Question

in the diagram below, ( overline{ad} ) bisects ( angle cab ), ( mangle adb = 97^{circ} ) and ( mangle cab = 66^{circ} ). find ( mangle b ).

Explanation:

Step1: Find the measure of ∠DAB

Since \(AD\) bisects \(\angle CAB\) and \(m\angle CAB = 66^{\circ}\), then \(m\angle DAB=\frac{1}{2}m\angle CAB\).
\(m\angle DAB=\frac{1}{2}\times66^{\circ}=33^{\circ}\)

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

In \(\triangle ABD\), we know that the sum of the interior angles of a triangle is \(180^{\circ}\), i.e., \(m\angle B + m\angle DAB+m\angle ADB = 180^{\circ}\).
We know \(m\angle DAB = 33^{\circ}\) and \(m\angle ADB=97^{\circ}\). Let \(m\angle B=x\). Then \(x + 33^{\circ}+97^{\circ}=180^{\circ}\).
\(x=180^{\circ}-(33^{\circ} + 97^{\circ})\)
\(x=180^{\circ}-130^{\circ}\)

Answer:

\(m\angle B = 50^{\circ}\)