Question

which angle’s measure is equal to the sum of the measures of ∠bac and ∠bca?
○ ∠cba
○ ∠cbe
○ ∠dac
○ ∠bcf

Explanation:

In triangle \(ABC\), by the triangle angle - sum property, the sum of the interior angles of a triangle is \(180^{\circ}\), so \(\angle BAC+\angle BCA+\angle CBA = 180^{\circ}\). Also, \(\angle CBE\) and \(\angle CBA\) are supplementary (they form a linear pair), so \(\angle CBE+\angle CBA=180^{\circ}\). From \(\angle BAC+\angle BCA+\angle CBA = 180^{\circ}\), we can get \(\angle BAC+\angle BCA=180^{\circ}-\angle CBA\), and from \(\angle CBE+\angle CBA = 180^{\circ}\), we get \(\angle CBE = 180^{\circ}-\angle CBA\). So \(\angle CBE=\angle BAC + \angle BCA\) (by the exterior - angle theorem, the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. \(\angle CBE\) is an exterior angle of \(\triangle ABC\) at vertex \(B\), and the two non - adjacent interior angles are \(\angle BAC\) and \(\angle BCA\)).

Answer:

\(\boldsymbol{\angle CBE}\)