which statements regarding the diagram of △ebc are true? choose three correct answers.
□ ∠bec is an exterior angle.
□ ∠dec is an exterior angle.
□ ∠abe and ∠ebc are supplementary angles.
□ ∠bec is a remote interior angle to exterior ∠bcf.
□ ∠bcf and ∠dec are supplementary angles.

Question

Accepted Answer

# Explanation:
## Step1: Recall Exterior Angle and Supplementary Angles
- Exterior angle of a triangle: An angle formed by one side of the triangle and the extension of another side. For $\triangle EBC$, an exterior angle should be adjacent to an interior angle and form a linear pair (supplementary) with it, and not be adjacent to the other two interior angles (remote interior angles).
- Supplementary angles: Two angles that add up to $180^\circ$ (form a linear pair or are adjacent on a straight line).

## Step2: Analyze Each Statement
1. **"$\angle BEC$ is an exterior angle."**
   An exterior angle of $\triangle EBC$ should be formed by extending a side. $\angle BEC$ is an interior angle of $\triangle EBC$ (it's inside the triangle at vertex $E$), so this is false.

2. **"$\angle DEC$ is an exterior angle."**
   Side $EB$ is extended to $D$, so $\angle DEC$ is formed by side $EC$ and the extension of $EB$ (from $E$ to $D$). It is adjacent to interior angle $\angle BEC$ (they form a linear pair, so supplementary) and is not adjacent to the other two interior angles ($\angle EBC$ and $\angle ECB$) of $\triangle EBC$. So $\angle DEC$ is an exterior angle. This is true.

3. **"$\angle ABE$ and $\angle EBC$ are supplementary angles."**
   Points $A$, $B$, and the line containing $EBC$ (since $A$ is on a ray from $B$): $\angle ABE$ and $\angle EBC$ form a linear pair (they are adjacent on a straight line $AB$ - $B$ - $E$ - $D$? Wait, actually, looking at the diagram, $A$ is on a ray from $B$, so $AB$ and $BE$ (or $BC$?) Wait, the straight line: $A$ is on a ray, $B$ is a vertex, and $EBC$ is a triangle. Wait, $\angle ABE$ and $\angle EBC$ share a common side $BE$ and their non - common sides form a straight line (since $A$ is on a ray from $B$, so $AB$ and $BC$ - no, $AB$ is a ray, $B$ is connected to $E$ and $C$. Wait, actually, $\angle ABE$ and $\angle EBC$ are adjacent and form a linear pair (sum to $180^\circ$) because they are on a straight line (the ray $BA$ and the segment $BE$ or $BC$? Wait, the diagram shows $A$ is on a ray from $B$, so $AB$ is a straight line with $BE$ (or $BC$?) No, $B$ is connected to $E$ and $C$, and $A$ is on a ray from $B$, so $\angle ABE$ and $\angle EBC$ are adjacent and form a linear pair, so they are supplementary. This is true.

4. **"$\angle BCF$ and $\angle DEC$ are supplementary angles."**
   $\angle BCF$ is formed by extending $BC$ to $F$, and $\angle DEC$ is an exterior angle at $E$. There's no indication they form a linear pair or sum to $180^\circ$. So this is false.

5. **"$\angle BEC$ is a remote interior angle to exterior $\angle BCF$."**
   Remote interior angles to an exterior angle are the two interior angles of the triangle that are not adjacent to the exterior angle. For exterior angle $\angle BCF$ (formed by extending $BC$ to $F$), the adjacent interior angle is $\angle ECB$, so the remote interior angles should be $\angle BEC$ and $\angle EBC$. Wait, $\angle BEC$ is an interior angle not adjacent to $\angle BCF$, so yes, $\angle BEC$ is a remote interior angle to exterior $\angle BCF$. Wait, let's re - check: Exterior angle $\angle BCF$ is adjacent to $\angle ECB$ (they form a linear pair, supplementary). The other two interior angles of $\triangle EBC$ are $\angle BEC$ and $\angle EBC$, so they are remote interior angles to $\angle BCF$. So this statement is true? Wait, no, the statement says "$\angle BEC$ is a remote interior angle to exterior $\angle BCF$". Yes, because remote interior angles are the two non - adjacent interior angles to the exterior angle. So $\angle BCF$ is exterior, adjacent to $\angle ECB$, so $\angle BEC$ and $\angle EBC$ are remote. So this is true? Wait, maybe I made a mistake earlier. Wait, let's re - evaluate each:

Wait, maybe the correct true statements are:
- $\angle DEC$ is an exterior angle (true).
- $\angle ABE$ and $\angle EBC$ are supplementary (true, since they form a linear pair).
- $\angle BEC$ is a remote interior angle to exterior $\angle BCF$ (true, because $\angle BCF$ is exterior, adjacent to $\angle ECB$, so $\angle BEC$ (and $\angle EBC$) are remote). Wait, but earlier I thought $\angle BEC$ was interior, which it is, but remote interior to the exterior angle $\angle BCF$.

Wait, let's re - check the diagram:
- $\triangle EBC$ has vertices $E$, $B$, $C$.
- Exterior angle at $C$: extending $BC$ to $F$, so $\angle BCF$ is exterior, adjacent to $\angle ECB$ (supplementary). The remote interior angles to $\angle BCF$ are the two interior angles not adjacent to it, which are $\angle BEC$ (at $E$) and $\angle EBC$ (at $B$). So $\angle BEC$ is a remote interior angle to $\angle BCF$ (exterior angle). So that statement is true.

Wait, but the first statement "$\angle BEC$ is an exterior angle" is false. The second "$\angle DEC$ is an exterior angle" is true. The third "$\angle ABE$ and $\angle EBC$ are supplementary" is true (linear pair). The fourth "$\angle BCF$ and $\angle DEC$ are supplementary" - no, they don't form a linear pair. The fifth "$\angle BEC$ is a remote interior angle to exterior $\angle BCF$" is true.

Wait, maybe the three correct ones are:
- $\angle DEC$ is an exterior angle.
- $\angle ABE$ and $\angle EBC$ are supplementary.
- $\angle BEC$ is a remote interior angle to exterior $\angle BCF$.

But let's confirm again:

1. $\angle BEC$ is interior, not exterior: false.
2. $\angle DEC$ is exterior (formed by extending $EB$ to $D$, adjacent to $\angle BEC$, supplementary, and remote from $\angle EBC$ and $\angle ECB$): true.
3. $\angle ABE$ and $\angle EBC$: $AB$ is a ray from $B$, so $AB$ and $BE$ (or $BC$?) form a straight line? Wait, $B$ is connected to $A$ (ray), $B$ is connected to $E$ and $C$. So $\angle ABE$ and $\angle EBC$ share $B$ and $BE$, and their other sides ($BA$ and $BC$?) Wait, no, $A$ is on a ray, so $BA$ is a straight line with $BE$? Maybe $BA$ and $BE$ are colinear? If $A$, $B$, $E$ are colinear (since $A$ is on a ray from $B$, and $E$ is on a line from $B$ to $D$), then $\angle ABE$ and $\angle EBC$ form a linear pair (supplementary). So this is true.

4. $\angle BCF$ and $\angle DEC$: no, they are not adjacent or forming a linear pair. False.

5. $\angle BEC$ is a remote interior angle to exterior $\angle BCF$: Exterior angle $\angle BCF$ is at $C$, adjacent to $\angle ECB$. The remote interior angles are the two angles not adjacent to $\angle BCF$, which are $\angle BEC$ (at $E$) and $\angle EBC$ (at $B$). So $\angle BEC$ is a remote interior angle to $\angle BCF$ (exterior). True.

So the three true statements are:
- $\angle DEC$ is an exterior angle.
- $\angle ABE$ and $\angle EBC$ are supplementary angles.
- $\angle BEC$ is a remote interior angle to exterior $\angle BCF$.

# Answer:
The three correct statements (to check the boxes for) are:
- $\angle DEC$ is an exterior angle.
- $\angle ABE$ and $\angle EBC$ are supplementary angles.
- $\angle BEC$ is a remote interior angle to exterior $\angle BCF$.

(In the diagram, the boxes corresponding to these statements should be checked. If we consider the labels:
- The second box (top middle: $\angle DEC$ is an exterior angle) - check.
- The first box (bottom left: $\angle ABE$ and $\angle EBC$ are supplementary) - check.
- The fifth box (bottom right: $\angle BEC$ is a remote interior angle to exterior $\angle BCF$) - check. )

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Exterior Angle and Supplementary Angles

Step2: Analyze Each Statement

Answer: