Question

The vertex angle of an isosceles triangle measures $${90}^{\circ }$$. Find the measure of the base angle.

Answer 1

Hint: Consider an isosceles triangle $$\mathrm{\Delta }ABC$$ with $$\mathrm{\angle }A$$ being the vertex angle. Use the fact that the angles opposite to equal sides are equal to prove that other two angles of isosceles angles are equal as their two sides are equal. Assume that the base angle is $$x^{\circ }$$. Use the fact that the sum of all three angles of a triangle is $${180}^{\circ }$$ to write an equation in ‘x’. Simplify the equation to find the measure of base angle.

Complete step-by-step answer:
We know that the vertex angle of an isosceles angle is $${90}^{\circ }$$. We have to calculate the measure of base angle.
Let’s consider an isosceles triangle $$\mathrm{\Delta }ABC$$ with $$\mathrm{\angle }A$$ being the vertex angle.

We know that sides adjoining vertex angles are of equal length. We also know that the angles opposite to equal sides are equal. Thus, the other two angles of the isosceles triangle are equal.
Let’s assume that the value of base angle is $$x^{\circ }$$.
We know that in any triangle, the sum of all the angles is $${180}^{\circ }$$. Thus, we have $$x^{\circ }+x^{\circ }+{90}^{\circ }={180}^{\circ }$$.
Simplifying the above equation, we have $$2x^{\circ }={180}^{\circ }-{90}^{\circ }={90}^{\circ }$$. Rearranging the terms, we have $$x={\displaystyle \frac{{90}^{\circ }}{2}}={45}^{\circ }$$.
Hence, the value of base angle is $$x={45}^{\circ }$$.

Note: An isosceles triangle is the one in which the length of at least two sides are equal. The equal sides of the isosceles triangle are called the legs and the third side is called the base. The angle included by the legs is called vertex angle and the angles that have a base as one of its sides are called the base angles. We must keep in mind that the measure of all the angles is in degrees; otherwise, we will get an incorrect answer.