# Triangles, Exterior Angle Theorem

The exterior angle theorem is one of the most fundamental theorems of triangles. Before we begin the discussion, let us have a look at what a triangle is. A polygon is defined as a plane figure bounded by a finite number of line segments to form a closed figure. Triangle is the polygon bounded by a least number of line segments, i.e. three. It has three edges and three vertices. Figure 1 below represents a triangle with three sides AB, BC, CA, and three vertices A, B and C. ∠ABC, ∠BCA and ∠CAB are the three interior angles of ∆ABC.

Fig. 1 Triangle ABC

One of the basic theorems explaining the properties of a triangle is the exterior angle theorem. Let us discuss this theorem in detail.

## Exterior Angle Theorem

Statement: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Fig. 2 Exterior Angle Theorem

The above statement can be explained using the figure provided as:

According to the Exterior Angle property of a triangle theorem, the sum of measures of ∠ABC and ∠CAB would be equal to the exterior angle ∠ACD.

General proof of this theorem is explained below:

Proof:

Consider a ∆ABC as shown in fig. 2, such that the side BC of ∆ABC is extended. A line, parallel to the side AB is drawn as shown in the figure.

Fig. 3 Exterior Angle Theorem

 S. No Statement Reason 1. ∠CAB = ∠ACE ⇒∠1=∠x Pair of alternate angles(($$\overline{BA}$$) ||($$\overline{CE}$$) and ($$\overline{AC}$$) is the transversal) 2. ∠ABC = ∠ECD ⇒∠2 = ∠y Corresponding angles (($$\overline{BA}$$) ||($$\overline{CE}$$) and ($$\overline{BD}$$)) is the transversal) 3. ⇒∠1+∠2 = ∠x+∠y From statements 1 and 2 4. ∠x+∠y = ∠ACD From fig. 3 5. ∠1+∠2 = ∠ACD From statements 3 and 4

Thus, from the above statements, it can be seen that exterior ∠ACD of ∆ABC is equal to the sum of two opposite interior angles i.e. ∠CAB and ∠ABC of the ∆ABC.

Hence proved.

## Frequently Asked Questions – FAQs

### What is the exterior angle theorem formula?

The below formulas can be stated from the exterior angle theorem.
The measure of exterior angle = Sum of two opposite interior angles’ measure

### What is the exterior angle Inequality Theorem?

According to the exterior angle inequality theorem, the measure of an exterior angle of a triangle is greater than either of its interior opposite angles.

### What is the exterior angle property?

If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

### Do all polygon exterior angles add up to 360?

Yes, the sum of exterior angles in a polygon is always added up to 360 degrees.

### What is the sum of the measures of the exterior angles of any triangle?

The sum of the measures of the exterior angles of any triangle is 360 degrees.

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