Exterior Angle Inequality

Definition: The measure of a triangle’s outside angle is greater than the measure of its opposite interior angles.

Angle d is greater than Angle a and Angle c
∠d   >   ∠a
∠d   >   ∠c

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Example: If ∠1 = 50° and ∠2 = 20°, which of the following is a possible measure of ∠4 according to the Exterior Angle Inequality Theorem?

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a) 45°          b) 27°          c) 70°          d) 50°

Solution: 

The Exterior Angle Inequality Theorem tells us that a triangle’s outside angle (∠4) is always greater than it’s opposite interior angles (∠1 and ∠2). We can use this information to choose the best option.

Step 1: Use the given information and the exterior angle inequality theorem to choose the best answer from the given options.

We know ∠1 = 50° and ∠2 = 20°, and we know the exterior angle must be greater than both of these angles.

Step 2: Based on this information, we can determine Option C, 70° to be the best answer, because 70° degrees is greater than both 50° and 20°.

Step 3: Use the Exterior Angle Theorem to check the answer. This theorem tells us that the outside angle of a triangle is equal to the sum of the opposite interior angles. We can write the following equation using this theorem:

∠1 + ∠2 = ∠4
50° + 20° = 70°

Option C is the best answer.

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