Question

If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled.

Answer 1

ANSWER:
Let ABC be the triangle.
Let ∠A<(∠B+∠C)
Then,
2∠A<(∠A+∠B+∠C )(Adding ∠A to both sides)

⇒2∠A<180° (∵∠A+∠B+∠C =180°)

⇒∠A<90°
Also, let ∠B<(∠A+∠C)
Then,
2∠B<(∠A+∠B+∠C ) (Adding ∠B to both sides)

⇒2∠B<180° ( ∵∠A+∠B+∠C =180°)

⇒∠B<90°
And let ∠C<(∠A+∠B)
Then,
2∠C<(∠A+∠B+∠C) ( Adding ∠C to both sides)

⇒2∠C<180° (∵∠A+∠B+∠C =180°)

⇒∠C<90°
Hence, each angle of the triangle is less than 90°.
Therefore, the triangle is acute-angled.