May 11th, 2023, 7:22 pm
Had to get an arctangent lookup.
Proof:
CDB = 135°, CBD = 30°.
Draw a line from point C to line extended from AB. Call the point of intersection F.
CDF is a 45°/45°/90° right triangle, and CBF is a 60°/30°/90° right triangle.
If we say the length of CF is 1, then the length of DF is also 1, and the length of BF is sqrt(3).
So the length of DB is sqrt(3) - 1, which is also the length of AD.
This makes the length of AF 1 - [sqrt(3) - 1], or 2 - sqrt(3).
Thus the arctangent of 2 - sqrt(3) is the measure of angle ACF, and this happens to be 15°.
Angle DCF is 45°, so this means angle ACD is 30°.