Question

1. the figure shows lines that intersect at point o. which pair of angles are vertical angles?

options: ∠aoc and ∠coe; ∠coe and ∠bod; ∠eob and ∠aop; ∠coe and ∠eod

Explanation:

Step1: Recall Vertical Angles Definition

Vertical angles are opposite angles formed by two intersecting lines, sharing a common vertex and no common side.

Step2: Analyze Each Option

• Option 1: $\angle AOC$ and $\angle COE$ share a common side $OC$ and vertex $O$, not vertical angles.
• Option 2: $\angle COE$ and $\angle BOD$: Lines $AB$ and $CD$ intersect at $O$, lines $EP$ is a straight line. Wait, re - check: Wait, actually, let's look at the lines. Lines $AC$ (wait, $AB$ and $CD$ intersect at $O$, and $EP$ is a vertical line. Wait, $\angle EOB$ and $\angle AOP$: Wait, no, let's re - evaluate. Wait, the correct vertical angles: When two lines intersect, vertical angles are opposite. Wait, the lines are $AB$ (with points $A$, $O$, $B$) and $CD$ (with points $C$, $O$, $D$), and $EP$ (with $E$, $O$, $P$). Wait, $\angle EOB$ and $\angle AOP$: Let's see, $\angle EOB$: vertex $O$, sides $OE$ and $OB$. $\angle AOP$: vertex $O$, sides $OA$ and $OP$. Since $OE$ and $OP$ are opposite rays (straight line $EP$), and $OA$ and $OB$ are opposite rays (straight line $AB$), so $\angle EOB$ and $\angle AOP$ are vertical angles? Wait, no, wait the options: Wait the third option is $\angle EOB$ and $\angle AOP$. Wait, let's check each:

Wait, first, vertical angles are formed by two intersecting lines. So for two intersecting lines, say line 1 and line 2 intersecting at $O$, the vertical angles are the pairs opposite each other.

Looking at the figure:

• $\angle AOC$ and $\angle BOD$ would be vertical, but that's not an option. Wait, the options are:

1. $\angle AOC$ and $\angle COE$: share side $OC$, adjacent, not vertical.

1. $\angle COE$ and $\angle BOD$: not opposite.

1. $\angle EOB$ and $\angle AOP$: Let's see, line $AB$ ( $A - O - B$) and line $EP$ ( $E - O - P$) intersect at $O$. So the angles formed are $\angle EOB$ (between $OE$ and $OB$) and $\angle AOP$ (between $OA$ and $OP$). Since $OE$ and $OP$ are a straight line, $OA$ and $OB$ are a straight line, so these two angles are opposite each other (vertical angles).

1. $\angle COE$ and $\angle EOD$: share side $OE$, adjacent, not vertical.

So the correct pair is $\angle EOB$ and $\angle AOP$.

Answer:

$\angle EOB$ and $\angle AOP$ (the third option: $\boldsymbol{\angle EOB}$ and $\boldsymbol{\angle AOP}$)