Question

what is m ∠abc?
○ m ∠abc = 15°
○ m ∠abc = 45°
○ m ∠abc = 60°
○ m ∠abc = 75° (diagram: points d, b, c are collinear; triangle abc with ∠a = 75°, exterior angle at c = 135°)

Explanation:

Step1: Find the interior angle at C

The angle of \(135^\circ\) is supplementary to the interior angle at \(C\) in \(\triangle ABC\). So, the interior angle at \(C\) is \(180^\circ - 135^\circ = 45^\circ\).

Step2: Use the triangle angle - sum property

The sum of the interior angles of a triangle is \(180^\circ\). Let \(\angle ABC=x\). We know that \(\angle A = 75^\circ\) and the interior angle at \(C\) is \(45^\circ\). So, \(x + 75^\circ+ 45^\circ=180^\circ\).
Simplify the left - hand side: \(x + 120^\circ=180^\circ\).
Subtract \(120^\circ\) from both sides: \(x=180^\circ - 120^\circ = 60^\circ\)? Wait, no, wait. Wait, the exterior angle at \(C\) is \(135^\circ\), so the interior angle at \(C\) is \(180 - 135=45^\circ\). Then in triangle \(ABC\), \(\angle A = 75^\circ\), \(\angle C = 45^\circ\), so \(\angle ABC=180-(75 + 45)=180 - 120 = 60^\circ\)? Wait, but let's re - check. Wait, maybe I made a mistake. Wait, the exterior angle at \(C\): the linear pair. So interior angle at \(C\) is \(180 - 135 = 45^\circ\). Then angle at \(A\) is \(75^\circ\), so angle at \(B\) ( \(\angle ABC\)) is \(180-(75 + 45)=60^\circ\). Wait, but let's check the options. The options are \(15^\circ\), \(45^\circ\), \(60^\circ\), \(75^\circ\). So \(m\angle ABC = 60^\circ\). Wait, no, wait another approach. The exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, the exterior angle at \(C\) is \(135^\circ\), which should be equal to \(\angle A+\angle ABC\). So \(135^\circ=75^\circ+\angle ABC\). Then \(\angle ABC = 135^\circ - 75^\circ=60^\circ\). Yes, that's correct. The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two remote (non - adjacent) interior angles. So if we consider the exterior angle at \(C\) ( \(135^\circ\)), it is equal to \(\angle A+\angle ABC\). So \(\angle ABC=135^\circ - 75^\circ = 60^\circ\).

Answer:

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