Question

prove that each angle of an equilaterla trianglesis 60^0

Answer 1

Given

• ΔABC is a equilateral triangle

To prove

• Each angle is 60°
• ∠A = ∠B = ∠C = 60°

Solution

• In ΔABC
• AB = AC = AC (Sides in a equilateral triangle are equal)
• If AB = BC , then ∠C = ∠A  - (i)
• If BC = AC , then ∠A = ∠B  - (ii)

☯ If two sides of a triangle are equal, then angles opposite are equal

• From (i) and (ii) :-
• ∠A = ∠B = ∠C
• In ΔABC
• ∠A + ∠B + ∠C = 180°    (Angle sum property)
• ∠A + ∠A + ∠A = 180°    (All angles are equal)
• 3∠A = 180°
• ∠A = ¹⁸⁰⁄₃
• ∠A = 60°
• So, ∠B = ∠C = 60°

☯ Hence, all the angles in a equilateral triangle is equal to 60°

Answer 2

Given :-

• An equilateral triangle.

─━─━─━─━─━─━─━─━─━─━─━─━─

To Prove :-

• Each angle of equilateral triangle is 60°.

─━─━─━─━─━─━─━─━─━─━─━─━─

Proof :-

☆ Let us consider an equilateral triangle ABC.

☆, Now as triangle is equilateral.

⇒ AB = BC = CA

☆ Now, AB = BC

⇒ ∠BAC = ∠BCA -----(i)

(Angle opposite to equal sides are always equal)

☆ Again, BC = AC

⇒ ∠ABC = ∠BAC --------(ii)

(Angle opposite to equal sides are always equal)

☆ From, (i) and (ii), we concluded that

☆ ∠BAC = ∠BCA = ∠ABC = x (say)

☆ In triangle ABC,

We know, sum of all angles of a triangle is 180°.

⇒ ∠BAC + ∠BCA + ∠ABC = 180°

⇒ x + x + x = 180°

⇒ 3x = 180°

⇒ x = 60°

$\begin{gathered}\begin{gathered}\bf \therefore \: angles \: are \: = \begin{cases} &\sf{ \angle \: ABC = 60 \degree} \\ &\sf{ \angle \: BCA \: = 60 \degree \: } \\ &\sf{ \angle \: BAC = 60 \degree \: } \end{cases}\end{gathered}\end{gathered}$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$