Find the angles of a right-angled triangle if the angle between the bisector

Find the angles of a right-angled triangle if the angle between the bisector and the height drawn from the vertex of the right angle is 15 degrees.

Consider a triangle ABC, where B is the right angle, BD is the height, BE is the bisector.

Since BE is a bisector, the angles ABE and EBC are equal. Hence,

angle ABE = angle EBC = angle ABC / 2 = 90 ° / 2 = 45 °.

According to the problem, the angle between the bisector and the height DBE = 15 °.

Then

angle ABD = angle ABE – angle DBE = 45 ° – 15 ° = 30 °.

Since the triangle ABD is right-angled and the angle BDA = 90 °, then

angle BAC = 180 ° – angle BDA – angle ABD = 180 ° – 90 ° – 30 ° = 60 °.

It’s obvious that

angle BCA = 180 ° – angle ABC – angle BAC = 180 ° – 90 ° – 60 ° = 30 °.

Answer: 60 ° and 30 °.