Question

in the diagram below, \\( \overline { a d } \\) bisects \\( m \angle b = 45 ^ { \circ } \\). find \\( m \angle c a b \\).

Explanation:

Step1: Find \(m\angle BAD\)

In \(\triangle ABD\), use the angle - sum property of a triangle (\(m\angle ADB + m\angle B+m\angle BAD=180^{\circ}\)).
Substitute \(m\angle ADB = 98^{\circ}\) and \(m\angle B = 45^{\circ}\) into the formula:
\(m\angle BAD=180^{\circ}-(m\angle ADB + m\angle B)\)
\(m\angle BAD=180^{\circ}-(98^{\circ}+ 45^{\circ})\)
\(m\angle BAD=180^{\circ}-143^{\circ}=37^{\circ}\)

Step2: Use the angle - bisector property

Since \(AD\) bisects \(\angle CAB\), by the angle - bisector definition (\(\angle CAD=\angle BAD\)).
And \(m\angle CAB=m\angle CAD + m\angle BAD\) (angle - addition postulate).
Since \(\angle CAD=\angle BAD = 37^{\circ}\), then \(m\angle CAB=37^{\circ}+37^{\circ}=74^{\circ}\)

Answer:

\(74^{\circ}\)