Question

what is m∠abc?
○ m∠abc = 60°
○ m∠abc = 67°
○ m∠abc = 120°
○ m∠abc = 127°

Explanation:

Step1: Find angle at D in triangle

In triangle \(BCD\), we know two angles: \(\angle C = 67^\circ\) and \(\angle CDB = 60^\circ\). The sum of angles in a triangle is \(180^\circ\), so \(\angle CBD = 180^\circ - 67^\circ - 60^\circ = 53^\circ\)? Wait, no, wait. Wait, \(A\), \(B\), \(D\) are colinear, so \(\angle ABC\) and \(\angle CBD\) are supplementary? Wait, no, let's correct. Wait, \(A - B - D\) is a straight line, so \(\angle ABD = 180^\circ\). In triangle \(BCD\), angles sum to \(180^\circ\), so \(\angle CBD = 180^\circ - 67^\circ - 60^\circ = 53^\circ\)? No, that's not right. Wait, no, the angle at \(D\) is \(60^\circ\), angle at \(C\) is \(67^\circ\), so angle at \(B\) (angle \(CBD\)) is \(180 - 67 - 60 = 53^\circ\)? But then angle \(ABC\) is supplementary to angle \(CBD\), so \(180 - 53 = 127^\circ\). Wait, let's do it again.

Step1: Sum of angles in triangle

In \(\triangle BCD\), \(\angle C + \angle CDB + \angle CBD = 180^\circ\). Given \(\angle C = 67^\circ\), \(\angle CDB = 60^\circ\), so \(\angle CBD = 180 - 67 - 60 = 53^\circ\).

Step2: Supplementary angles

Since \(A\), \(B\), \(D\) are on a straight line, \(\angle ABC + \angle CBD = 180^\circ\). So \(\angle ABC = 180 - 53 = 127^\circ\).

Answer:

\(m\angle ABC = 127^\circ\) (the option: \(m\angle ABC = 127^\circ\))