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.
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.