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∠P and ∠Q are vertically opposite angles, and ∠Q and ∠R form a linear pair. If ∠P = 126°, find ∠R.

Class 7Lines and AnglesLines and Angles
  • A126°
  • B36°
  • C54°
  • D234°

Step-by-Step Solution

Vertically opposite angles are equal, so ∠Q = ∠P = 126°. A linear pair is supplementary, so ∠R = 180 − 126 = 54°. Trap: 126° is what you get by carrying the equality one step too far — ∠R is a linear-pair partner, not another vertical opposite; 36° is 90 − 54, the complement rather than the supplement.
Answer: C) 54°

Key idea: Vertically opposite angles

When two straight lines cross, four angles form at the intersection. The two angles directly **across** the vertex (sharing only the vertex, not a side) are called **vertically opposite** — and they are always equal.

If one angle is 70°, then:
- Its vertical opposite is also 70°.
- The other two angles are each 110° (linear pair with 70°).

Why? Each pair forms a linear pair with a common neighbour, both summing to 180° — so the opposites end up equal.

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