$ are supplementary? No, wait, $\angle MKH$ and $\angle JKM$ are supplementary, and $\angle JKM$ is equal to $\angle HKG$ (vertical angles), and $\angle HKG$ and $\angle JKG$ are supplementary. So $\angle HKG=180^{\circ}-\angle JKG = 180 - 155 = 25^{\circ}$. Then $\angle JKM=\angle HKG = 25^{\circ}$ (vertical angles). Then $\angle MKH = 180^{\circ}-\angle JKM = 180 - 25=155^{\circ}$? No, that's not right. Wait, no, I think I mixed up the angles. Wait, actually, $\angle MKH$ and the $155^{\circ}$ angle are vertical angles? No, vertical angles are equal. Wait, maybe the $155^{\circ}$ angle and $\angle MKH$ are vertical angles. Wait, if we look at the intersection of two lines, the opposite angles are equal. So if $\angle JKG = 155^{\circ}$, then $\angle MKH$ (which is opposite to $\angle JKG$) is also $155^{\circ}$? But that seems wrong. Wait, no, let's draw it mentally. Line $JH$: $J$---$K$---$H$. Line $MG$: $M$---$K$---$G$. So the four angles at $K$: $\angle JKM$ (between $J$ and $M$), $\angle MKH$ (between $M$ and $H$), $\angle HKG$ (between $H$ and $G$), $\angle GKJ$ (between $G$ and $J$). So $\angle GKJ = 155^{\circ}$, $\angle HKG$ is adjacent to it, so $\angle HKG = 180 - 155 = 25^{\circ}$. $\angle HKG$ and $\angle JKM$ are vertical angles, so $\angle JKM = 25^{\circ}$. $\angle JKM$ and $\angle MKH$ are adjacent (on line $JH$), so $\angle MKH=180 - 25 = 155^{\circ}$. Wait, but that would mean $\angle MKH = 155^{\circ}$, but that seems to contradict the first sub - question. Wait, no, maybe I made a mistake. Wait, no, in the first sub - question, the angle was $130^{\circ}$, and we found $50^{\circ}$. In the second sub - question, the angle is $155^{\circ}$, so the adjacent angle (linear pair) is $25^{\circ}$, but $\angle MKH$: wait, no, maybe $\angle MKH$ is equal to the $155^{\circ}$ angle because they are vertical angles. Wait, no, vertical angles are opposite each other. So $\angle GKJ$ (155°) and $\angle MKH$: are they opposite? Let's see, $\angle GKJ$ is between $G$ and $J$, $\angle MKH$ is between $M$ and $H$. So if you look at the intersection, $\angle GKJ$ and $\angle MKH$ are vertical angles? Yes! Because the two lines are $JH$ and $MG$, so the vertical angles are $\angle GKJ$ and $\angle MKH$, and $\angle JKM$ and $\angle HKG$. So vertical angles are equal, so $\angle MKH=\angle GKJ = 155^{\circ}$. Wait, that makes sense. Because when two lines intersect, vertical angles are congruent. So $\angle MKH$ and $\angle JKG$ (which is $155^{\circ}$) are vertical angles, so they are equal.
## Step 2: Conclude the measure of $\angle MKH$
Since $\angle MKH$ and $\angle JKG$ are vertical angles, and $\angle JKG = 155^{\circ}$, then $\angle MKH = 155^{\circ}$. Wait, but that seems to go against the linear pair idea. Wait, no, if $\angle MKH = 155^{\circ}$, then the adjacent angle (on line $JH$) is $180 - 155 = 25^{\circ}$, which is $\angle JKM$, and $\angle JKM$ is vertical to $\angle HKG$, so $\angle HKG = 25^{\circ}$, which is consistent with the first sub - question's logic (where we had a $130^{\circ}$ angle and found $50^{\circ}$, which is $180 - 130$). Wait, no, in the first sub - question, the angle we found was $\angle GKH$, which is $\angle HKG$ in the second sub - question's notation. So in the first sub - question, $\angle HKG=50^{\circ}$ (since $180 - 130 = 50$), and in the second sub - question, $\angle HKG = 25^{\circ}$ (since $180 - 155 = 25$), and $\angle MKH$ is equal to $\angle JKG = 155^{\circ}$ because they are vertical angles.
# Answer:
The measure of $\angle MKH$ is $155^{\circ}$.

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QUESTION IMAGE

Question

Explanation:

First Sub - question: Find the measure of $\angle GKH$

Step 1: Recall the property of supplementary angles

Step 2: Calculate the measure of $\angle GKH$

Step 1: Recall the property of vertical angles or supplementary angles (also, note the linear pair and vertical angle relationships)

Answer:

Second Sub - question: Find the measure of $\angle MKH$