# Exterior Angle of a Triangle and Its Property

#### description

• An exterior angle of a triangle is equal to the sum of its interior opposite angles.
• The sum of exterior angles of a triangle is 360°.

#### definition

• Exterior angle: On extending one side of a triangle, the angle obtained which forms a linear pair with the adjacent interior angle of the triangle is called an exterior angle of that triangle.

# Exterior angle of a triangle Property:

• An exterior angle of a triangle is formed when a side of a triangle is produced.

• At each vertex, you have two ways of forming an exterior angle.

• Observe the angle ACD formed at point C. This angle lies in the exterior of ∆ABC. We call it an exterior angle of the ∆ABC formed at vertex C.
• Clearly ∠BCA is an adjacent angle to ∠ACD. The remaining two angles of the triangle namely ∠A and ∠B are called the two interior opposite angles or the two remote interior angles of ∠ACD.
• m∠ACD = m ∠A + m ∠B
• In the figure alongside, all exterior angles of a triangle are shown. a, b, c, d, e, f are the exterior angles of ∆PQR. In the same way, every triangle has six exterior angles.

# A property of exterior angles:

The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.

Given: Consider △ ABC. ∠ ACD is an exterior angle.

To Show: m∠ ACD = m∠ A + m∠ B. Through C draw bar"CE", parallel to bar"BA".

### Justification:

 Steps Reasons (a) ∠1 = ∠x bar"BA" || bar"CE" and bar"AC" is a transversal. Therefore, alternate angles should be equal. (b) ∠2 = ∠y bar"BA" || bar"CE" and bar"BD" is a transversal.Therefore, the corresponding angles should be equal. (c) ∠ 1 + ∠ 2 = ∠ x + ∠ y (d) Now, ∠ x + ∠ y = m∠ ACD From given Figure Hence, ∠ 1 + ∠ 2 = ∠ ACD

Thus, The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.

#### Example

Find angle x in Fig

Sum of interior opposite angles = Exterior angle
or     50° + x = 110°
or     x = 60°

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