Question

The measure of the obtuse angle in the isosceles triangle is two and a half times the measure of one base angle. What are the measures of all the angles?

Answer 1

Hint: Here we have to find the measure of all the angles of an isosceles triangle. An isosceles is a triangle in which two sides are equal in length. In an isosceles triangle two angles are equal and one is different which is an obtuse angle. In order to solve this question we form an equation and then we will find the measure of all the angles of an isosceles triangle.

Complete step by step answer:
In an isosceles triangle two angles are equal and one is different which is an obtuse angle. obtuse angles are the angles which are greater than $90^\circ $. So, in an isosceles triangle one angle is obtuse angle and two angles are acute which are equal in measure.

Let the two acute angles be $x$ and the obtuse angle be $y$.We know that the sum of all the angles of a triangle is $180^\circ $.Therefore, the sum of these angles is equal to $180^\circ $. So, we can form an equation by putting all these together which is,
$x + x + y = 180^\circ $
$ \Rightarrow 2x + y = 180 \ldots \ldots (1)$
It is given that the measure of the obtuse angle in an isosceles triangle is two and a half times the measure of one base angle.So, the above statement can be written as
$y = 2.5x$

Putting $y = 2.5x$ in equation $(1)$. We get,
$ \Rightarrow 2x + 2.5x = 180^\circ $
Adding $x$ terms. We get,
$ \Rightarrow 4.5x = 180^\circ $
$ \Rightarrow x = \dfrac{{180}}{{4.5}}$
$ \Rightarrow x = 40^\circ $
Putting $x = 40$ in the equation $y = 2.5x$. We get,
$ \Rightarrow y = 2.5 \times 40$
$ \therefore y = 100^\circ $

Therefore, the measure of all the three angles of an isosceles triangle is $100^\circ ,\,\,40^\circ ,\,\,40^\circ $.

Note: A triangle that has two sides of equal length is called the isosceles triangle. In an isosceles triangle, the angles that are opposite to the equal sides are equal. The third unequal angle can be acute or obtuse. Triangles are of three types on the basis of their angles they are acute angled triangle whose all interior angles are less than $90^\circ $, right- angled triangle whose one interior angle is of $90^\circ $ and obtuse angled triangle whose one interior angle is more than $90^\circ $.