Question

what is ( mangle abc )?
( mangle abc = 60^circ )
( mangle abc = 67^circ )
( mangle abc = 120^circ )
( mangle abc = 127^circ )

Explanation:

Step1: Recall triangle angle sum and linear pair

In triangle \(BCD\), we know two angles? Wait, no, first, the sum of angles in a triangle is \(180^\circ\), but also, \(\angle BDC = 60^\circ\), \(\angle BCD = 67^\circ\), so we can find \(\angle CBD\) first. Wait, no, actually, \(\angle ABC\) and \(\angle CBD\) are supplementary (linear pair), so first find \(\angle CBD\) using triangle angle sum.

Step2: Calculate \(\angle CBD\)

In \(\triangle BCD\), sum of angles is \(180^\circ\). So \(\angle CBD=180^\circ - 67^\circ - 60^\circ=53^\circ\)? Wait, no, that's wrong. Wait, no, the line \(AD\) is straight, so \(\angle ABC + \angle CBD = 180^\circ\) (linear pair). But first, in \(\triangle BCD\), angles are \(\angle C = 67^\circ\), \(\angle D = 60^\circ\), so \(\angle CBD = 180^\circ - 67^\circ - 60^\circ = 53^\circ\)? No, that can't be. Wait, no, maybe I misread the diagram. Wait, the triangle is \(ABC\)? No, the points are \(A - B - D\) on a straight line, and \(C\) above \(B\) and \(D\). So triangle \(BCD\) has \(\angle C = 67^\circ\), \(\angle D = 60^\circ\), so \(\angle CBD = 180 - 67 - 60 = 53\), but then \(\angle ABC = 180 - 53 = 127\)? Wait, let's check again.

Wait, the options include \(127^\circ\). Let's recalculate:

Sum of angles in triangle: \(67^\circ + 60^\circ + \angle CBD = 180^\circ\)

So \(\angle CBD = 180 - 67 - 60 = 53^\circ\)

Then, since \(A - B - D\) is a straight line, \(\angle ABC + \angle CBD = 180^\circ\) (linear pair)

So \(\angle ABC = 180 - 53 = 127^\circ\)

Yes, that makes sense. So \(m\angle ABC = 127^\circ\)

Answer:

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