NCERT Solutions for Class 9 Maths Exercise 7.2 Question 8
Welcome, students! In this final question of Exercise 7.2, we will prove a fundamental property of equilateral triangles: that all of their interior angles are &&60^\circ&&. This is a classic proof that combines two of the most important theorems about triangles.
| Given Information | An equilateral triangle &&\triangle ABC&&, which means &&AB = BC = AC&&. |
|---|---|
| To Prove | &&\angle A = \angle B = \angle C = 60^\circ&&. |
| Key Concepts Used | Isosceles Triangle Theorem and the Angle Sum Property of a Triangle. |
Question 8: Show that the angles of an equilateral triangle are &&60^\circ&& each.
How to Prove All Angles of an Equilateral Triangle are 60° 🤔
This proof is a logical process that can be broken down into a few simple steps.
- Step 1: Use the Isosceles Triangle Theorem for a Pair of Sides
Consider an equilateral triangle ABC, where &&AB = BC = AC&&. First, take the equality &&AB = AC&&. According to the isosceles triangle theorem, angles opposite to equal sides are equal. Therefore, &&\angle C = \angle B&&. - Step 2: Use the Theorem Again for Another Pair of Sides
Now, take another pair of equal sides, for example, &&BC = AC&&. Applying the same theorem, the angles opposite to these sides must also be equal. Therefore, &&\angle A = \angle B&&. - Step 3: Conclude that All Angles are Equal
From the first two steps, we have established that &&\angle C = \angle B&& and &&\angle A = \angle B&&. By combining these results, we can conclude that all three angles are equal: &&\angle A = \angle B = \angle C&&. - Step 4: Apply the Angle Sum Property and Solve
The sum of the interior angles of any triangle is &&180^\circ&&. So, &&\angle A + \angle B + \angle C = 180^\circ&&. Since all three angles are equal, we can rewrite this as &&3\angle A = 180^\circ&&. Solving for &&\angle A&& gives us &&\angle A = 60^\circ&&. Because all angles are equal, it follows that &&\angle A = \angle B = \angle C = 60^\circ&&.
Detailed Step-by-Step Proof 📝
Here is the formal proof, written step-by-step.
Given:
Let &&\triangle ABC&& be an equilateral triangle. By definition, all its sides are equal.
So, &&AB = BC = AC&&.
To Prove:
&&\angle A = \angle B = \angle C = 60^\circ&&.
Proof:
First, let’s establish that all angles are equal.
In &&\triangle ABC&&, we have:
&&AB = AC&& (Sides of an equilateral triangle)
&&\implies \angle C = \angle B&& (Theorem: Angles opposite to equal sides are equal) … (Equation 1)
Similarly, we also have:
&&BC = AC&& (Sides of an equilateral triangle)
&&\implies \angle A = \angle B&& (Theorem: Angles opposite to equal sides are equal) … (Equation 2)
From Equation 1 and Equation 2, we can see that &&\angle A = \angle B&& and &&\angle C = \angle B&&. This means all three angles are equal.
&&\angle A = \angle B = \angle C&&
Now, we use the Angle Sum Property of a Triangle, which states that the sum of all interior angles of a triangle is &&180^\circ&&.
&&\angle A + \angle B + \angle C = 180^\circ&&
Since all angles are equal, we can substitute &&\angle B&& and &&\angle C&& with &&\angle A&&:
&&\implies \angle A + \angle A + \angle A = 180^\circ&&
&&\implies 3\angle A = 180^\circ&&
&&\implies \angle A = \frac{180^\circ}{3} = 60^\circ&&
Conclusion:
Since &&\angle A = \angle B = \angle C&&, and &&\angle A = 60^\circ&&, we have:
&&\angle A = \angle B = \angle C = 60^\circ&&
Hence, proved that each angle of an equilateral triangle is &&60^\circ&&.
A Fundamental Geometric Fact ✅
The result of this proof is a core property of Euclidean geometry: an equilateral triangle (equal sides) is always an equiangular triangle (equal angles), and those angles are always &&60^\circ&&. This is a fact you should memorize as it is used frequently in other problems.
FAQ (Frequently Asked Questions)
Q: What is an equilateral triangle?
A: An equilateral triangle is a triangle in which all three sides have the same length. As this proof shows, it also means all three interior angles are equal to &&60&& degrees.
Q: What is the key theorem used in this proof?
A: The key theorem is the Isosceles Triangle Theorem, which states that ‘angles opposite to equal sides of a triangle are equal.’ This theorem is used twice to show that all three angles are equal to each other.
Q: Is the converse of this property also true?
A: Yes, the converse is true. If all three angles of a triangle are equal (an equiangular triangle), then all three sides must also be equal, making it an equilateral triangle.
Q: Why is the Angle Sum Property necessary for this proof?
A: The Isosceles Triangle Theorem helps us establish that all angles are equal (&&\angle A = \angle B = \angle C&&). However, it doesn’t tell us their actual measure. We need the Angle Sum Property (&&\angle A + \angle B + \angle C = 180^\circ&&) to set up an equation that allows us to solve for the specific value of the angles, which is &&60^\circ&&.
