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.
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.
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:
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.
|1.||∠CAB = ∠ACE
|Pair of alternate angles(BA || CE) and (AC) is the transversal)|
|2.||∠ABC = ∠ECD
⇒∠2 = ∠y
|Corresponding angles (BA) ||(CE) and (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 the exterior ∠ACD of ∆ABC is equal to the sum of two opposite interior angles i.e. ∠CAB and ∠ABC of the ∆ABC.
To know more about triangles and the properties of triangles, download BYJU’S-The Learning App from Google Play Store.
Frequently Asked Questions – FAQs
What is the exterior angle theorem formula?
The measure of exterior angle = Sum of two opposite interior angles’ measure