Let's find the measures of the numbered angles one at a time.
Measure of ∠1
Before we begin, let's name the vertices of our kite. We will call it quadrilateral ABCD. The angle that forms a linear pair with ∠1 we will label M.
If a quadrilateral is a kite, then its diagonals are perpendicular. Therefore, ∠1 is right and its measure is 90.
Measure of ∠2
We have a triangle with the angles ∠M, ∠2, and the angle with measure 65. By the Triangle Angle-Sum Theorem we know that the measures of the angles add to 180.
45+ m∠M+m∠2=180
Since ∠M is formed by the diagonals it is right and its measure is 90. Let's substitute 90 for m∠M and solve the equation.
