Exterior Angle Theorem: Properties, Proof & Examples

Exterior angle theorem states that, if the side of a triangle is produced further, the exterior angle formed is equal to the sum of the opposite two interior angles. Exterior angle theorem defines the relationship between the exterior angles and interior angles and can be proved easily with the help of angle sum property of the triangle. 

Key takeaways: Exterior angle property of triangle, Angle sum property of triangle, Linear angle sum property.

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Exterior Angle Theorem

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Exterior angle property of triangle states that if the side of the angle is produced, the exterior angle of the triangle formed is equal to the sum of the opposite 2 interior angles.

Exterior Angle Theorem

Let’s look at the proof of this theorem. 

Read More: Surface areas

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Proof

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The following theorem can be proved in two alternative ways. 

To Prove

• Angle sum property of triangle: The sum of all the interior angles of a triangle is equal to 180°.
• The alternate interior angles are equal in case of parallel lines.
• The corresponding angles of a transverse are equal in case of parallel lines.

Given: A triangle ABC with side CB extended till D. The exterior angle formed is ∠ABD. ∠BAC and ∠ACB are two opposite interior angles. 

To prove: ∠ABD=∠BAC+∠BCA

Figure-1

Proof - 1

ABC is a triangle. (see figure-1)

According to the angle sum property of triangle, 

∠ABC + ∠BCA + ∠BAC = 180°

I.e- ∠ABC=180°- (∠BCA+ ∠BAC)….. (a)

Also, by linear pair, ∠ABC+∠ABD=180°…..(b)

From (a) and (b), 

180°- (∠BCA+ ∠BAC) + ∠ABD=180°

180°- (∠BCA+ ∠BAC) + ∠ABD=180° 

I.e- ∠BCA + ∠BAC = ∠ABD

Hence the exterior angle theorem is proved.

Proof - 2 

A triangle ABC with line CB extended will D. 

BE is parallel to CA and AB is the transverse. (see figure-2)

∠1 = ∠3 {alternate angles, as CA parallel to BE and AB is the transverse}... (A)

∠2 = ∠4 {corresponding angles as CA parallel to BE and CD is the transverse}...(B)

Figure-2

Adding (A) and (B)

∠1 + ∠2 = ∠3 + ∠4

I.e- ∠BAC + ∠BCA = ∠ABD

Hence the exterior angle theorem of the triangle is proved.

Also Read:

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Things to Remember

• The sum of all the interior angles of a triangle is equal to 180°.
• The exterior angle formed is equal to the sum of two opposite interior angles.
• The exterior angle formed is always greater than either of the two interior angles.
• The sum of all exterior angles of a triangle is 360°.
• Vertically opposite angles are equal.

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