Question

Suppose ABC is an equiangular triangle. Prove that it is equilateral. (You have seen earlier that an equilateral triangle is equiangular. Thus for triangles equiangularity is equivalent to equilaterality).

Answer 1

Hint: We have given that ABC is an equiangular triangle meaning all the angles of the triangle are equal and we have to prove that it is equilateral also. To prove the equilateral triangle we have to show that all the sides of the triangle are equal which we are going to show by dropping a perpendicular from any one of the vertices on to the opposite side of the triangle then the triangle split into 2 triangles. After that we will prove that the two triangles are congruent to each other and using CPCT (Corresponding part of the congruent triangles) the two sides of the main triangle are equal. Similarly, we can show that all the sides of the triangle are equal.

Complete step-by-step answer:
We have been given in the question that a triangle ABC is an equiangular triangle meaning all the angles of the triangle are equal.
In the below diagram, we have shown a triangle ABC with all its angles are equal.

Now, we are asked to show that the above triangle is an equilateral triangle so we have to show that all the sides of the above triangle are equal which we are going to show by dropping a perpendicular from vertex A onto the side BC.

In the above figure, as you can see that we have dropped a perpendicular from vertex A on the side BC then triangle ABC is divided into two triangles. Now, we are going to show the congruency of the two divided triangles.
We are going to prove the congruence of triangles ADB and ADC.
In $\Delta ADB \And \Delta ADC$ we have,
$\angle B=\angle C$
The above angles are equal because it is given that all the angles of triangle ABC are equal.
$\angle ADB=\angle ADC$
The above angles are equal because both the angles are equal to ${{90}^{\circ }}$.
And the side AD is common in both the triangles ADB and ADC.
Hence, $\Delta ADB \And \Delta ADC$ are congruent to each other by the AAS axiom of congruency.
Now, as we have shown the two triangles $\Delta ADB\And \Delta ADC$ are congruent then by CPCT (Corresponding part of the congruent triangles) sides AB and AC are equal.
$\Delta ADB\cong \vartriangle ADC$
$AB=AC$ [By CPCT]
Hence, we have shown the side AB is equal to side BC.
Similarly, we can also show side AC is equal to BC by dropping a perpendicular from vertex C onto the side AB and then prove the congruence of the divided triangle and hence, by CPCT can show that side AC equals BC.
Now, the following sides are equal that we have shown above:
$\begin{align}
  & AB=AC \\
 & AC=BC \\
\end{align}$
From the above equality of two sides, we have found the relation between the sides AB, BC and AC as:
$AB=BC=AC$
From the above, we have shown that all the sides of the triangles are equal. Hence, we have shown that triangle ABC is an equilateral triangle.

Note: You might be thinking when you solve the question by yourself that if all the angles of a triangle are given as equal then how to show that triangle is an equilateral triangle. The answer to this question is that, equilateral means all the sides are equal or if we spit the word “equilateral” then “equi” meaning equal and “lateral” meaning sides so to prove the triangle as an equilateral triangle we have to show that all its sides are equal. That’s why we have also proved the triangle as equilateral by showing all its sides as equal.