Question

determining angle relationships in a diagram
which statements are true regarding the diagram?
$angle$gnh and $angle$hnj are complementary.
$angle$jnk and $angle$knl are supplementary.
$angle$knl and $angle$lnm are complementary.
$mangle hnk + mangle knl = 180^circ$
$mangle mng + mangle gnh = 90^circ$

Explanation:

Step1: Define complementary/supplementary

Complementary: sum = $90^\circ$; Supplementary: sum = $180^\circ$. $\overleftrightarrow{GK} \perp \overleftrightarrow{JM}$, so $\angle JNK = \angle JNG = \angle GNM = \angle MNK = 90^\circ$.

Step2: Check $\angle GNH$ & $\angle HNJ$

$\angle GNH + \angle HNJ = \angle GNJ = 90^\circ$. They are complementary.

Step3: Check $\angle JNK$ & $\angle KNL$

$\angle JNK = 90^\circ$, $\angle KNL < 90^\circ$, sum $<180^\circ$. Not supplementary.

Step4: Check $\angle KNL$ & $\angle LNM$

$\angle KNL + \angle LNM = \angle MNK = 90^\circ$. They are complementary.

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

$\angle HNK + \angle KNL = \angle HNL$, a straight angle. Sum = $180^\circ$.

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

$\angle MNG = 90^\circ$, so sum $>90^\circ$. Not equal to $90^\circ$.

Answer:

• $\angle GNH$ and $\angle HNJ$ are complementary.
• $\angle KNL$ and $\angle LNM$ are complementary.
• $m\angle HNK + m\angle KNL = 180^\circ$