Question

The exterior angles of a triangle are (2x+10),(3x+15)and (4x+20). What is the value of x and what is the largest interior angle of the triangle?

Answer 1

x = 35 and 100°

Explanation:

the sum of the exterior angles of a triangle = 360° , then

2x + 10 + 3x + 15 + 4x + 20 = 360 , that is

9x + 45 = 360 ( subtract 45 from both sides )

9x = 315 ( divide both sides by 9 )

x = 35

then each exterior angle is

2x + 10 = 2(35) + 10 = 70 + 10 = 80°

3x + 15 = 3(35) + 15 = 105 + 15 = 120°

4x + 20 = 4(35) + 20 = 140 + 20 = 160°

the sum of an exterior angle and corresponding interior angle = 180°

then

180° - 80° = 100°

180° - 120° = 60°

180° - 160° = 20°

the largest interior angle of the triangle is 100°

Answer 2

x = 35

largest interior angle = 100°

Explanation:

The exterior angles of a triangle sum to 360°

⇒ (2x + 10) + (3x + 15) + (4x + 20) = 360

⇒ 2x + 10 + 3x + 15 + 4x + 20 = 360

⇒ 9x + 45 = 360

⇒ 9x = 315

⇒ x = 35

Therefore, the three exterior angles are:

• (2x + 10) = 2(35) + 10 = 80°
• (3x + 15) = 3(35) + 15 = 120°
• (4x + 20) = 4(35) + 20 = 160°

If the side of a triangle is extended, the angle formed outside the triangle is the exterior angle. As angles on a straight line sum to 180°:

⇒ interior angle + exterior = 180°

So the largest interior angle will be the supplementary angle to the smallest exterior angle.

The smallest exterior angle is 80°, so:

⇒ largest interior angle + 80° = 180°

⇒ largest interior angle = 180° - 80°

⇒ largest interior angle = 100°