In triangle ABC, angle C = 45 °, AD bisector, CAD = 30 °. Find Angle B.

The bisector of an angle is a ray starting at the apex of the angle that divides the angle into two equal parts.

The bisector AD divides the ΔABC triangle into two triangles ΔABD and ΔADC.

Consider a triangle ΔADC.

Let’s calculate the degree measure of the angle ∠ADC. Since the sum of all the angles of the triangle is 180º, then:

∠ADС = 180º – ∠DСА – ∠DАС;

∠ADC = 180º – 45º – 30º = 105º.

Since the angle ∠ВDC is a deployed angle (equal to 180º), then:

∠ВDA = 180º – ADС;

∠ВDA = 180º – 105º = 75º.

Since the bisector of an angle divides it in half, then:

∠ВАD = ∠DАС = 30º.

Since the sum of all the angles of the triangle is 180º, then:

∠АВD = 180º – 30º – 75º = 75º.

Answer: the degree measure of the angle ∠В is equal to 75º.