# Congruence Theorem (Triangles)

There are five ways of finding two similar triangles. But the main question is how to find out that two triangles are congruent to each other? Thus, in simple words, we can say that two triangles are said to be congruent if they have the same three sides and angles. But we need not have to check out all these three angles and sides for knowing its congruence, just three of all these six is fine. Thus the five theorems of congruent triangles are SSS, SAS, AAS, HL, and ASA.

## SSS – side, side, and side

This ‘SSS’ means side, side, and side which clearly states that if the three sides of both triangles are equal then, both triangles are congruent to each other.

For instance, triangle ABC has the side of 2cm * 3cm * 5cm and triangle PQR has sides of 2cm * 3cm * 5cm, then according to the SSS theorem, both these triangles ABC and QPR are congruent to each other.

In simple words, if all the three sides of one triangle are equal to all the three sides of another triangle, then both the triangles are congruent to each other.

## SAS – side, angle, and side

This ‘SAS’ means side, angle, and side which clearly states that any of the two sides and one angle of both triangles are the same, then both triangles are considered congruent to each other.

For instance, if the triangle ABC measures 2cm * 3cm * 6cm with the angle of 75 degrees, and the triangle PQR measures 4cm * 3cm * 6cm with the angle of 75 degrees, then according to the SAS theorem of the triangle, both these triangles will be considered congruent triangles.

In simple words, if the two sides and one angle of one triangle is similar to the other triangle, then they are known as congruent to each other.

## ASA – angle, side, and angle

This ‘ASA’ means angle, side, and angle which clearly states that two angles and one side of both triangles are the same, then these two triangles are said to be congruent to each other.

For instance, triangle ABC measures angle of 35 and 45 degree with one side of 5 cm and triangle PQR measures angle of 35 and 45 degree with one side of 5 cm, then both these triangles have the same measures, which as per the theorem of ASA, these two triangles are congruent to each other.

In simple words, if two angles and one side of one triangle is equal to the other triangle, then both these triangles are congruent.

## AAS – angle, angle, and side

This ‘AAS’ means angle, angle, and sides which clearly states that two angles and one side of both triangles are the same, then these two triangles are said to be congruent to each other.

For instance, triangle ABC measures angle of 35 and 45 degree with one side of 5 cm and triangle PQR measures angle of 35 and 45 degree with one side of 5 cm, then both these triangles have the same measures, which as per the theorem of AAS, these two triangles are congruent to each other.

In simple words, if two angles and one side of one triangle is equal to the other triangle, then both these triangles are congruent.

## HL – hypotenuse and leg

This ‘HL’ means hypotenuse and leg which clearly states that if hypotenuse and leg of both the triangles are the same, then these two triangles are said to be congruent to each other.

For instance, triangle ABC measures hypotenuse of 5 cm and leg of 4 cm and triangle PQR measures hypotenuse of 5 cm and leg of 4 cm, then both these triangles have the same measures, which as per the theorem of HL, these two triangles are congruent to each other. Geo 5.5-5.7 Notes