Class 9 Some Properties of a Triangle for CBSE-NCERT

### Topic covered

color{red} ♦ Some Properties of a Triangle

### Some Properties of a Triangle

In the above section you have studied two criteria for congruence of triangles. Let us now apply these results to study some properties related to a triangle whose two sides are 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 (see Fig. 7.24). You have done such constructions in earlier classes.

Do you remember what is such a triangle called?

A triangle in which two sides are equal is called an isosceles triangle. So, Delta ABC of Fig. 7.24 is an isosceles triangle with AB = AC.

Now, measure ∠ B and ∠ C. What do you observe ?

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.

 color{blue} text(Theorem 7.2 :)

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.

 color{blue} text(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 (see Fig. 7.25).

In Delta BAD and Delta CAD,

AB = AC (Given)

∠ BAD = ∠ CAD (By construction)

AD = AD (Common)

So, Delta BAD ≅ Delta 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 (see Fig. 7.26)

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:

 color{blue} text ( Theorem 7.3) :

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.
Q 3210178010

In Delta ABC, the bisector AD of ∠ A is perpendicular to side BC (see Fig. 7.27). Show that AB = AC and Delta ABC is isosceles.

Class 9 Chapter 7 Example 4
Solution:

In Delta ABD and Delta ACD,

∠ BAD = ∠ CAD (Given)

AD = AD (Common)

∠ ADB = ∠ ADC = 90° (Given)

So, Delta ABD ≅ Delta ACD (ASA rule)

So, AB = AC (CPCT)

or, Delta ABC is an isosceles triangle.
Q 3220178011

E and F are respectively the mid-points of equal sides AB and AC of Delta ABC (see Fig. 7.28). Show that BF = CE.

Class 9 Chapter 7 Example 5
Solution:

In Delta ABF and Delta ACE,

AB = AC (Given)

∠ A = ∠ A (Common)

AF = AE (Halves of equal sides)

So, Delta ABF ≅ Delta ACE (SAS rule)

Therefore, BF = CE (CPCT)
Q 3230178012

In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD (see Fig. 7.29). Show that AD = AE.

Class 9 Chapter 7 Example 6
Solution:

In Delta ABD and Delta ACE,

AB = AC (Given) (1)

∠ B = ∠ C

(Angles opposite to equal sides) (2)

Also, BE = CD

So, BE – DE = CD – DE

That is, BD = CE (3)

So, Delta ABD ≅ Delta ACE

(Using (1), (2), (3) and SAS rule).

This gives AD = A E(CPCT)