Question

line jm intersects line gk at point n. which statements are true about the figure? select two options. \\( \angle \mathrm { gnj } \\) is complementary to \\( \angle \mathrm { jnk } \\). \\( \angle \mathrm { mnl } \\) is complementary to \\( \angle \mathrm { knl } \\). \\( \angle \mathrm { mng } \\) is complementary to \\( \angle \mathrm { gnj } \\). \\( \angle \mathrm { knj } \\) is supplementary to \\( \angle \mathrm { mnl } \\). \\( \angle \mathrm { gnm } \\) is supplementary to \\( \angle \mathrm { jnk } \\).

Explanation:

Step1: Recall the definitions of complementary and supplementary angles

Complementary angles: Two angles whose sum is \(90^{\circ}\). Supplementary angles: Two angles whose sum is \(180^{\circ}\).

Step2: Analyze each option

• For \(\angle GNJ\) and \(\angle JNK\): \(\angle GNJ+\angle JNK

eq90^{\circ}\), so this is false.

• For \(\angle MNL\) and \(\angle KNL\): Since \(\angle MNK = 90^{\circ}\) (from the right - angle in the figure), and \(\angle MNK=\angle MNL+\angle KNL\), so \(\angle MNL+\angle KNL = 90^{\circ}\). This is true.
• For \(\angle MNG\) and \(\angle GNJ\): \(\angle MNG+\angle GNJ = 180^{\circ}\) (linear pair), not \(90^{\circ}\). So this is false.
• For \(\angle KNJ\) and \(\angle MNL\): There is no relation that \(\angle KNJ+\angle MNL=180^{\circ}\). So this is false.
• For \(\angle GNM\) and \(\angle JNK\): \(\angle GNM+\angle JNK

eq180^{\circ}\). So this is false.

Answer:

\(\angle MNL\) is complementary to \(\angle KNL\)