Chapter 6, Problem 14CR

Solution

To Prove: HMKJ is a kite

Given information: $\frac{\overline{GH}}{\overline{HM}}\cong \frac{\overline{GK}}{\overline{KM}}$ is provided in the question which will help to find the solution.

Proof: GH = HK (given)

Hence, $\angle GMH=\angle GMK$ (angle opposite to equal sides are equal)

And, HM = KM (given)

Hence, $\angle HJM=\angle MJK$ (angle opposite to equal sides are equal)

Here, $\angle GMH=\angle GMK$

  $\angle GMH+\angle HMJ=\angle GMK+\angle HMJ$

  $\angle HMJ+\angle GMK={180}^{\circ }\mathrm{.........}(1)$

Also, $\angle GMK+\angle KMJ={180}^{\circ }\mathrm{.........}(2)$

Equate $\left(1\right)$ and $\left(2\right)$

So, $\angle GMK+\angle HMJ=\angle GMK+\angle KMJ$

  $\angle HMJ=\angle KMJ$

So that, HJ = JK (sides opposite to equal angles are equal)

Hence, HM = KM and HJ = JK

So that, we can say that HMKJ is a kite or rhombus.