Triangles, Exterior Angle Theorem

Exterior angle theorem is one of the most basic 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 which is bounded by finite number of line segments to form a closed figure.  The smallest polygon is a triangle as three line segments bound it. Triangle is the smallest polygon bounded by three line segments. It has three edges and three vertices. The figure 1 given 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.

Exterior Angle

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

The exterior angle theorem states that if a triangle’s side gets an extension, then the resultant exterior angle would be equal to the total of the two opposite interior angles of the triangle.

Exterior Angle

Fig. 2 Exterior Angle Theorem

According to the Exterior Angle 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 side BC of ∆ABC is extended. A line, parallel to the side AB is drawn as shown in the figure.

Exterior Angle

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 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.

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Practise This Question

Kushagra is drawing a railway track on paper as a part of his project. He asked Sarosh to draw a line parallel to the given line. Sarosh said that we can only construct a line parallel to the given line using alternate angles concept. Is this true?