Exterior angle inequality

key notes :

Definition of Exterior Angle:

• An exterior angle of a triangle is formed when a side of the triangle is extended.
• It is the angle between the extended side and the adjacent side of the triangle.

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Exterior Angle Inequality Theorem:

• The Exterior Angle Inequality Theorem states that the measure of an exterior angle of a triangle is greater than the measure of either of the two remote interior angles.
• Mathematically, if ∠ABC is an exterior angle of triangle ABC, then

• where ∠ACB and ∠CAB are the remote interior angles.

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Properties:

• The exterior angle is always greater than the remote interior angles but not equal to the sum of the interior angles.
• This inequality helps in solving for unknown angles in triangles, especially when dealing with exterior angles.

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Application of Exterior Angle Inequality:

• Solving for angles: This inequality can be used to find missing angles in various geometric problems.
• Understanding the relationship between angles in a triangle: It helps students understand that exterior angles are larger than the interior angles opposite them.

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Example:

• In triangle ABC, if ∠ABC is an exterior angle and the remote interior angles are ∠ACB = 40° and ∠CAB = 50°, then by the exterior angle inequality, we know that:

Therefore, ∠ABC must be greater than both 40° and 50°.

Learn with an example

Let’s practice!