# Congruency In Triangles

## Theorems

- Two figures are congruent, if they are of the same shape and of the same size.
- Two circles of the same radii are congruent.
- Two squares of the same sides are congruent.
- If two triangles ABC and PQR are congruent under the correspondence A – P, B-Q and C-R, then symbolically, it is expressed as Δ ABC Δ PQR.

### SAS Congruence Rule

If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent. (Axiom: This result cannot be proved with the help of previously known results.)

### ASA Congruence Rule

If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent (ASA Congruence Rule).

**Construction:** Two triangles are given as follows in which:

`∠ABC=∠DEF` and `∠ACB=∠DEF`

Sides `AB=DE`

**To prove:** `ΔABC≅ ΔDEF`

**Proof:** `∠ABC=∠DEF` (given)

`AB=DE`

`AC=DF`

(Sides opposite to corresponding angles are in the same ratio as ratio of angles)

Hence, `ΔABC≅ ΔDEF` (SAS rule)

### AAS Congruence Rule

If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.

This theorem can be proved in similar way as the previous one.

- Angles opposite to equal sides of a triangle are equal.
- Sides opposite to equal angles of a triangle are equal.
- Each angle of an equilateral triangle is of 60°.

### SSS Congruence Rule

If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent.

### RHS Congruence Rule

If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent (RHS Congruence Rule).

- In a triangle, angle opposite to the longer side is larger (greater).
- In a triangle, side opposite to the larger (greater) angle is longer.
- Sum of any two sides of a triangle is greater than the third side.

**Theorem:** Angles opposite to equal sides of an isosceles triangle are equal.

**Theorem:** The sides opposite to equal angles of a triangle are equal.

**Theorem:** If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater).

**Theorem:** In any triangle, the side opposite to the larger (greater) angle is longer.

**Theorem:** The sum of any two sides of a triangle is greater than the third side.