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))
(squareangle cef) is a straight angle
(squareangle aef) is a right angle

Question

Accepted Answer

# Answer:
- $m\angle CEA = 90^{\circ}$
- $\angle CEF$ is a straight angle
- $\angle AEF$ is a right angle

# Explanation:
## Step 1: Analyze $\angle CEA$
Since $EA\perp EC$ (right - angle symbol at $E$), by the definition of a right - angle, $m\angle CEA=90^{\circ}$

## Step 2: Analyze $\angle CEF$
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 ($m\angle CEF = 180^{\circ}$)

## Step 3: Analyze $\angle AEF$
$\angle AEF=\angle AEC+\angle CEF$. Since $\angle AEC = 90^{\circ}$ and $\angle CEF=180^{\circ}$ is wrong. Wait, no. Since $EA\perp EC$ and $EC$ and $EF$ are opposite rays ($\angle CEF = 180^{\circ}$), $\angle AEF=\angle AEC+\angle CEF$ is wrong. Wait, actually, $\angle AEF$: $EA\perp EC$ and $EC$ and $EF$ form a straight line. $\angle AEF=\angle AEC+\angle CEF$ is wrong. Wait, no. $\angle AEF$: $EA\perp EC$ ( $m\angle AEC = 90^{\circ}$), and $EC$ and $EF$ are opposite rays ($\angle CEF=180^{\circ}$). But actually, $\angle AEF$: $EA$ and $EF$ - since $EA\perp EC$ and $EC$ and $EF$ are collinear, $\angle AEF = 90^{\circ}$ (because $\angle AEC+\angle CEF$ is wrong. Wait, no. $\angle AEF$: Since $EA\perp EC$ ($m\angle AEC=90^{\circ}$) and $EC$ and $EF$ are opposite rays ($\angle CEF = 180^{\circ}$), but actually, $\angle AEF=\angle AEC+\angle CEF$ is wrong. Wait, no. Looking at the figure, $EA\perp EC$ (right - angle at $E$ between $EA$ and $EC$), and $EC$ and $EF$ are a straight line. So $\angle AEF$ is composed of $\angle AEC$ ( $90^{\circ}$) and $\angle CEF$ ( $90^{\circ}$)? No. Wait, no. Wait, $EA\perp EC$, and $EC$ and $EF$ are opposite rays. So $\angle AEF = 90^{\circ}$ (because $\angle AEC = 90^{\circ}$ and $\angle CEF$ is a straight angle, but $\angle AEF$ is adjacent to $\angle AEC$ and $\angle CEF$ in a way that $\angle AEF=90^{\circ}$ (from the right - angle symbol in the figure).

For $m\angle CEB = 2(m\angle CEA)$: If $EB$ bisects $\angle CEA$, then $m\angle CEB=\frac{1}{2}m\angle CEA$ (since it's a bisector), so this is wrong.

For $m\angle CEF=m\angle CEA + m\angle BEF$: There is no information to support this additive relationship in the way it's written. Since $EB$ bisects $\angle CEA$, but $\angle CEF$ is a straight angle ($180^{\circ}$) and $\angle CEA=90^{\circ}$, this equation $m\angle CEF=m\angle CEA + m\angle BEF$ is not valid.

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QUESTION IMAGE

Question

Explanation:

Step 1: Analyze \(\angle CEA\)

Step 2: Analyze \(\angle CEF\)

Step 3: Analyze \(\angle AEF\)

Answer: