Question

find the measures of the numbered angles in the kite. the figure is not drawn to scale. m∠1 = (type an integer or a decimal.)

Explanation:

Step1: Recall kite - angle properties

In a kite, one pair of opposite angles are equal and the diagonals of a kite are perpendicular to each other.

Step2: Use the angle - sum property of a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. Let the known angles be $34^{\circ}$ and $90^{\circ}$. Let $\angle1$ and $\angle2$ be the unknown angles.
We know that one of the properties of a kite gives us some angle - relationships. Since the diagonals of a kite are perpendicular, we can also use the fact that in the non - right - angled triangles formed within the kite.
We know that $\angle1 = 146^{\circ}$ because the sum of the angles in a quadrilateral is $360^{\circ}$. Let's assume the four - angle measures of the kite are $A = 34^{\circ}$, $B=90^{\circ}$, $C = \angle1$, and $D=\angle2$.
We know that $A + B+C + D=360^{\circ}$. Also, due to the symmetry properties of a kite, we can find $\angle1$ as follows:
The sum of the two non - right angles adjacent to the $90^{\circ}$ angle in the kite is $360^{\circ}- 90^{\circ}-90^{\circ}=180^{\circ}$.
If one of the non - right angles is $34^{\circ}$, then $\angle1=180^{\circ}-34^{\circ}=146^{\circ}$.

Answer:

$146$