Question

given that $overrightarrow{eb}$ bisects $angle cea$, which statements must be true? select three options.$square mangle cea = 90^circ$$square mangle cef = mangle cea + mangle bef$$square mangle ceb = 2(mangle cea)$$square angle cef$ is a straight angle.$square angle aef$ is a right angle.

Explanation:

Step1: Identify right angle ∠CEA

From the diagram, $\angle CEA$ has a right angle symbol, so $m\angle CEA = 90^\circ$.

Step2: Verify straight angle ∠CEF

Points C, E, F are colinear (vertical line), so $\angle CEF$ is a straight angle ($180^\circ$).

Step3: Check right angle ∠AEF

$\angle AEF = \angle CEA + \angle CEF - \angle CEA = 180^\circ - 90^\circ = 90^\circ$, so it is a right angle.

Step4: Eliminate incorrect statements

• $m\angle CEF = m\angle CEA + m\angle BEF$ is false: $\angle CEF = 180^\circ$, $m\angle CEA + m\angle BEF = 90^\circ + m\angle BEF < 180^\circ$.
• $m\angle CEB = 2(m\angle CEA)$ is false: $\overrightarrow{EB}$ bisects $\angle CEA$, so $m\angle CEB = 45^\circ$, $2(m\angle CEA)=180^\circ$, which is not equal.

Answer:

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