Question

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 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 $\angle GNH$ and $\angle HN J$

Since $\angle GNK = 90^{\circ}$ (right - angle), and $\angle GNK=\angle GNH+\angle HN J$. So $\angle GNH+\angle HN J = 90^{\circ}$, 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 angle).

Step4: Analyze $\angle KNL$ and $\angle LNM$

Since $\angle MNK = 90^{\circ}$ (right - angle), and $\angle MNK=\angle KNL+\angle LNM$. So $\angle KNL+\angle LNM = 90^{\circ}$, they are complementary.

Step5: Analyze $m\angle HNK + m\angle KNL$

$\angle HNK+\angle KNL
eq180^{\circ}$ (they do not form a linear - pair).

Step6: Analyze $m\angle MNG + m\angle GNH$

Since $\angle MNG+\angle GNH=\angle MNH$, and $\angle MNH = 90^{\circ}$ (right - angle formed by the intersection of lines), so $m\angle MNG + m\angle GNH=90^{\circ}$.

Answer:

• $\angle GNH$ and $\angle HN J$ are complementary
• $\angle KNL$ and $\angle LNM$ are complementary
• $m\angle MNG + m\angle GNH=90^{\circ}$