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. The sum of the interior angles of a quadrilateral is 360°.

Step2: Set up the angle - sum equation

Let's call the known non - right angles \(34^{\circ}\) and the right angles \(90^{\circ}\). We know that \(\angle1\) and the \(34^{\circ}\) angle are opposite angles in the kite, so \(\angle1 = 34^{\circ}\).
To find \(\angle2\), we use the angle - sum formula for a quadrilateral \(A + B+C + D=360^{\circ}\). Substituting the known values: \(34^{\circ}+90^{\circ}+34^{\circ}+\angle2 = 360^{\circ}\).
First, add the known angles on the left - hand side: \(34^{\circ}+90^{\circ}+34^{\circ}=158^{\circ}\). Then, we can solve for \(\angle2\) as \(\angle2=360^{\circ}-158^{\circ}-90^{\circ} = 112^{\circ}\). But we can also use the fact that in a kite, the non - congruent adjacent angles are supplementary to the right - angled corner. Since the sum of angles around the intersection of the diagonals is 360° and the diagonals are perpendicular, we can also note that \(\angle2 = 112^{\circ}\).

Answer:

\(m\angle1 = 34\)