Prove that a triangle must have atleast two acute angles
Solution:
Consider a triangle ABC
We have to prove that a triangle must have atleast two acute angles.
∠A + ∠B + ∠C = 180° ---------------- (1)
Case (1): consider ∠B = 90° and ∠C = 90°
From (1),
∠A + 90° + 90° = 180°
∠A + 180° = 180°
∠A = 180° - 180°
∠A = 0°
Therefore, no triangle can be formed.
Case (2): when two angles are obtuse
Consider ∠B and ∠C are greater than 90°
Let ∠B = 100° and ∠C = 95°
From (1),
∠A + 100° + 95° = 180°
∠A + 195° = 180°
∠A = 180° - 195°
∠A = - 15°
Angle A is negative, which is not possible.
Therefore, no triangle can be formed.
Case (3) : when one angle is 90 and other angle is obtuse
Let ∠B = 90° and ∠C = 100°
From (1),
∠A + 90° + 100° = 180°
∠A + 190° = 180°
∠A = 180° - 190°
∠A = -10°
Angle A is negative, which is not possible.
Therefore, no triangle can be formed.
Case (4) : when two angles are acute
Consider ∠B and ∠C are less than 90
Let ∠B = 80° and ∠C = 60°
∠A + 80° + 60° = 180°
∠A + 140° = 180°
∠A = 180° - 140°
∠A = 40°
Verification:
∠A + ∠B + ∠C = 180°
LHS : 40° + 80° + 60°
= 100° + 80°
= 180°
= RHS
Therefore, triangles can be formed.
✦ Try This: Prove that a triangle cannot have two right angles.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 6
NCERT Exemplar Class 9 Maths Exercise 6.4 Problem 6
Prove that a triangle must have atleast two acute angles
Summary:
It is proven that a triangle must have atleast two acute angles. Acute angle is an angle which measures less than 90 degrees. Obtuse angle is an angle which measures greater than 90 degrees
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