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 property

An angle - bisector divides an angle into two equal parts. Given $\overrightarrow{EB}$ bisects $\angle CEA$, we have $\angle CEB=\angle BEA$.

Step2: Analyze each option

• Option 1: Just because $\overrightarrow{EB}$ bisects $\angle CEA$, we cannot say $m\angle CEA = 90^{\circ}$ without further information. But if we assume from the figure (if it is a right - angled situation), it can be true.
• Option 2: $m\angle CEF=m\angle CEA + m\angle BEF$ is incorrect. Since $\angle CEF$ is a straight - angle and $\angle CEA$ and $\angle BEF$ are not related in this way. In fact, $\angle CEF = 180^{\circ}$ and $\angle CEA$ and $\angle BEF$ are parts of other angle relations.
• Option 3: Since $\overrightarrow{EB}$ bisects $\angle CEA$, $m\angle CEB=\frac{1}{2}m\angle CEA$, not $m\angle CEB = 2(m\angle CEA)$.
• Option 4: $\angle CEF$ is a straight angle. By the definition of a straight - line, the angle formed by points $C$, $E$, and $F$ on a straight - line is $180^{\circ}$, so $\angle CEF$ is a straight angle.
• Option 5: Since $\angle AEF$ is composed of two right - angles (assuming the perpendicular lines in the figure), $\angle AEF$ is a right - angle.

Answer:

$m\angle CEA = 90^{\circ}$, $\angle CEF$ is a straight angle, $\angle AEF$ is a right angle.