- A59
- B65
- C71
- D130
Step-by-Step Solution
Exterior angle = sum of two opposite interior angles. 120 = ∠ABC + ∠BAC = 55 + ∠BAC → ∠BAC = 65°.
Answer: B) 65
In ΔABC, side BC is extended to D. If ∠ABC = 55° and ∠ACD (exterior) = 120°, find ∠BAC.
Class 9Lines and AnglesGeometry — Lines and Angles
Key idea: Exterior angle theorem
**Exterior angle**: extend one side of a triangle past a vertex; the angle outside is the exterior angle.
**Theorem**: The exterior angle of a triangle equals the sum of the two **opposite interior** angles.
Proof: Let interior angles be ∠A, ∠B, ∠C. Extend BC to D forming exterior ∠ACD.
- ∠ACD + ∠ACB = 180° (linear pair)
- ∠A + ∠B + ∠C = 180° (angle sum)
- Subtract: ∠ACD = ∠A + ∠B. ✓
Useful corollary: an exterior angle is always GREATER than either non-adjacent interior angle (since they're positive).
**Theorem**: The exterior angle of a triangle equals the sum of the two **opposite interior** angles.
Proof: Let interior angles be ∠A, ∠B, ∠C. Extend BC to D forming exterior ∠ACD.
- ∠ACD + ∠ACB = 180° (linear pair)
- ∠A + ∠B + ∠C = 180° (angle sum)
- Subtract: ∠ACD = ∠A + ∠B. ✓
Useful corollary: an exterior angle is always GREATER than either non-adjacent interior angle (since they're positive).
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