Question

va v. geometry sem a geometry
ndations of euclidean geometry
line jm intersects line gk at point n
which statements are true about the figure? choose two correct answers.
∠mng is complementary to ∠gnj.
∠gnm is supplementary to ∠jnk.
∠mnl is complementary to ∠knl.
∠knj is supplementary to ∠mnl.
∠gnj is complementary to ∠jnk.

Explanation:

Step1: Recall angle - relation definitions

Complementary angles add up to 90°, supplementary angles add up to 180°.

Step2: Analyze each option

• $\angle MNG$ and $\angle GNJ$: $\angle MNG+\angle GNJ = 180^{\circ}$ (linear - pair), so they are supplementary, not complementary.
• $\angle GNM$ and $\angle JNK$: $\angle GNM+\angle JNK= 180^{\circ}$ (vertical - angles and linear - pair relations). Since $\angle GNM$ and $\angle JNK$ are vertical angles to angles that form a linear pair, $\angle GNM+\angle JNK = 180^{\circ}$, they are supplementary.
• $\angle MNL$ and $\angle KNL$: $\angle MNL+\angle KNL=\angle MNK$. Since $\angle MNK = 90^{\circ}$ (given by the right - angle symbol), $\angle MNL$ and $\angle KNL$ are complementary.
• $\angle KNJ$ and $\angle MNL$: There is no fixed sum relationship between them.
• $\angle GNJ$ and $\angle JNK$: There is no fixed sum relationship between them.

Answer:

$\angle GNM$ is supplementary to $\angle JNK$, $\angle MNL$ is complementary to $\angle KNL$