line jm intersects line gk at point n. which statements are true about the figure? select two options. □ ∠gnj is complementary to ∠jnk. □ ∠mnl is complementary to ∠knl. □ ∠mng is complementary to ∠gnj. □ ∠knj is supplementary to ∠mnl. □ ∠gnm is supplementary to ∠jnk.

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Accepted Answer

# Explanation:
## Step1: Recall angle definitions
Complementary angles sum to $90^\circ$, supplementary angles sum to $180^\circ$. From the diagram, $\angle JNK = 90^\circ$ (right angle), and lines $JM\perp GK$ (implied by the right angle symbol).

## Step2: Analyze each option
- **Option 1 ($\angle GNJ$ and $\angle JNK$)**: $\angle GNJ + \angle JNK=\angle GNK$. But $\angle JNK = 90^\circ$, and $\angle GNJ$ is adjacent, but their sum is not $90^\circ$ (since $\angle GNK$ is a straight line? Wait, no—wait, $JM$ and $GK$ intersect at $N$, with $\angle JNK = 90^\circ$, so $JM\perp GK$. So $\angle GNJ + \angle JNK = \angle GNK$? No, actually, $\angle GNJ$ and $\angle JNK$: $\angle JNK = 90^\circ$, $\angle GNJ$ is also $90^\circ$? Wait, no—wait, $GK$ is a straight line, $JM$ is perpendicular to $GK$, so $\angle JNG = 90^\circ$ (same as $\angle JNK$). Wait, maybe I mislabel. Let's re-express: $JM$ is vertical, $GK$ is horizontal, intersect at $N$, right angle at $JNK$ (so $JN\perp KN$). Then:
  - $\angle MNL$ and $\angle KNL$: $\angle MNL + \angle KNL=\angle MNK$. Since $JN\perp KN$, $\angle MNK = 90^\circ$ (because $JM$ is straight, so $\angle JNK + \angle MNK = 180^\circ$, but $\angle JNK = 90^\circ$, so $\angle MNK = 90^\circ$). Thus, $\angle MNL + \angle KNL = 90^\circ$ (complementary). So this option is true.
  - $\angle MNG$ and $\angle GNJ$: $\angle MNG$ is a straight angle? Wait, $G - N - K$ is straight, $J - N - M$ is straight. $\angle MNG$ and $\angle GNJ$: $\angle MNG = 180^\circ - \angle JNG$? No, $\angle JNG = 90^\circ$ (since $JM\perp GK$), so $\angle MNG = 90^\circ$ (wait, $J - N - M$ is straight, so $\angle JNG + \angle MNG = 180^\circ$? No, no—$GK$ is horizontal, $JM$ is vertical, intersect at $N$. So $\angle JNG = 90^\circ$ (between $JN$ (up) and $GN$ (left)), $\angle MNG$ is between $MN$ (down) and $GN$ (left), so $\angle JNG + \angle MNG = 180^\circ$ (supplementary), not complementary. So this is false.
  - $\angle KNJ$ is $90^\circ$, $\angle MNL$: $\angle KNJ + \angle MNL$—wait, $\angle KNJ = 90^\circ$, $\angle MNL$ is part of $\angle MNK = 90^\circ$. Wait, $\angle KNJ$ and $\angle MNL$: $\angle KNJ + \angle MNL + \angle KNL = 180^\circ$? No, earlier we saw $\angle MNK = 90^\circ$, so $\angle MNL + \angle KNL = 90^\circ$. $\angle KNJ = 90^\circ$, so $\angle KNJ + \angle MNL$ would be $90^\circ + \angle MNL$, which is not $180^\circ$ unless $\angle MNL = 90^\circ$, which it's not. Wait, maybe another approach: $\angle GNM$ and $\angle JNK$: $\angle GNM$ is a straight angle? No, $\angle GNM$ is $180^\circ - \angle JNG = 90^\circ$? Wait, no—$G - N - K$ is straight ($180^\circ$), $J - N - M$ is straight ($180^\circ$). $\angle GNM$: points $G, N, M$—so from $G$ to $N$ to $M$, that's a straight line? No, $G$ is left, $M$ is down, so $\angle GNM$ is $90^\circ$? Wait, I think I messed up. Let's use the right angle: $JN\perp KN$ (so $\angle JNK = 90^\circ$). Then:
    - $\angle MNL$ and $\angle KNL$: since $\angle MNK = 90^\circ$ (because $JN\perp KN$ and $JM$ is straight, so $\angle JNK + \angle MNK = 180^\circ$ ⇒ $\angle MNK = 90^\circ$), so $\angle MNL + \angle KNL = \angle MNK = 90^\circ$ (complementary). So this is true.
    - $\angle GNM$ and $\angle JNK$: $\angle GNM$ is $180^\circ - \angle JNG$, but $\angle JNG = 90^\circ$ (since $JN\perp GK$), so $\angle GNM = 90^\circ$? Wait, no—$G - N - K$ is $180^\circ$, $J - N - M$ is $180^\circ$. $\angle GNM$: from $G$ to $N$ to $M$, that's a right angle? Wait, $GN$ is left, $MN$ is down, so yes, $\angle GNM = 90^\circ$? No, that can't be. Wait, maybe $\angle GNM$ is $180^\circ - \angle JNG$, but $\angle JNG = 90^\circ$, so $\angle GNM = 90^\circ$? Then $\angle GNM + \angle JNK = 90^\circ + 90^\circ = 180^\circ$ (supplementary). So $\angle GNM$ is supplementary to $\angle JNK$. That's true.
    - Wait, let's recheck:
      - $\angle MNL$ and $\angle KNL$: sum to $90^\circ$ (complementary) – true.
      - $\angle GNM$ and $\angle JNK$: $\angle GNM = 180^\circ - \angle JNG$, but $\angle JNG = 90^\circ$ (since $JN\perp GK$), so $\angle GNM = 90^\circ$? No, $\angle GNM$ is actually a straight angle? Wait, no—$G$, $N$, $M$: $GN$ is horizontal left, $MN$ is vertical down, so the angle between them is $90^\circ$? No, horizontal and vertical make $90^\circ$, but $GN$ is left (horizontal), $MN$ is down (vertical), so $\angle GNM = 90^\circ$. $\angle JNK = 90^\circ$, so $90 + 90 = 180$ – supplementary. So that's true.
      - Wait, the options:
        - $\angle MNL$ complementary to $\angle KNL$ – true (sum to $90^\circ$).
        - $\angle GNM$ supplementary to $\angle JNK$ – true (sum to $180^\circ$).

## Step3: Confirm the two true options
After analyzing, $\angle MNL$ and $\angle KNL$ are complementary, and $\angle GNM$ and $\angle JNK$ are supplementary.

# Answer:
B. $\angle MNL$ is complementary to $\angle KNL$
E. $\angle GNM$ is supplementary to $\angle JNK$

(Note: Assuming the options are labeled as A: $\angle GNJ$ complementary to $\angle JNK$, B: $\angle MNL$ complementary to $\angle KNL$, C: $\angle MNG$ complementary to $\angle GNJ$, D: $\angle KNJ$ supplementary to $\angle MNL$, E: $\angle GNM$ supplementary to $\angle JNK$)

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Question

Explanation:

Step1: Recall angle definitions

Step2: Analyze each option

Answer: