Question

quadrilateral abcd is a kite. what is m∠d?
a 61°
c 77°
m∠d = □ °

Explanation:

Step1: Recall angle - sum property of quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. In a kite, one pair of opposite angles are equal. Here, $\angle B$ and $\angle D$ are the pair of non - congruent angles, and $\angle A$ and $\angle C$ are given.

Step2: Set up the equation

Let $m\angle D=x$. We know that $m\angle A = 61^{\circ}$ and $m\angle C=77^{\circ}$. Using the angle - sum formula for a quadrilateral $\angle A+\angle B+\angle C+\angle D = 360^{\circ}$. Since in a kite, the non - vertex angles (the angles between the non - congruent sides) are equal, we have $61^{\circ}+x + 77^{\circ}+x=360^{\circ}$. Combining like terms gives $2x+138^{\circ}=360^{\circ}$.

Step3: Solve for $x$

First, subtract $138^{\circ}$ from both sides of the equation: $2x=360^{\circ}- 138^{\circ}=222^{\circ}$. Then divide both sides by 2: $x = 111^{\circ}$.

Answer:

$111$