- A64°
- B58°
- C122°
- D90°
Step-by-Step Solution
. So and , giving . The bisectors of a linear pair are always perpendicular.
Answer: D) 90°
AOB is a straight line and ∠AOC = 64°. The blue ray OD bisects ∠AOC and the blue ray OE bisects ∠BOC. Find the orange angle = ∠DOE.
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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