Joseph F. answered • 05/12/20
Joe's Math, Science and Chess
Hi Becky!
I have two facts that will help you make short work of this. One is that the angles of any quadrilateral all dd up to 360 degrees. How do I prove that? Well, any quadrilateral (i.e., four flat sides) can be broken up into two triangles by drawing kitty-corner across it from one vertex to the one that it cannot reach via only one side. And since the angles of each of the two triangles add up to 180 degrees, the angles of the quadrilateral add up to 360 degrees.
OK, you say, that's fine; you still have two variables; how do you find x and y with only one equation? That's where my other fact comes in: your quadrilateral has all its vertices on a circle. And if any angle has its vertex upon a circle, the measure of the arc swept out by the two sides of the angle in the circle is twice the measure of the angle. Try this: draw a square inside a circle. The two kitty-corner vertices of the square sit at the ends of right angles. And the two right angles that make up the square, together, sweep out the entire 360 degrees of the circle, right? But because they are two right angles, they only sum up to 180 degrees, and this is a special case of this fact.
To prove my second fact more generally, draw a "V" with its bottom point at the bottom of a circle. Now bisect this angle with a radius from the vertex running up to the center of the circle. Draw radii from the center out to each of the points where the lines of the "V" intersect the circle. Now, notice that you have drawn two isosceles triangles; they are isosceles because two of each of their sides have the same length. That means you can call the angle of the "V" theta, and the angles of the two base angles of each of the two isosceles triangles are theta/2. That means the angles between the radii at the center (the vertex angle of each triangle) are each (180 - 2*(theta/2)) = 180 - theta. You have two of those, and the sum of these angles is 360 - 2*theta. That means that what is left between the radii above these triangles must be 2*theta.
OK, so, armed with this second fact, you can sum up your kitty-corner angles and set the sum to 180, because between them, these angles sweep out the entire circle. Solve for x and y, and choose the correct answer.
Cheers,
Joe
