Question

ow, \\( \overline { a d } \\) bisects \\( \angle c a b \\), \\( \mathrm { m } \angle a d b = 99 ^ { \circ } \\) and \\( \mathrm { m } \angle c a d = 31 ^ { \circ } \\). find \\( \mathrm { m } \angle b \\).
you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

Explanation:

Step1: Find the measure of ∠BAD

Since \( \overline{AD} \) bisects \( \angle CAB \), then \( \angle BAD=\angle CAD \). Given \( \angle CAD = 31^{\circ} \), so \( \angle BAD=31^{\circ} \).

Step2: Use the triangle - angle sum theorem in \( \triangle ABD \)

The sum of the interior angles of a triangle is \( 180^{\circ} \). In \( \triangle ABD \), we know that \( \angle ADB = 99^{\circ} \) and \( \angle BAD=31^{\circ} \). Let \( \angle B=x \).
By the formula \( \angle BAD+\angle B+\angle ADB = 180^{\circ} \), we substitute the known values: \( 31^{\circ}+x + 99^{\circ}=180^{\circ} \).
Simplify the left - hand side: \( x+130^{\circ}=180^{\circ} \).
Subtract \( 130^{\circ} \) from both sides: \( x=\angle B=180^{\circ}-130^{\circ} \).

Answer:

\( 59^{\circ} \)