Question

in the diagram, which angles are vertical angles? choose two correct answers. ∠abc and ∠cbd ∠abe and ∠cbd ∠abe and ∠abc ∠abe and ∠ebd ∠abc and ∠ebd

Explanation:

Step1: Recall Vertical Angles Definition

Vertical angles are opposite angles formed by two intersecting lines, and they are equal in measure.

Step2: Analyze Each Option

- Option 1: $\angle ABC$ and $\angle EBD$ – Check if they are opposite angles from intersecting lines. Lines $AE$ and $CD$ intersect at $B$. $\angle ABC$ and $\angle EBD$: Wait, let's re - examine. Wait, lines $AB - BD$ and $CB - BE$ intersect at $B$. Wait, correct vertical angles: When two lines intersect, the vertical angles are the non - adjacent angles. Let's list the angles:
- For $\angle ABC$: Its vertical angle should be $\angle EBD$? Wait no, wait the lines are $AB$ and $BD$ (one line) and $CB$ and $BE$ (another line). Wait, actually, the two intersecting lines are $AE$ (with points $A$, $B$, $E$) and $CD$ (with points $C$, $B$, $D$). So the vertical angles are: $\angle ABC$ and $\angle EBD$? Wait no, let's see: $\angle ABC$ is formed by $BC$ and $BA$, and $\angle EBD$ is formed by $BE$ and $BD$. Wait, no, the correct vertical angles: when two lines intersect, say line 1: $A - B - D$ and line 2: $C - B - E$. Then the vertical angles are $\angle ABC$ (between $BC$ and $BA$) and $\angle EBD$ (between $BE$ and $BD$)? Wait, no, another pair: $\angle ABE$ (between $BA$ and $BE$) and $\angle CBD$ (between $BC$ and $BD$). Wait, let's check the options:
- Option $\angle ABC$ and $\angle CBD$: No, they are adjacent.
- Option $\angle ABE$ and $\angle CBD$: Yes, because lines $AE$ and $CD$ intersect at $B$, so $\angle ABE$ and $\angle CBD$ are vertical angles.
- Option $\angle ABC$ and $\angle EBD$: Let's see, $\angle ABC$: sides $BC$ and $BA$; $\angle EBD$: sides $BE$ and $BD$. Since $BA$ and $BD$ are opposite rays (along line $A - B - D$), and $BC$ and $BE$ are opposite rays (along line $C - B - E$), so $\angle ABC$ and $\angle EBD$ are vertical angles. Wait, the options given:
Looking at the options (from the diagram's choices):
1. $\angle ABC$ and $\angle CBD$ – adjacent, not vertical.
2. $\angle ABE$ and $\angle CBD$ – vertical (lines $AE$ and $CD$ intersect, so $\angle ABE$ (between $BA$ and $BE$) and $\angle CBD$ (between $BC$ and $BD$) are vertical).
3. $\angle ABE$ and $\angle ABC$ – adjacent (share side $BA$ and $BE$, $BC$), not vertical.
4. $\angle ABE$ and $\angle EBD$ – adjacent (share side $BE$), not vertical.
5. $\angle ABC$ and $\angle EBD$ – vertical (as $BA$ and $BD$ are opposite rays, $BC$ and $BE$ are opposite rays, so the angles formed are vertical).

Wait, the problem says "Choose two correct answers". So the two correct are $\angle ABC$ and $\angle EBD$ and $\angle ABE$ and $\angle CBD$? Wait, looking at the options in the image (the five options):

From the user's image, the options are:

1. $\angle ABC$ and $\angle CBD$

2. $\angle ABE$ and $\angle CBD$

3. $\angle ABE$ and $\angle ABC$

4. $\angle ABE$ and $\angle EBD$

5. $\angle ABC$ and $\angle EBD$

So the correct two are $\angle ABE$ and $\angle CBD$ (because lines $AE$ and $CD$ intersect at $B$, so vertical angles) and $\angle ABC$ and $\angle EBD$ (because lines $AE$ and $CD$ intersect at $B$, so vertical angles).

Wait, let's confirm the definition of vertical angles: two angles are vertical angles if their sides are opposite rays.

For $\angle ABE$ and $\angle CBD$:
- Sides of $\angle ABE$: $\overrightarrow{BA}$ and $\overrightarrow{BE}$
- Sides of $\angle CBD$: $\overrightarrow{BC}$ and $\overrightarrow{BD}$
- $\overrightarrow{BA}$ and $\overrightarrow{BD}$ are opposite rays (since $A - B - D$ is a straight line)
- $\overrightarrow{BE}$ and $\overrightarrow{BC}$ are opposite rays (since $C - B - E$ is a straight line)
So $\angle ABE$ and $\angle CBD$ are vertical angles.

For $\angle ABC$ and $\angle EBD$:
- Sides of $\angle ABC$: $\overrightarrow{BC}$ and $\overrightarrow{BA}$
- Sides of $\angle EBD$: $\overrightarrow{BE}$ and $\overrightarrow{BD}$
- $\overrightarrow{BA}$ and $\overrightarrow{BD}$ are opposite rays
- $\overrightarrow{BC}$ and $\overrightarrow{BE}$ are opposite rays
So $\angle ABC$ and $\angle EBD$ are vertical angles.

So the two correct answers are $\angle ABC$ and $\angle EBD$ (fifth option) and $\angle ABE$ and $\angle CBD$ (second option).

Answer:

$\angle ABC$ and $\angle EBD$, $\angle ABE$ and $\angle CBD$ (corresponding to the options: the fifth option $\angle ABC$ and $\angle EBD$ and the second option $\angle ABE$ and $\angle CBD$)