- A17°
- B127°
- C53°
- D55°
Step-by-Step Solution
Linear pairs give and . So .
Answer: C) 53°
Side BC of triangle ABC is produced both ways, to E and to D. The exterior angles are ∠ABE = 108° and ∠ACD = 125°. 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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