Math՚s: Congruence: Angles Opposite to Equal Sides of a Triangle and Vice Versa (For CBSE, ICSE, IAS, NET, NRA 2022)

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Angles Opposite to Equal Sides of a Triangle and Vice Versa

Based on the criteria for congruence of triangles, there are some important theorems related to the isosceles triangles which are summarized as follows-

Theorem 1- The angles opposite to the equal sides of a triangle are equal.

Angle Opposite to Equal Side

Given: A triangle ABC in which

To prove:

Construction: Draw bisector of meeting at .

Proof: In and ,

(Given)

(By construction)

and (Common)

(SAS)

Hence (c. p. c. t)

Hence the theorem.

Theorem 2- The sides opposite to the equal angles of a triangle are equal.

Angle Opposite to Equal Side

Given: A triangle in which

To Prove:

Construction: Draw bisector of meeting at .

Proof: In and ,

(Given)

(By construction)

and (Common)

(SAS)

Hence (c. p. c. t)

Hence the theorem.

Theorem 3- Perpendiculars (altitudes) drawn to equal sides, from opposite vertices of an isosceles triangle are equal.

Angle Opposite to Equal Side

Given: In and

To Prove:

Proof: In and

(Angles opposite equal sides of a triangle)

(Common)

(AAS)

Hence, (c. p. c. t.)

Inequalities in a Triangle

There are some relations among sides and angles of a triangle when they are unequal. These are summarized as theorems

Ineualities in a Triangle

Theorem 1- If two sides of a triangle are unequal, then the longer side has the greater angle opposite to it.

Given: A triangle in which .

To Prove:

Construction: Make a point on the side such that and join .

Proof: In ,

(Angles opposite equal sides)

But (Exterior angle of a triangle is greater than opposite

interior angle)

Again (Point D lies in the interior of the ∠ ACB) .

Theorem 2- Sum of any two sides in a triangle is greater than the third side.

Two Sides of Triangle

Given: is a triangle

To Prove: and

Construction: Produce to such that , Join and

Proof: … . (1) Since,

… . (2)

Therefore,

from (1) & (2)

In (Greater angle has greater side opposite to it)

()

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