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 angle - sum property of a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. Let the given angles be $34^{\circ}$, $90^{\circ}$, and the two unknown angles $\angle1$ and $\angle2$. Since one of the properties of a kite is that one pair of non - adjacent angles are equal. Here, assume the non - adjacent angles are equal.
We know that $34^{\circ}+90^{\circ}+\angle1+\angle2 = 360^{\circ}$. Also, since the diagonals of a kite are perpendicular, we can find the relationship between the angles. In this case, we can use the fact that the sum of angles around the intersection of the diagonals is $360^{\circ}$ and the right - angle property of the diagonals.
We know that $\angle1$ and the angle adjacent to the $34^{\circ}$ angle are equal.
We use the fact that the sum of angles in a triangle formed by the diagonals and the sides of the kite. Consider the triangle with the $34^{\circ}$ angle and the right - angle (formed by the diagonals).
The third angle in this triangle is $180^{\circ}-(90^{\circ} + 34^{\circ})=56^{\circ}$.
Since the kite has symmetry properties related to its diagonals, $\angle1 = 56^{\circ}$.

Answer:

$56$