Question

10. Show that ABC is equilateral if
(i) A = x +30°, B = 3x - 30° and C = 4x -60°.

​

Answer 1

Question:

Show that ∆ABC is equilateral if;

A = x + 30°

B = 3x - 30°

C = 4x - 60°

Note:

• Angel sum property of a triangle:

In any triangle, the sum of its all the three interior angles is 180°.

• Equilateral triangle: A triangle is said to be equilateral if all of its three sides are equal.

• In an equilateral triangle, each of its interior angle is 60°.

Proof:

In ∆ABC , we have;

A = x + 30°

B = 3x - 30°

C = 4x - 60°

As per the angle sum property of the triangle,

We have;

=> A + B + C = 180°

=> x + 30° + 3x - 30° + 4x - 60° = 180°

=> 8x - 60° = 180°

=> 8x = 180° + 60°

=> 8x = 240°

=> x = 240°/8

=> x = 30°

Now,

=> A = x + 30°

=> A = 30° + 30°

=> A = 60°

Also,

=> B = 3x - 30°

=> B = 3•30° - 30°

=> B = 90° - 30°

=> B = 60°

Also,

=> C = 4x - 60°

=> C = 4•30° - 60°

=> C = 120° - 60°

=> C = 60°

Here,

We get that , all the interior angles of the ∆ABC are of 60° .

ie; A = B = C = 60°

Hence,

The given ∆ABC is an equilateral triangle.

Answer 2

Question :-

Given above ↑

To prove :-

→ ∆ABC is a equilateral triangle ( all are same angles ) Io

Proof :-

Given that

• A = x + 30°
• B = 3x - 30°
• C = 4x - 60°

We know that

Sum of angles of triangle is 180°

→ A + B + C = 180°

→ x + 30° + 3x - 30° + 4x -60° = 180°

→8x = 180° + 60°

→ 8x = 240°

→ x = 30°

Now ,

→ A = x + 30° = 30° + 30° = 60°

→ B = 3x -30° = 90° - 30° = 60°

→ C = 4x - 60° = 120° -60° = 60°

because Angle A, B and C are 60° hence it is an equilateral triangle .

hence proved