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# Special Right Triangle Problems and Solutions

## Problem 1

Q: In △ABC, side AB has length 4, and ∠A = ∠C = 45°. Find BC and AC.

A: A triangle always has 180 degrees in Euclidean geometry. Since ∠A and ∠C are both 45°, the third angle of the triangle, ∠B has to be 180 - (45 + 45) = 90°. Triangles with angle measures of 45°, 45°, and 90° are a type of special right triangle. The two legs of the triangle will always be equal, due to the triangle sum theorem, and the hypotenuse will be √2 times larger than a leg. Since side AB of the triangle is 4 units, and AB is a leg, BC should also be 4 units since it is a leg as well. (It may help to draw the triangle on a sheet of paper). Since AC is the hypotenuse, or longest side of the right triangle, it has length 4√2.

So, BC = 4 and AC = 4√2

## Problem 2

Q: In △XYZ, side YZ has length 6, and ∠X = 30°, ∠Y = 90°. Find XY and XZ.