- A90
- B180
- C81
- D99
Step-by-Step Solution
If the linear-pair angles are a and , the half-angles are and . Their sum (the angle between bisectors) = °.
Answer: A) 90
Bisectors of two angles forming a linear pair meet at the vertex. What is the angle between the two bisectors?
Class 9Lines and AnglesGeometry — Lines and Angles
Key idea: Linear pair and vertically opposite angles
When two straight lines intersect, four angles are formed. They satisfy two key properties:
**Linear Pair Axiom**: If a ray stands on a line, the sum of the two adjacent angles formed is **180°**.
**Vertically Opposite Angles Theorem**: When two lines cross, the angles directly opposite each other are equal.
**Quick proof of vertically opposite angles**:
Let two lines intersect, forming angles ∠1, ∠2, ∠3, ∠4 (in order around the point).
- ∠1 + ∠2 = 180° (linear pair)
- ∠2 + ∠3 = 180° (linear pair)
- So ∠1 = ∠3 (subtract ∠2 from both). ✓
**Linear Pair Axiom**: If a ray stands on a line, the sum of the two adjacent angles formed is **180°**.
**Vertically Opposite Angles Theorem**: When two lines cross, the angles directly opposite each other are equal.
**Quick proof of vertically opposite angles**:
Let two lines intersect, forming angles ∠1, ∠2, ∠3, ∠4 (in order around the point).
- ∠1 + ∠2 = 180° (linear pair)
- ∠2 + ∠3 = 180° (linear pair)
- So ∠1 = ∠3 (subtract ∠2 from both). ✓
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