Question

chapter 2 lines, angles, and quadrilaterals
in the diagram below, name the second angle in each pair of vertical angles.

1. $\angle ypv = \underline{quadquad}$ 4. $\angle vpt = \underline{quadquad}$ 7. $\angle mln = \underline{quadquad}$ 10. $\angle glm = \underline{quadquad}$
2. $\angle qpr = \underline{quadquad}$ 5. $\angle rpt = \underline{quadquad}$ 8. $\angle klh = \underline{quadquad}$ 11. $\angle klm = \underline{quadquad}$
3. $\angle spt = \underline{quadquad}$ 6. $\angle vps = \underline{quadquad}$ 9. $\angle gln = \underline{quadquad}$ 12. $\angle hlg = \underline{quadquad}$

Explanation:

Step1: Recall vertical angles definition

Vertical angles are opposite, congruent angles formed by intersecting lines.

Step2: Match each angle to its vertical pair

For each given angle, identify the angle directly opposite it at the intersection point.

1. $\angle YPV$: Opposite is $\angle RPS$
2. $\angle QPR$: Opposite is $\angle TPV$
3. $\angle SPT$: Opposite is $\angle YPQ$
4. $\angle VPT$: Opposite is $\angle QPS$
5. $\angle RPT$: Opposite is $\angle YPQ$ (correction: $\angle RPT$ is opposite $\angle YPQ$ → no, $\angle RPT$ opposite is $\angle YPQ$? No, $\angle RPT$ intersects at P: $\angle RPT$ opposite is $\angle YPQ$ → correct. Wait, $\angle RPT$: lines YS and QT intersect at P? No, lines VR and QT intersect at P. $\angle RPT$ opposite is $\angle YPQ$.

Wait, correct pairs:

1. $\angle YPV$ (lines YS and VR intersect at P) → opposite $\angle RPS$
2. $\angle QPR$ (lines QT and VR intersect at P) → opposite $\angle TPV$
3. $\angle SPT$ (lines YS and QT intersect at P) → opposite $\angle YPQ$
4. $\angle VPT$ (lines VR and QT intersect at P) → opposite $\angle QPS$
5. $\angle RPT$ (lines VR and QT intersect at P) → opposite $\angle YPQ$
6. $\angle VPS$ (lines VR and YS intersect at P) → opposite $\angle RPY$

For the right diagram (intersection at L):

1. $\angle MLN$ (lines MG and JG intersect at L) → opposite $\angle KLH$
2. $\angle KLH$ (lines KJ and MH intersect at L) → opposite $\angle MLN$
3. $\angle GLN$ (lines JG and MH intersect at L) → opposite $\angle JLM$
4. $\angle GLM$ (lines MG and JG intersect at L) → opposite $\angle JLK$
5. $\angle KLM$ (lines KJ and MG intersect at L) → opposite $\angle HLG$
6. $\angle HLG$ (lines MH and JG intersect at L) → opposite $\angle KLM$

Wait, correcting all pairs properly:

1. $\angle YPV$: vertical angle is $\angle RPS$
2. $\angle QPR$: vertical angle is $\angle TPV$
3. $\angle SPT$: vertical angle is $\angle YPQ$
4. $\angle VPT$: vertical angle is $\angle QPS$
5. $\angle RPT$: vertical angle is $\angle YPQ$ (no, $\angle RPT$ is formed by PR and PT, so opposite is $\angle YPQ$ (YP and PQ) → yes.
6. $\angle VPS$: vertical angle is $\angle RPY$
7. $\angle MLN$: vertical angle is $\angle KLH$
8. $\angle KLH$: vertical angle is $\angle MLN$
9. $\angle GLN$: vertical angle is $\angle JLM$
10. $\angle GLM$: vertical angle is $\angle JLK$
11. $\angle KLM$: vertical angle is $\angle HLG$
12. $\angle HLG$: vertical angle is $\angle KLM$

Answer:

1. $\boldsymbol{\angle RPS}$
2. $\boldsymbol{\angle TPV}$
3. $\boldsymbol{\angle YPQ}$
4. $\boldsymbol{\angle QPS}$
5. $\boldsymbol{\angle YPQ}$
6. $\boldsymbol{\angle RPY}$
7. $\boldsymbol{\angle KLH}$
8. $\boldsymbol{\angle MLN}$
9. $\boldsymbol{\angle JLM}$
10. $\boldsymbol{\angle JLK}$
11. $\boldsymbol{\angle HLG}$
12. $\boldsymbol{\angle KLM}$