Question

determining angle relationships in a diagram
which statements are true regarding the diagram?
□ ∠gnh and ∠hnj are complementary.
□ ∠jnk and ∠knl are supplementary.
□ ∠knl and ∠lnm are complementary.
□ m∠hnk + m∠knl = 180°
□ m∠mng + m∠gnh = 90°

Explanation:

Step1: Recall the definitions

• Complementary angles: sum to \(90^{\circ}\).
• Supplementary angles: sum to \(180^{\circ}\).

Step2: Analyze \(\angle GNH\) and \(\angle HN J\)

Since \(\angle GNH+\angle HN J = 90^{\circ}\) (as \(GJ\) is a right - angled line at \(N\)), they are complementary.

Step3: Analyze \(\angle JNK\) and \(\angle KNL\)

\(\angle JNK+\angle KNL
eq180^{\circ}\) (they are not adjacent to form a straight - line).

Step4: Analyze \(\angle KNL\) and \(\angle LNM\)

Since \(\angle KNL+\angle LNM = 90^{\circ}\) (as \(MK\) is a right - angled line at \(N\)), they are complementary.

Step5: Analyze \(m\angle HNK + m\angle KNL\)

\(\angle HNK\) and \(\angle KNL\) are not adjacent to form a straight - line, so \(m\angle HNK + m\angle KNL
eq180^{\circ}\).

Step6: Analyze \(m\angle MNG + m\angle GNH\)

Since \(\angle MNG+\angle GNH = 90^{\circ}\) (as \(GM\) is a right - angled line at \(N\)), \(m\angle MNG + m\angle GNH = 90^{\circ}\).

Answer:

\(\angle GNH\) and \(\angle HN J\) are complementary; \(\angle KNL\) and \(\angle LNM\) are complementary; \(m\angle MNG + m\angle GNH = 90^{\circ}\)