Triangle $ABC$ is a right triangle with right angle at $A$. Suppose $\overline{AX}$ is an altitude of the triangle, $\overline{AY}$ is an angle bisector of the triangle, and $\overline{AZ}$ is a median of the triangle, and $\angle XAY = 13^\circ$. If $X$ is on $\overline{BY}$, then what is the measure of $\angle ZAC$?
After eliminating all the weird symbols, I got:
Triangle ABC is a right triangle with right angle at A. Suppose AX is an altitude of the triangle, AY is an angle bisector of the triangle, and AZ is a median of the triangle, and angle XAY = 13°. If X is on BY, then what is the measure of angle ZAC ?
In my sketch I made AC considerably longer than AB to get a good separation between X, Y, and Z.
Since AY is the bisector of angle A, and A is 90°, then angle YAB = 45°, which makes angle XAB = (45-13)° = 32°
One of the properties you should know is that if a median is drawn to the hypotenuse of a right-angled triangle, that median equals half the hypotenuse, that is,
AY = BY = CY
In the original triangle ABC, angle B = 58° and thus angle C = 32°
But since triangle AZC is isosceles, angle CAZ = 32°
Therefore angle ZAY = 32°