Answer
Verified
403.8k+ views
Hint: We will use the concept that angles opposite to equal sides of a triangle are equal and we know that all the sides of an equilateral triangle are equal. We will also use the property that the sum of all the three angles of a triangle is equal to 180˚.
Complete step by step solution:
It is given in the question that we have to prove that equilateral triangle is equiangular. Here, equiangular means all the three angles of the equilateral triangle have the same measure.
Now, let us assume an equilateral triangle ABC. We know that all the sides of an equilateral triangle are equal, it means that in triangle ABC, we have AB = BC = AC.
We know that the angles opposite to the equal sides of a triangle are equal. So, here we have the side AB equal to the side AC, it means that $\angle B=\angle C.........\left( i \right)$.
Similarly, we have the side AB equal to the side BC, it means that $\angle C=\angle A.........\left( ii \right)$.
So, from equations (i) and (ii), we get,
$\angle A=\angle B=\angle C$
This means that all the angles of an equilateral triangle are equal.
Now, we know that the sum of all angles of a triangle is equal to 180˚. As all the angles are equal, we can write the equation as,
$\begin{align}
& \angle A+\angle A+\angle A={{180}^{\circ }} \\
& 3\angle A={{180}^{\circ }} \\
& \angle A=\dfrac{{{180}^{\circ }}}{3}={{60}^{\circ }} \\
\end{align}$
As all the angles are equal it means that $\angle A=\angle B=\angle C={{60}^{\circ }}$
Thus, it is proved that the equilateral triangle is equiangular.
Note: Most of the students make mistakes while taking angles while applying the concept that angles opposite to the equal sides of a triangle are equal. For example, we have the sides AB and BC equal, so according to the concept, we will get $\angle A=\angle C$, but some students may write it as $\angle A=\angle B$ which is wrong.
Complete step by step solution:
It is given in the question that we have to prove that equilateral triangle is equiangular. Here, equiangular means all the three angles of the equilateral triangle have the same measure.
Now, let us assume an equilateral triangle ABC. We know that all the sides of an equilateral triangle are equal, it means that in triangle ABC, we have AB = BC = AC.
We know that the angles opposite to the equal sides of a triangle are equal. So, here we have the side AB equal to the side AC, it means that $\angle B=\angle C.........\left( i \right)$.
Similarly, we have the side AB equal to the side BC, it means that $\angle C=\angle A.........\left( ii \right)$.
So, from equations (i) and (ii), we get,
$\angle A=\angle B=\angle C$
This means that all the angles of an equilateral triangle are equal.
Now, we know that the sum of all angles of a triangle is equal to 180˚. As all the angles are equal, we can write the equation as,
$\begin{align}
& \angle A+\angle A+\angle A={{180}^{\circ }} \\
& 3\angle A={{180}^{\circ }} \\
& \angle A=\dfrac{{{180}^{\circ }}}{3}={{60}^{\circ }} \\
\end{align}$
As all the angles are equal it means that $\angle A=\angle B=\angle C={{60}^{\circ }}$
Thus, it is proved that the equilateral triangle is equiangular.
Note: Most of the students make mistakes while taking angles while applying the concept that angles opposite to the equal sides of a triangle are equal. For example, we have the sides AB and BC equal, so according to the concept, we will get $\angle A=\angle C$, but some students may write it as $\angle A=\angle B$ which is wrong.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE