I figured this out without trigonometry, but the video shows both a trig and a non-trig method.
My answer will be in the comments.
Tuesday, May 16, 2023
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My non-trig method:
ReplyDeleteDraw a line segment connecting the vertex of the supplemental angle to y with the vertex of the supplemental angle to x. You've just divided the blue region into two triangles: one isosceles (I'll justify this in a second), and one right.
The white triangle on the left has a hypotenuse of 13 (Pythagorean triple: 5:12:13). The segment connecting x and y also has a length of 13 for the same reason, so the bigger of the two blue-region triangles is a 45-45-90 right triangle (along with being isosceles).
The smaller blue-region right triangle (upper-right corner) has the same side lengths and angle measures as the left-hand white right triangle. This means the larger acute angle inside the smaller blue right triangle also measures y degrees.
So angle x remains mysterious, but we don't have to solve for x—only for x + y. We know that x + y + 45 (from the 45-45-90 triangle) = 180, so x + y = 135º.
QED. I did this on paper, then belatedly realized I could have done most or all of this in my head.
The trig method shown in the video is a bit above my pay grade.