Take Note Corollary: Corollary to the Triangle Exterior Angle Theorem

Corollary

The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

If …

∠ 1 angle 1  is an exterior angle

Then …

m ∠ 1 > m ∠ 2 m angle , 1 greater than , m angle 2  and m ∠ 1 > m ∠ 3 m angle , 1 greater than , m angle 3

Proof

Proof of the Corollary

Given: ∠ 1 angle 1  is an exterior angle of the triangle.

Prove: m ∠ 1 > m ∠ 2 m angle , 1 greater than , m angle 2  and m ∠ 1 > m ∠ 3 . m angle , 1 greater than , m angle 3 .

Proof: By the Triangle Exterior Angle Theorem, m ∠ 1 = m ∠ 2 + m ∠ 3 . m angle , 1 equals , m angle , 2 plus , m angle 3 .  Since m ∠ 2 > 0 m angle , 2 greater than 0  and m ∠ 3 > 0 , m angle , 3 greater than 0 , comma  you can apply the Comparison Property of Inequality and conclude that m ∠ 1 > m ∠ 2 m angle , 1 greater than , m angle 2  and m ∠ 1 > m ∠ 3 . m angle , 1 greater than , m angle 3 .

Problem 1 Applying the Corollary

Use the figure below. Why is m ∠ 2 > m ∠ 3 ? bold italic m angle 2 greater than bold italic m angle 3 question mark

Image Long Description

In Δ A C D , C B ¯ ≅ C D ¯ , cap delta eh c d comma , c b bar , approximately equal to . c d bar , comma  so by the Isosceles Triangle Theorem, m ∠ 1 = m ∠ 2 . ∠ 1 m angle , 1 equals , m angle 2 . angle 1  is an exterior angle of Δ A B D , cap delta eh b d comma  so by the Corollary to the Triangle Exterior Angle Theorem, m ∠ 1 > m ∠ 3 . m angle , 1 greater than , m angle 3 .  Then m ∠ 2 > m ∠ 3 m angle , 2 greater than , m angle 3  by substitution.

✓ Got It?

1. Why is m ∠ 5 > m ∠ C ? m angle , 5 greater than , m angle c question mark

Take Note Theorem 5-10

Theorem

If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

If …

X Z > X Y x z greater than x y

Then …

m ∠ Y > m ∠ Z m angle , y greater than , m angle z

You will prove Theorem 5-10 in Exercise 40.