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.
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.
