Question

In mathematics, a kite is a special type of quadrilateral. Two pairs of sides are congruent, but opposite sides are not congruent. Exactly one pair of opposite angles are congruent. In kite ABCD shown, the measure of $\angle A$ is twice the measure of $\angle C$, and $\angle B$ has a measure of $114°$. Find the measure of $\angle A$, $\angle C$, and $\angle D$.

Short answer

$\angle A=88{}^{\circ },\angle C=44{}^{\circ },\angle D=114°$

Step-by-step solution

Step 1. Given Information.

The figure is shown below.

Step 2. Formula used.

Angle sum property of quadrilateral:

$\angle 1+\angle 2+\angle 3+\angle 4=360°$

Step 3. Calculation.

The sides AB and AD are congruent, and BC and DC are congruent. Therefore, the angles between them are congruent, too. Hence,

$\angle B=\angle D$

Therefore,

$\angle D=114°$

Let, the $\angle C$ be “x”.

Since, $\angle A$is twice$\angle C$,

$\angle A=2x$

Substituting the values of all the angles in the equation of angle sum property of quadrilateral,

$\begin{array}{l}2x+114+x+114=360\\ 3x+228=360\end{array}$

Shifting the right-hand side and left-hand side values,

$\begin{array}{l}3x=360-228\\ 3x=132\end{array}$

Cross-multiplying the values,

$\begin{array}{l}x=\frac{132}{3}\\ x=44\end{array}$

Hence,

$\begin{array}{l}\angle C={44}^{°}\\ \angle A=2x\\ \angle A=2\times 44\\ \angle A={88}^{°}\end{array}$

Therefore, $\angle A=88{}^{\circ },\angle C=44{}^{\circ },\angle D=114°$.