Question

which angles measure is equal to the sum of the measures of ∠bac and ∠bca?
∠cba
∠cbe
∠dac
∠bcf

Explanation:

Step1: Recall the exterior - angle theorem

The exterior - angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Identify the triangle and angles

In \(\triangle ABC\), \(\angle CBE\) is an exterior angle. The two non - adjacent interior angles to \(\angle CBE\) are \(\angle BAC\) and \(\angle BCA\).

By the exterior - angle theorem, \(\angle CBE=\angle BAC + \angle BCA\)

For \(\angle CBA\): In \(\triangle ABC\), \(\angle BAC+\angle BCA+\angle CBA = 180^{\circ}\) (angle - sum property of a triangle), so \(\angle CBA=180^{\circ}-(\angle BAC + \angle BCA)\)

For \(\angle DAC\): \(\angle DAC\) and \(\angle BAC\) form a linear pair (\(\angle DAC+\angle BAC = 180^{\circ}\)), and there is no relation to directly sum \(\angle BAC\) and \(\angle BCA\) to get \(\angle DAC\)

For \(\angle BCF\): There is no geometric relationship (such as exterior - angle or linear - pair) that would make \(\angle BCF=\angle BAC+\angle BCA\) without additional information about parallel lines or other angle - equalities not indicated in the basic triangle - angle relationships.

Answer:

B. \( \angle CBE \)