Exterior Angle Theorem - Definition, Proof, Examples - Cuemath

Exterior Angle Theorem

The exterior angle theorem states that when a triangle's side is extended, the resultant exterior angle formed is equal to the sum of the measures of the two opposite interior angles of the triangle. The theorem can be used to find the measure of an unknown angle in a triangle. To apply the theorem, we first need to identify the exterior angle and then the associated two remote interior angles of the triangle.

1.	What is Exterior Angle Theorem?
2.	Proof of Exterior Angle Theorem
3.	Exterior Angle Inequality Theorem
4.	FAQs on Exterior Angle Theorem

What is Exterior Angle Theorem?

The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two opposite(remote) interior angles of the triangle. Let us recall a few common properties about the angles of a triangle: A triangle has 3 internal angles which always sum up to 180 degrees. It has 6 exterior angles and this theorem gets applied to each of the exterior angles. Note that an exterior angle is supplementary to its adjacent interior angle as they form a linear pair of angles. Exterior angles are defined as the angles formed between the side of the polygon and the extended adjacent side of the polygon.

We can verify the exterior angle theorem with the known properties of a triangle. Consider a Δ ABC.

The three angles a + b + c = 180 (angle sum property of a triangle) ----- Equation 1

c= 180 - (a+b) ----- Equation 2 (rewriting equation 1)

e = 180 - c----- Equation 3 (linear pair of angles)

Substituting the value of c in equation 3, we get

e = 180 - [180 - (a + b)]

e = 180 - 180 + (a + b)

e = a + b

Hence verified.

Proof of Exterior Angle Theorem

Consider a ΔABC. a, b and c are the angles formed. Extend the side BC to D. Now an exterior angle ∠ACD is formed. Draw a line CE parallel to AB. Now x and y are the angles formed, where, ∠ACD = ∠x + ∠y

Statement	Reason
∠a = ∠x	Pair of alternate angles. (Since BA is parallel to CE and AC is the transversal).
∠b = ∠y	Pair of corresponding angles. (Since BA is parallel to CE and BD is the transversal).
∠a + ∠b = ∠x + ∠y	From the above statements
∠ACD = ∠x + ∠y	From the construction of CE
∠a + ∠b = ∠ACD	From the above statements

Hence proved that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Exterior Angle Inequality Theorem

The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles. This condition is satisfied by all the six external angles of a triangle.

Related Articles

Check out a few interesting articles related to Exterior Angle Theorem.

• Exterior Angle Formula
• Exterior Angle Theorem Worksheets
• Alternate Exterior Angles
• How to find the measure of each exterior angle of a regular pentagon?
• Properties of Triangle
• Interior and Exterior Angles Worksheets
• Sum of Exterior Angles Formula

Important notes

• The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle.
• The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than either of the opposite interior angles.
• The exterior angle and the adjacent interior angle are supplementary. All the exterior angles of a triangle sum up to 360º.