Theorems

Theorem

Theorem 7.1  (ASA congruence rule) : Two triangles are congruent if two angles and  the included side of one triangle are equal to two angles and the included side of other  triangle.
So, two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. We may call it as the AAS Congruence Rule.

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. 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, Δ ABC
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.
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.
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.
Let us take some examples to apply these results.

Some More Criteria for Congruence of Triangles

You have seen earlier in this chapter that equality of three angles of one triangle to three angles of the other is not sufficient for the congruence of the two triangles. You may wonder whether equality of three sides of one triangle to three sides of another triangle is enough for congruence of the two triangles. You have already verified in earlier classes that this is indeed true.
To be sure, construct two triangles with sides 4 cm, 3.5 cm and 4.5 cm. Cut them out and place them on each other. What do you observe? They cover each other completely, if the equal sides are placed on each other. So, the triangles are congruent.

Repeat this activity with some more triangles. We arrive at another rule for congruence.
Theorem 7.4 (SSS congruence rule) : If three sides of one triangle are equal to the  three sides of another triangle, then the two triangles are congruent.
This theorem can be proved using a suitable construction.
You have already seen that in the SAS congruence rule, the pair of equal angles has to be the included angle between the pairs of corresponding pair of equal sides and if this is not so, the two triangles may not be congruent.
Perform this activity:
Construct two right angled triangles with hypotenuse equal to 5 cm and one side equal to  4 cm each

Cut them out and place one triangle over the other with equal side placed on each other. Turn the triangles, if necessary. What do you observe?
The two triangles cover each other completely and so they are congruent. Repeat this activity with other pairs of right triangles. What do you observe?
You will find that two right triangles are congruent if one pair of sides and the hypotenuse are equal. You have verified this in earlier classes.
Note that, the right angle is not the included angle in this case.
So, you arrive at the following congruence rule:
Theorem 7.5 (RHS congruence rule): If in two right triangles the hypotenuse and  one  side of one triangle are equal to the hypotenuse and one side of the other  triangle, then the two triangles are congruent.
Note that RHS stands for Right angle - Hypotenuse - Side.
Let us now take some examples.