An obtuse triangle is a triangle in which one of the angles is larger than 90°. There is only one such angle possible, since the sum of angles in a triangle is 180°.

It’s easy to show that in an obtuse triangle, the obtuse angle is larger than the sum of the other two angles, and also the converse- that if an angle in a triangle is larger than the sum of the other two, then the triangle is obtuse. Both of these follow directly from the fact that the sum of angles in a triangle is 180°.

## Problem

Show that in an obtuse triangle, the obtuse angle is larger than the sum of the other two angles, and the converse- that if an angle in a triangle is larger than the sum of the other two, then the triangle is obtuse.

## Strategy

Both of these are direct consequence of the fact that the sum of all three angles in a triangle is exactly 180° , so we will prove them using arithmetic manipulation of the sum of the angles.

## Proof

(1) m∠1 + m∠2 + m∠3= 180° //sum of angles in a triangle is 180°

(2) m∠2>90° // Given, ΔABC is obtuse

(3) m∠1 + m∠3 < 90° // subtract (2) from (1)

(4) m∠1 + m∠3 < m∠2 //(2), (3)

Now lets do the converse, show that if m∠1 + m∠3 < m∠2, then ΔABC is obtuse:

(5) m∠1 + m∠2 + m∠3= 180° //sum of angles in a triangle is 180°

(6) m∠1 + m∠3 < m∠2 // Given

(7) m∠2 >180° -m∠2 // subtract (6) from (5)

(8) 2 * m∠2 > 180° //Add m∠2 to both sides

(9) m∠2 > 90° //divide by 2

(10) ΔABC is obtuse //(9) , defintion of obtuse triangle