Question

what is ( mangle abc )?
( mangle abc = 15^{circ} )
( mangle abc = 45^{circ} )
( mangle abc = 60^{circ} )
( mangle abc = 75^{circ} )

Explanation:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In \(\triangle ABC\), the exterior angle at \(C\) is \(135^{\circ}\), and the two non - adjacent interior angles are \(\angle A\) and \(\angle ABC\). Let \(m\angle ABC=x\). Then, by the exterior - angle theorem, \(135^{\circ}=\angle A + x\).

Step2: Substitute the value of \(\angle A\)

We know that \(\angle A = 75^{\circ}\). Substituting \(\angle A = 75^{\circ}\) into the equation \(135^{\circ}=\angle A + x\), we get \(135^{\circ}=75^{\circ}+x\).

Step3: Solve for \(x\)

To find \(x\) (which is \(m\angle ABC\)), we use the subtraction property of equality. Subtract \(75^{\circ}\) from both sides of the equation: \(x=135^{\circ}-75^{\circ}\).

$$x = 60^{\circ}$$

Answer:

\(m\angle ABC = 60^{\circ}\)