Question

find the indicated measure in ▱hijk. explain.

1. hi
2. kh
3. gh
4. hj
5. m∠kih
6. m∠jih
7. m∠kji
8. m∠hki

Explanation:

Step1: Recall properties of parallelogram

In parallelogram $HIJK$, opposite - sides are equal.

Step2: Find $HI$

Since $KJ$ and $HI$ are opposite - sides, and $KJ = 16$, then $HI=16$.

Step3: Find $KH$

Since $IJ$ and $KH$ are opposite - sides, and $IJ = 10$, then $KH = 10$.

Step4: Use triangle properties for $GH$

In $\triangle HKI$, we don't have enough information to find $GH$ directly from the given parallelogram properties. Assume $G$ is the mid - point of $HI$ and $KJ$ (if not given otherwise, we can't solve). If $G$ is the mid - point of $HI$, and $HI = 16$, then $GH=\frac{HI}{2}=8$.

Step5: Find $HJ$

We can use the law of cosines in $\triangle HKI$. In parallelogram $HIJK$, $\angle KHI=84^{\circ}$, $KH = 10$, $HI = 16$. In $\triangle HKI$, by the law of cosines, $HJ^{2}=KH^{2}+HI^{2}-2\cdot KH\cdot HI\cdot\cos\angle KHI$.
\[

$$\begin{align*} HJ^{2}&=10^{2}+16^{2}-2\times10\times16\times\cos84^{\circ}\\ &=100 + 256-320\times0.1045\\ &=100 + 256-33.44\\ &=322.56\\ HJ&\approx17.96 \end{align*}$$

\]

Step6: Find $\angle KIH$

In $\triangle HKI$, we know $\angle KHI = 84^{\circ}$ and $\angle HKI=28^{\circ}$. Using the angle - sum property of a triangle ($\angle KHI+\angle HKI+\angle KIH = 180^{\circ}$), then $\angle KIH=180^{\circ}-84^{\circ}-28^{\circ}=68^{\circ}$.

Step7: Find $\angle JIH$

Since $\angle KIH = 68^{\circ}$ and $\angle KIJ$ is a straight - line angle with $\angle KIH$, and in a parallelogram adjacent angles are supplementary, $\angle JIH = 180^{\circ}-\angle KIH=112^{\circ}$.

Step8: Find $\angle KJI$

In parallelogram $HIJK$, $\angle KJI$ and $\angle KHI$ are opposite angles, so $\angle KJI = 84^{\circ}$.

Step9: Find $\angle HKI$

Given $\angle HKI = 28^{\circ}$.

Answer:

1. $HI = 16$
2. $KH = 10$
3. If $G$ is the mid - point of $HI$, $GH = 8$
4. $HJ\approx17.96$
5. $m\angle KIH = 68^{\circ}$
6. $m\angle JIH = 112^{\circ}$
7. $m\angle KJI = 84^{\circ}$
8. $m\angle HKI = 28^{\circ}$