Question

A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options. m∠5 + m∠3 = m∠4 m∠3 + m∠4 + m∠5 = 180° m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6 m∠2 + m∠3 + m∠5 = 180°

Answer 1

The three options are

• m∠5 + m∠6 = 180° Linear Pair
• ∠ 2+ ∠ 3 = ∠ 6 Exterior angle Property of Triangle
• m∠2 + m∠3 + m∠5 = 180° Triangle Sum Property

Consider Δ ABC with Exterior angles as

∠ 1 , ∠ 4 ,and ∠ 6

Exterior angle Property of Triangle:

An exterior angle of a triangle is equal to the sum of the opposite interior angles.

For Exterior ∠ 1 we have

∠ 1 = ∠ 5 + ∠ 3 Exterior angle Property of Triangle

Similarly,

For Exterior ∠ 4 we have

∠ 4 = ∠ 5 + ∠ 2 Exterior angle Property of Triangle

Similarly,

For Exterior ∠ 6 we have

∠ 6 = ∠ 2 + ∠ 3 Exterior angle Property of Triangle

Triangle Sum Property:

In a triangle sum of the measures of angles is equal to 180°

∴ m∠2 + m∠3 + m∠5 = 180° Triangle Sum Property

Linear Pair:

The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.

∴ m∠5 + m∠6 = 180° Linear Pair