One important property of kites to remember is that the diagonals of a kite form four right angles. The diagonal between the vertex angles (the angles formed by two congruent sides) also bisect these angles of the kite.
Additionally, they contains two pairs of adjacent, congruent sides. As a result, it is possible to find the measure of corresponding angles in the kite, since the diagonals form two pairs of congruent triangles.
We can use what we know about kites to solve for missing angles let’s start with W
Well if I look at W it’s in this triangle right here where I don’t know any of the angles. So it doesn’t look like we can start with W. I’m going to start with y because I know Y if I go back to what I know about kites, is part of this intersection of the diagonals. This intersection will always be 90 degrees. Which means Y is a right angle, so I’m going to write Y equals 90 degrees and I'm going to erase Y and I’m going to draw in my symbol for a right angle. Which means if I do a little problem solving here I see that 90 plus 20 plus this angle must add up to 180 degrees.
Another way of saying that is that these two angles must be complementary so this angle is 70 degrees. Two congruent triangles are created here which means 70 degrees corresponds to W, so now I can figure out what W is. W must be 70 degrees. We’ve got a couple more variables we’ve got X and we’ve got Z.
Well if I start with 60 degrees here and I know that these two angles are vertical, which makes this a 90 degree angle, that means that this angle down here must be 30 degrees, because 30 plus 60 is 90. I also know that something exists between 60 and Z.
Let’s go back to our diagram here and I see that this vertex angle is bisected by that diagonal creating two congruent angles. Which means Z must be congruent to 60 degrees. By the same token we know that if Z and 60 are corresponding and congruent then x and 30 must be corresponding and congruent. So X is 30 degrees.
Key things that we use here was that this vertex angle was bisected by the diagonal and that the two diagonals meet at a right angle.