Question

State true or false.
An exterior angle of a triangle is greater than the opposite interior angles of a triangle.

Answer 1

Hint:Here we use the theorem which states that the exterior angle of a triangle is equal to the sum of opposite interior angles of a triangle and check if the statement holds true or false.
Exterior angle of a triangle is the angle formed by extending the side of the triangle.

Complete step-by-step answer:
First we draw a diagram for the question

In \[\vartriangle ABC\] , exterior angles are the angles which are drawn outside the triangle.
Now, from the theorem we can say that the measure of exterior angle is always equal to the sum of opposite interior angles.
Let us denote the exterior angle by the extension \[ext\]
\[\angle A,\angle B\]are opposite interior angles to the exterior angle \[ext\angle C\]
\[\angle B,\angle C\] are opposite interior angles to the exterior angle \[ext\angle A\]
\[\angle C,\angle A\] are opposite interior angles to the exterior angle \[ext\angle B\]
So from the theorem we can say
\[\angle A + \angle B = ext\angle C\]
\[\angle B + \angle C = ext\angle A\]
\[\angle C + \angle A = ext\angle B\]
If the sum of two values on LHS of the equation equals to the value on RHS of the equation, then the value on RHS is greater than both the values taken on the LHS of the equation individually.
So, from the above equations,
\[\angle A,\angle B < ext\angle C\]
\[\angle B,\angle C < ext\angle A\]
\[\angle C,\angle A < ext\angle B\]
Therefore, an exterior angle of a triangle is always greater than the opposite interior angles of that particular exterior angle.
So, the statement is TRUE.

Note:Students usually make mistake when they are confused about which angle is exterior angle and which are the opposite interior angles to that exterior angles, so keep in mind that opposite interior angles to an exterior angles are those angles which are not on the same line with the exterior angle or which are not supplementary to the exterior angles.