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

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

Hence proved.

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