Question

In triangle $\text{ }\!\!\Delta\!\!\text{ ABC}$, $AB=AC$ and $\angle A:\angle B=8:5$. Then the measure of $\angle A$ in degrees is:

Answer 1

Hint: In this question we have been given with a triangle $\text{ }\!\!\Delta\!\!\text{ ABC}$ for which we have been given with the ration of the two angles in it. We have $\angle A:\angle B=8:5$ and we have also been given that the sides $AB=AC$ . We have to find the measure of $\angle A$ from the given data. We will consider the ratio of the two given angles and then use the property of the isosceles triangle to get the ratio of the third angle. We will then use the property of a triangle that the sum of all angles is ${{180}^{\circ }}$ and then we will find the value of $x$ and then multiply it with the ratio to get the required solution.

Complete step by step solution:
We have been given with $\text{ }\!\!\Delta\!\!\text{ ABC}$ as:

We know that $\angle A:\angle B=8:5$ therefore, we can write it as:
$\Rightarrow \dfrac{\angle A}{\angle B}=\dfrac{8x}{5x}$
Therefore, we can write:
$\Rightarrow \angle A=8x\to \left( 1 \right)$
$\Rightarrow \angle B=5x\to \left( 2 \right)$
Now we have been given with the data that $AB=AC$.
Now since both the sides are equal, we will use the isosceles triangle property which states that when two sides are equal in a triangle, the corresponding angles of the sides are the same. Therefore, we can state that $\angle B=\angle C$
Now from equation $\left( 2 \right)$, we get:
$\Rightarrow \angle C=5x\to \left( 3 \right)$
Now the sum of angles in a triangle is always equal to ${{180}^{\circ }}$, therefore from equations $\left( 1 \right),\left( 2 \right)$ and $\left( 3 \right)$, we get:
$\Rightarrow 8x+5x+5x=180$
On adding the terms, we get:
$\Rightarrow 18x=180$
On transferring $18$ from the left-hand side to the right-hand side, we get:
$\Rightarrow x=\dfrac{180}{18}$
On simplifying, we get:
$\Rightarrow x=10$
Now we have to find the value of $\angle A$.
From equation $\left( 1 \right)$, we get:
$\Rightarrow \angle A=8x$
On substituting $x=10$, we get:
$\Rightarrow \angle A=8\times 10$
On simplifying, we get:
$\Rightarrow \angle A={{80}^{\circ }}$, which is the required solution.

Note: It is to be remembered that an isosceles triangle is a triangle which has two sides of the triangle the same. There also exists triangles which are equilateral triangles which have all the three sides and the three angles same. Which means all the angles of the triangle are ${{60}^{\circ }}$.