Question

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.

Explanation:

Step1: Recall the definitions of complementary and supplementary angles

Complementary angles sum to \(90^{\circ}\), and supplementary angles sum to \(180^{\circ}\).

Step2: Analyze each option

• For \(\angle GNJ\) and \(\angle JNK\): \(\angle GNJ+\angle JNK = 90^{\circ}\) (they form a right - angle at \(N\) from the perpendicular lines \(JM\) and \(GK\) in the figure), so \(\angle GNJ\) is complementary to \(\angle JNK\).
• For \(\angle MNL\) and \(\angle KNL\): There is no indication that their sum is \(90^{\circ}\).
• For \(\angle MNG\) and \(\angle GNJ\): \(\angle MNG+\angle GNJ=180^{\circ}\) (they form a linear pair), so they are supplementary, not complementary.
• For \(\angle KNJ\) and \(\angle MNL\): There is no indication that their sum is \(180^{\circ}\).
• For \(\angle GNM\) and \(\angle JNK\): \(\angle GNM = 180^{\circ}-\angle JNK\) (since \(\angle GNM+\angle JNK=180^{\circ}\) as \(\angle GNM\) and \(\angle JNK\) form a linear pair).

Answer:

\(\angle GNJ\) is complementary to \(\angle JNK\) and \(\angle GNM\) is supplementary to \(\angle JNK\)