# Exterior Angles of a Triangle

If the side (s) of a triangle is produced to form a ray, then the angle formed is called exterior angle of the triangle.

**Example :** When side BC of Î” ABC is produced to form ray BD, then âˆ ACD is called exterior angle of Î” ABC at C and is denoted by ext. âˆ ACD. With respect to ext. âˆ ACD of Î” ABC at C, the angles A and B are called remote interior angles or interior opposite angles.

Now, if we produce AC to form a ray AE, then âˆ BCE will be the exterior angle of Î” ABC at C. Thus at each vertex of a triangle, there are two exterior angles of the triangle and these angles will be equal because they are vertically opposite angles.

**Theorem 9**

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

**Given: ** In Î” ABC, BC is produced to form an angle âˆ ACD,

**To Prove: ** âˆ ACD = âˆ BAC + âˆ ABC i.e. âˆ 4 = âˆ 1 + âˆ 2

**Proof:** In Î” ABC, we have

âˆ 1 + âˆ 2 + âˆ 3 = 180Â° (i)

Also âˆ 3 + âˆ 4 = 180Â° (ii) (linear pair)

From equation (i) and equation (ii), we get

âˆ 1 + âˆ 2 + âˆ 3 = âˆ 3 + âˆ 4

âˆ´ âˆ 4 = âˆ 1 + âˆ 2

[Note: Since âˆ 4 is the sum of âˆ 1 and âˆ 2, it is clear that âˆ 4 > âˆ 2 and âˆ 4 > âˆ 1. Such a result, which is an easy consequence of a theorem, is called corollary of that theorem. Now we may state that:

**Corollary**

An exterior angle of a triangle is greater than either of the interior opposite angles. This is also called as the Exterior Angle Theorem.

An exterior angle of a triangle is 120Â°, and one of the interior opposite angles is 35Â°. Find the other two angles of the triangle.

Let ABC be a triangle whose BC side is produced to form exterior âˆ ACD.

âˆ´ âˆ ACD = 120Â° and âˆ ABC = 35Â°

Now by exterior angle theorem, we have

âˆ ACD = âˆ ABC + âˆ BAC

âˆ´ 120Â° = 35Â° + âˆ BAC

âˆ´ âˆ BAC = 120Â° - 35Â° = 85Â°

Now in Î” ABC, we have

âˆ A + âˆ B + âˆ ACB = 180Â°

âˆ´ 85Â° + 35Â° + âˆ ACB = 180Â°

âˆ´ âˆ ACB = 180Â° - 85Â° - 35Â° = 60Â°

Hence two other angles are 85Â° and 60Â°.