Question

given that eb bisects ∠cea, which statements must be true? select three options.
□m∠cea = 90°
□m∠cef = m∠cea + m∠bef
□m∠ceb = 2(m∠cea)
□∠cef is a straight angle.
□∠aef is a right angle.

Explanation:

Step1: Recall angle - bisector definition

If $\overrightarrow{EB}$ bisects $\angle CEA$, then $\angle CEB=\angle BEA=\frac{1}{2}\angle CEA$. But there is no information given to say $\angle CEA = 90^{\circ}$, so $m\angle CEA=90^{\circ}$ is not necessarily true.

Step2: Analyze angle - addition relationships

$m\angle CEF=m\angle CEA + m\angle AEF$, not $m\angle CEA + m\angle BEF$, so $m\angle CEF=m\angle CEA + m\angle BEF$ is false. Also, $m\angle CEB=\frac{1}{2}(m\angle CEA)$, not $m\angle CEB = 2(m\angle CEA)$, so this is false.

Step3: Consider straight - angle and right - angle properties

Since points $C$, $E$, and $F$ are collinear, $\angle CEF$ is a straight angle, so $m\angle CEF = 180^{\circ}$. Also, since $\overrightarrow{EA}\perp\overrightarrow{EC}$ (as indicated by the right - angle symbol at $\angle AEC$), $\angle AEF$ is a right angle, $m\angle AEF=90^{\circ}$.

Answer:

$\angle CEF$ is a straight angle., $\angle AEF$ is a right angle.