Triangle ABC is a right triangle with angle BAC= 90 degrees and angle B > angle . Suppose AP is an altitude of the triangle, AQ is an angle bisector of the triangle, and AR is a median of the triangle, and angle PAQ = 18 degrees. If P is on BQ, then what is the measure of angle RAC?
PAQ = 18 degrees, since angle AQ bisects 90 degrees, angle BAQ is 45 degrees. Using subtraction, angle BAP is 45 - 18 = 27 degrees. By interior angle sum of a triangle, angle B is 180 - 90 - 27 = 63 degrees. Since AR is the median of a right triangle branching from the 90 degree angle, we have BR = AR = RC. Now we can notice that triangle RAC is isosceles because RA = RC, so angle RAC = angle RCA, and angle RCA is 27 degrees from similar triangles or you can get it with interior angle sum from angle B and angle A. Thus, angle RAC = angle RCA = 27 degrees.