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 pairs of non-adjacent angles formed by two intersecting lines, they are congruent and opposite each other.

Step2: Match each angle to its vertical pair (Diagram 1, point P)

1. $\angle YPV$ is opposite $\angle RPS$
2. $\angle QPR$ is opposite $\angle VPT$ (correction: $\angle QPR$ is opposite $\angle SPV$? No, correct: $\angle QPR$ ↔ $\angle TPV$? No, recheck: Lines Y-S and Q-T intersect at P, Lines V-R and Q-T intersect at P.

• $\angle YPV$ (formed by Y-P and V-P) ↔ $\angle RPS$ (R-P and S-P)
• $\angle QPR$ (Q-P and R-P) ↔ $\angle TPV$ (T-P and V-P)
• $\angle SPT$ (S-P and T-P) ↔ $\angle YPQ$ (Y-P and Q-P)
• $\angle VPT$ (V-P and T-P) ↔ $\angle QPR$ (Q-P and R-P)
• $\angle RPT$ (R-P and T-P) ↔ $\angle VPQ$ (V-P and Q-P)
• $\angle VPS$ (V-P and S-P) ↔ $\angle YPR$ (Y-P and R-P)

Step3: Match each angle to its vertical pair (Diagram 2, point L)

1. $\angle MLN$ (M-L and N-L) ↔ $\angle KLH$ (K-L and H-L)
2. $\angle KLH$ (K-L and H-L) ↔ $\angle MLN$ (M-L and N-L)
3. $\angle GLN$ (G-L and N-L) ↔ $\angle JLM$ (J-L and M-L)
4. $\angle GLM$ (G-L and M-L) ↔ $\angle JLN$ (J-L and N-L)
5. $\angle KLM$ (K-L and M-L) ↔ $\angle HLG$ (H-L and G-L)
6. $\angle HLG$ (H-L and G-L) ↔ $\angle KLM$ (K-L and M-L)

Answer:

1. $\boldsymbol{\angle RPS}$
2. $\boldsymbol{\angle TPV}$
3. $\boldsymbol{\angle YPQ}$
4. $\boldsymbol{\angle QPR}$
5. $\boldsymbol{\angle VPQ}$
6. $\boldsymbol{\angle YPR}$
7. $\boldsymbol{\angle KLH}$
8. $\boldsymbol{\angle MLN}$
9. $\boldsymbol{\angle JLM}$
10. $\boldsymbol{\angle JLN}$
11. $\boldsymbol{\angle HLG}$
12. $\boldsymbol{\angle KLM}$