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

In a triangle, an exterior angle is equal to the sum of the two non - adjacent interior angles.

Step2: Analyze each option

• For \(\angle CBA\): In \(\triangle ABC\), \(\angle BAC+\angle BCA+\angle CBA = 180^{\circ}\) (angle - sum property of a triangle). So \(\angle CBA=180^{\circ}-(\angle BAC + \angle BCA)\), not \(\angle CBA=\angle BAC+\angle BCA\).
• For \(\angle CBE\): \(\angle CBE\) is an exterior angle of \(\triangle ABC\). By the exterior - angle theorem, \(\angle CBE=\angle BAC+\angle BCA\) (since \(\angle BAC\) and \(\angle BCA\) are non - adjacent to \(\angle CBE\)).
• For \(\angle DAC\): \(\angle DAC\) and \(\angle BAC\) form a linear pair (\(\angle DAC+\angle BAC = 180^{\circ}\)), and there is no relation to show \(\angle DAC=\angle BAC+\angle BCA\).
• For \(\angle BCF\): There is no geometric relation (like the exterior - angle theorem) that would make \(\angle BCF=\angle BAC+\angle BCA\) based on the given figure.

Answer:

\(\angle CBE\)