Question

find the measure of $\angle gkh$. what is the measure of $\angle gkj$?

Explanation:

Step1: Identify supplementary angles

∠GKH and the 157° angle are supplementary (they form a linear pair), so their sum is 180°. Let \( x = \angle GKH \). Then \( x + 157^\circ = 180^\circ \).

Step2: Solve for x

Subtract 157° from both sides: \( x = 180^\circ - 157^\circ \).
\( x = 23^\circ \)? Wait, no—wait, the question in the image (the gray box) is "What is the measure of ∠GKJ?" Wait, maybe I misread. Wait, the original problem (the first line) is "Find the measure of ∠GKH", but the gray box is about ∠GKJ. Wait, let's re-examine the diagram:

- Line JH is horizontal, with K in the middle.
- Line GM (with M and G) intersects JH at K.
- The angle between KH and KM is 157°, so ∠HKM = 157°.
- ∠GKH and ∠HKM: Wait, no—∠GKH and ∠JK M? Wait, J---K---H (horizontal), M---K---G (the other line). So ∠HKM is 157°, so ∠GKH and ∠HKM are adjacent? No, ∠GKH and ∠JK M? Wait, maybe ∠GKJ: J---K---G? Wait, J is left of K, G is above K (or in the diagram, the line GK goes up-right, MK goes down-left). So ∠GKJ: angle at K between GK and JK.

Wait, the angle ∠HKM is 157°, so ∠GKJ is equal to ∠HKM? No, wait, vertical angles or linear pairs. Wait, ∠GKJ and the 157° angle: are they supplementary? Wait, no—JKH is a straight line (180°), so ∠GKJ + ∠GKH = 180°? No, maybe I messed up. Wait, let's clarify the diagram:

- J---K---H: straight line (180°).
- M---K---G: straight line (180°).
- So ∠HKM = 157°, so ∠GKH is adjacent to ∠HKM? Wait, no—∠HKM is between KH and KM, so ∠GKH is between KG and KH. So ∠GKH + ∠HKM = 180°? No, because KG and KM are a straight line (M---K---G), so ∠MKG is 180°. So ∠HKM (157°) and ∠GKH: ∠HKM + ∠GKH = 180°? Wait, no—∠HKM is 157°, so ∠GKH = 180° - 157° = 23°? But the gray box is about ∠GKJ. Wait, ∠GKJ: angle at K between GK and JK (J is left of K, G is up-right). So ∠GKJ: J---K---G. Since J---K---H is straight (180°), and ∠GKH is 23°, then ∠GKJ would be equal to ∠HKM? Wait, no—vertical angles. Wait, ∠GKJ and ∠HKM: are they vertical angles? Wait, ∠GKJ is at K, between JK and GK; ∠HKM is at K, between HK and KM. Since JK is opposite to HK? No, J and H are on the same line, so JK is the same as KH but opposite direction. Wait, JK is from J to K, KH is from K to H, so JK and KH are a straight line (180°). Similarly, GK and MK are a straight line (180°). So ∠GKJ (angle between JK and GK) and ∠HKM (angle between HK and MK) are vertical angles? Wait, no—vertical angles are opposite each other when two lines intersect. So the two lines are JH (J-K-H) and GM (G-K-M). So the vertical angles are ∠GKJ and ∠HKM, and ∠GKH and ∠JKM. Wait, no—when two lines intersect, the vertical angles are equal. So ∠GKJ and ∠HKM: are they vertical? Let's see:

- Intersection at K: lines JH and GM.
- Angles formed: ∠GKJ (between JK and GK), ∠GKH (between GK and HK), ∠HKM (between HK and MK), ∠JKM (between MK and JK).

So vertical angles: ∠GKJ and ∠HKM (opposite), ∠GKH and ∠JKM (opposite). Therefore, ∠GKJ = ∠HKM = 157°? Wait, no—wait, ∠HKM is 157°, so if ∠GKJ is opposite to ∠HKM, then they are equal? But that can't be, because JK and HK are a straight line (180°), so ∠GKJ + ∠GKH = 180°. Wait, maybe I made a mistake in the angle label. Let's check the diagram again (as per the user's image):

- The angle marked 157° is between KH (right along JH) and KM (down-left along GM). So ∠HKM = 157°.
- Then ∠GKH is the angle between KG (up-right along GM) and KH (right along JH). Since GM is a straight line (KG and KM are colinear), ∠GKH + ∠HKM = 180°, so ∠GKH = 180° - 157° = 23°. But the gray box asks for ∠GKJ, which is the angle between KG (up-right) and JK (left along JH). Since JK and KH are a straight line (180°), ∠GKJ + ∠GKH = 180°? No, ∠GKJ is adjacent to ∠GKH? Wait, JK is left of K, KH is right of K, so JK and KH are a straight line (180°). KG is the line going up-right from K. So ∠GKJ is the angle between JK (left) and GK (up-right), and ∠GKH is the angle between GK (up-right) and KH (right). So ∠GKJ + ∠GKH = 180°, because JK to KH is 180°. But ∠GKH is 23°, so ∠GKJ would be 180° - 23° = 157°? Wait, that's the same as ∠HKM. Alternatively, since ∠GKJ and ∠HKM are vertical angles (because they are opposite each other when two lines intersect), they should be equal. So ∠GKJ = ∠HKM = 157°? But that contradicts the earlier calculation. Wait, no—let's use the linear pair and vertical angles properly.

When two lines intersect, vertical angles are equal, and linear pairs are supplementary (sum to 180°). So lines JH (J-K-H) and GM (G-K-M) intersect at K. Therefore:

- ∠GKJ and ∠HKM are vertical angles, so ∠GKJ = ∠HKM = 157°.
- ∠GKH and ∠JKM are vertical angles, so ∠GKH = ∠JKM.
- Also, ∠GKJ + ∠GKH = 180° (linear pair), so 157° + ∠GKH = 180° ⇒ ∠GKH = 23°.

But the gray box (the question in the image) is "What is the measure of ∠GKJ?", so that's 157°? Wait, but the original problem (the first line) is "Find the measure of ∠GKH", which would be 23°. But maybe the user's question (the image) has a typo, or I misread. Wait, let's check the user's image again:

- The top line: "Find the measure of ∠GKH."
- The gray box: "What is the measure of ∠GKJ? [blank]°"

So perhaps the user intended to ask about ∠GKJ, or there's a mix-up. Let's resolve both:

For ∠GKH:

- ∠GKH and ∠HKM (157°) are supplementary (linear pair), so ∠GKH = 180° - 157° = 23°.

For ∠GKJ (the gray box):

- ∠GKJ and ∠HKM are vertical angles, so ∠GKJ = 157° (since vertical angles are equal). Alternatively, ∠GKJ and ∠GKH are supplementary: ∠GKJ = 180° - 23° = 157°.

But the original problem (the first line) is "Find the measure of ∠GKH", so that's 23°. Wait, but maybe I made a mistake in the angle labels. Let's confirm with the diagram:

- J---K---H (horizontal, J left, H right)
- M---K---G (M down-left, G up-right)
- Angle between KH (right) and KM (down-left) is 157°, so ∠HKM = 157°.
- ∠GKH is between KG (up-right) and KH (right), so it's adjacent to ∠HKM. Since KG and KM are a straight line (180°), ∠GKH + ∠HKM = 180° ⇒ ∠GKH = 180 - 157 = 23°.

Yes, that makes sense. So the measure of ∠GKH is 23 degrees.

Answer:

23°