Question

State and prove theorem 7.2​

Answer 1

Answer:

Objective:

This topic gives an overview of the theorems;

Angles opposite to equal sides of an isosceles triangle are equal.

The sides opposite to equal angles of a triangle are equal.

SSS congruence rule

RHS congruence rule

Some Properties of a Triangle

Let us apply the results of congruence of triangles to study some properties related to a triangle whose two sidesare equal.

Perform the activity given below:

Construct a triangle in which two sides are equal, say each equal to 3.5 cm and the third side equal to 5 cm. You have done such constructions in earlier classes.

A triangle in which two sides are equal is called an isosceles triangle. So, Δ ABC is an isosceles triangle with AB = AC. Now, measure ∠ B and ∠ C. Repeat this activity with other isosceles triangles with different sides. You may observe that in each such triangle, the angles opposite to the equal sides are equal. This is a very important result and is indeed true for any isosceles triangle. It can be proved as shown below.

Theorem : Angles opposite to equal sides of an isosceles triangle are equal.

This result can be proved in many ways. One of the proofs is given here.

Proof :

We are given an isosceles triangle ABC in which AB = AC. We need to prove that ∠B = ∠C.

Let us draw the bisector of ∠ A and let D be the point of intersection of this bisector of ∠ A and BC.

In Δ BAD and Δ CAD,

AB = AC (Given)

∠BAD = ∠CAD (By construction)

AD = AD (Common)

So, Δ BAD ≅ Δ CAD (By SAS rule)

So, ∠ABD = ∠ACD, since they are corresponding angles of congruent triangles.

So, ∠B = ∠C

Is the converse also true?That is: If two angles of any triangle are equal, can we conclude that the sides opposite to them are also equal?

Perform the following activity.

Construct a triangle ABC with BC of any length and ∠ B = ∠ C = 50°. Draw the bisector of ∠ A and let it intersect BC at D .Cut out the triangle from the sheet of paper and fold it along AD so that vertex C falls on vertex B.What can you say about sides AC and AB? Observe that AC covers AB completely So, AC = AB

Repeat this activity with some more triangles. Each time you will observe that the sides opposite to equal angles are equal. So we have the following:

Theorem : The sides opposite to equal angles of a triangle are equal

This is the converse of Theorem 7.2. You can prove this theorem by ASA congruence rule.Let us take some examples to apply these results.

Example: In Δ ABC, the bisector AD of ∠ A is perpendicular to side BC.Show that AB = AC and Δ ABC is isosceles.

Solution : In ΔABD and ΔACD,

∠BAD = ∠CAD (Given)

AD = AD (Common)

∠ADB = ∠ADC = 90° (Given)

So, Δ ABD ≅ Δ ACD (ASA rule)

So, AB = AC (CPCT) or, Δ ABC is an isosceles triangle.

Answer 2

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