Question

select all the pairs of vertical angles.
∠lkn and ∠jki
∠onk and ∠mnp
∠lki and ∠jkn
∠jki and ∠mnk

Explanation:

Step1: Recall Vertical Angles Definition

Vertical angles are formed by two intersecting lines and are opposite each other, sharing a common vertex. They are equal in measure.

Step2: Analyze Each Pair

• ∠LKN and ∠JKI: Check if they are opposite when lines intersect. Lines \( LJ \) and \( PI \) intersect at \( K \). \( \angle LKN \) and \( \angle JKI \) are opposite (vertical) angles.
• ∠ONK and ∠MNP: Lines \( MO \) and \( PI \) intersect at \( N \). \( \angle ONK \) and \( \angle MNP \) are opposite (vertical) angles.
• ∠LKI and ∠JKN: Lines \( LJ \) and \( PI \) intersect at \( K \). \( \angle LKI \) (straight angle - \( \angle LKN + \angle NKI \)) and \( \angle JKN \) (straight angle - \( \angle JKL + \angle LKN \))? Wait, no—actually, when \( LJ \) and \( PI \) intersect at \( K \), \( \angle LKI \) and \( \angle JKN \) are vertical? Wait, \( \angle LKI \) is adjacent to \( \angle LKN \), and \( \angle JKN \) is adjacent to \( \angle JKI \). Wait, no, let's re - check. The two lines intersecting at \( K \) are \( LJ \) and \( PI \). So the vertical angle pairs at \( K \) should be \( \angle LKN \) & \( \angle JKI \), and \( \angle LKI \) & \( \angle JKN \) (since \( \angle LKI + \angle LKN = 180^\circ \), \( \angle JKN + \angle JKI = 180^\circ \), and \( \angle LKN=\angle JKI \), so \( \angle LKI=\angle JKN \)). Wait, maybe I made a mistake earlier. Let's correct: When two lines intersect, there are two pairs of vertical angles. So at \( K \), lines \( LJ \) and \( PI \) intersect, so \( \angle LKN \) and \( \angle JKI \) (vertical), and \( \angle LKI \) and \( \angle JKN \) (vertical). At \( N \), lines \( MO \) and \( PI \) intersect, so \( \angle ONK \) and \( \angle MNP \) (vertical).
• ∠JKI and ∠MNK: \( \angle JKI \) is at \( K \), \( \angle MNK \) is at \( N \). Different vertices, so not vertical angles.

Answer:

The pairs of vertical angles are \( \angle LKN \) and \( \angle JKI \), \( \angle ONK \) and \( \angle MNP \), \( \angle LKI \) and \( \angle JKN \).