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 \(\angle BAC\) are supplementary (\(\angle DAC+\angle BAC = 180^{\circ}\)) and has no relation to \(\angle BCA\) based on the given figure.
• For \(\angle BCF\): There is no direct relationship with \(\angle BAC\) and \(\angle BCA\) as per the basic angle properties in a simple geometric figure (not an exterior angle related to \(\triangle ABC\) in the required way)

Answer:

B. \( \angle CBE \)