Question

foundations of euclidean geometry. given that eb bisects ∠cea, which statements must be true? choose three correct answers. m∠cef = m∠cea + m∠bef. ∠aef is a right angle. ∠cef is a straight angle. m∠cea = 90°. m∠ceb = 2(m∠cea)

Explanation:

Step1: Recall angle - bisector property

An angle - bisector divides an angle into two equal parts. So if $\overrightarrow{EB}$ bisects $\angle CEA$, then $m\angle CEB=m\angle BEA=\frac{1}{2}m\angle CEA$.

Step2: Analyze each statement

• For $m\angle CEF=m\angle CEA + m\angle BEF$: There is no information to support this relationship from the given that $\overrightarrow{EB}$ bisects $\angle CEA$.
• $\angle AEF$ is a right - angle: There is no information in the problem about the perpendicularity of lines to conclude that $\angle AEF$ is a right - angle.
• $\angle CEF$ is a straight angle: Since points $C$, $E$, and $F$ are collinear, by the definition of a straight angle (an angle whose measure is $180^{\circ}$), $\angle CEF$ is a straight angle.
• $m\angle CEA = 90^{\circ}$: There is no information to suggest that $\angle CEA$ is a right - angle just because $\overrightarrow{EB}$ bisects it.
• $m\angle CEB=\frac{1}{2}m\angle CEA$ (not $m\angle CEB = 2m\angle CEA$ as given in the option, which is incorrect).

Answer:

$\angle CEF$ is a straight angle.