Question

An exterior angle of a triangle is 100 and its interior opposite angles are equal to each other. Find the measure of each angle of the triangle.

Answer 1

Hint: The Angle Sum Theorem tells us that the sum of the measures of the interior angles of a triangle is 180 degrees. We can prove this by using properties of parallel lines cut by transversals. This creates alternate interior angles that are congruent.
An exterior angle of a triangle is equal to the sum of the opposite interior angles. For more on this see Triangle external angle theorem.
Using the theorem that if the side of a triangle is produced then the interior angle is equal to the sum of two interior opposite angles of the triangle.

Complete Step-by-step Solution

Let be a triangle where the side BC is produced.
By using angle sum theorem, we have
$\mathrm{\angle }A+\mathrm{\angle }B={100}^{\circ }$…………………………………………………………… (1)
It is given that the interior opposite angles are equal i.e., $$\mathrm{\angle }A=\mathrm{\angle }B.$$
$$\therefore$$Equation (1) becomes
$$2\mathrm{\angle }A={100}^{\circ }$$
$$\mathrm{\angle }A={50}^{\circ }$$
$$\Rightarrow \mathrm{\angle }B={50}^{\circ }$$
Using angle sum property of a triangle,
Sum of all angles of a triangle =180o
$$\Rightarrow \mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C={180}^{\circ }$$
$$\Rightarrow 100+\mathrm{\angle }C={180}^{\circ }$$ ………………………………………………… [from (1)]
$$\Rightarrow \mathrm{\angle }C={180}^{\circ }-{110}^{\circ }$$
$$\Rightarrow \mathrm{\angle }C={80}^{\circ }$$
Hence $$\mathrm{\angle }A={50}^{\circ }$$
$$\mathrm{\angle }B={50}^{\circ }$$
$$\mathrm{\angle }C={80}^{\circ }$$

Note: Student should keep in mind that Angle sum property is very important, it describes to you that the sum of angles of a triangle is always equal to ${180}^{\circ }$.
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. (Non-adjacent interior angles may also be referred to as remote interior angles.)