Question

what is ( mangle abc )? ( \bigcirc mangle abc = 15^circ ) ( \bigcirc mangle abc = 45^circ ) ( \bigcirc mangle abc = 60^circ ) ( \bigcirc mangle abc = 75^circ ) (image shows triangle ( abc ) with ( angle bac = 75^circ ), point ( c ) on a straight line with an exterior angle of ( 135^circ ), and points ( d, b, c ) collinear)

Explanation:

Step1: Find the interior angle at C

The angle at C adjacent to \(135^\circ\) is supplementary, so \(180^\circ - 135^\circ = 45^\circ\).

Step2: Use triangle angle sum

In \(\triangle ABC\), sum of angles is \(180^\circ\). Let \(\angle ABC = x\), \(\angle A = 75^\circ\), \(\angle ACB = 45^\circ\). So \(75^\circ + 45^\circ + x = 180^\circ\).

Step3: Solve for x

\(120^\circ + x = 180^\circ\), so \(x = 180^\circ - 120^\circ = 60^\circ\).

Answer:

\(m\angle ABC = 60^\circ\) (corresponding to the option \(m\angle ABC = 60^\circ\))