In triangle ABC, angle C is 20 degrees, AD is the bisector of angle A, and ADB is 58 degrees.

In triangle ABC, angle C is 20 degrees, AD is the bisector of angle A, and ADB is 58 degrees. Find the degree measure of angle B.

The degree measure of the angle ADB given by the condition of the problem is equal to the sum of two interior angles of triangle ADC not adjacent to it. We get:
∠ ADC = ∠ DAC + ∠ C → ∠ DAC = ∠ ADC – ∠ C = 58 ° – 20 ° = 38 °.
The degree measure of the angle DAC is equal to half of the angle A (AD is the bisector by condition).
∠ A = 2 * 38 ° = 76 °.
Find angle B:
∠ B = 180 ° – (∠ A + ∠ C) = 180 ° – 96 ° = 84 °.
The degree measure of angle B can also be found from the triangle ABD:
∠ B = 180 ° – (∠ ADB + ∠ BAD) = 180 ° – (58 ° + 38 °) = 84 °.
Answer: The degree measure of angle B is 84 °.