# Congruent Triangles

If two triangles are equal in all respects, they are said to be**Congruent triangles**.

Thus two congruent-triangles have the same shape and same size.

Let ΔABC and ΔPQR be two triangles. Then we can superimpose ΔABC on ΔPQR, so as to cover exactly.

**Due to this superimposition :**

Vertex A falls on Vertex P

Vertex B falls on Vertex Q

Vertex C falls on Vertex R

AB = PQ ∠A = ∠P

BC =QR ∠B = ∠Q

AC = PR ∠C = ∠R

Hence triangles ABC and PQR are congruent to each other.

**Note :**1) Congruent-triangles are similar but the similar triangles are not always congruent.

2) The symbol

**≅**reads " is congruent to ".

If two triangles are congruent then there is one to one correspondence (↔) between the two triangles.

**ΔABC ↔ ΔPQR**then

∠A ≅ ∠P , ∠B ≅ ∠Q and ∠C ≅ ∠R

∠AB ≅ ∠PQ , ∠BC ≅ ∠QR and ∠AC ≅ ∠PR .

**Note :**If two triangles are congruent then their corresponding parts are congruent.

**C**orresponding

**P**arts of

**C**ongruent

**T**riangles are

**C**ongruent

⇒ C. P .C. T. C

**Congruence Relation**

1) Every triangle is congruent to itself. ΔABC ≅ ΔABC.

2) If ΔABC ≅ ΔPQR then ΔPQR = ΔABC.

3) If ΔABC ≅ ΔPQR and ΔABC ≅ ΔDEF then ΔPQR ≅ ΔDEF.

**Examples :**

**In the following pairs of triangles, find out whether the triangles in each pair are congruent or not.**

1) ΔABC : AB = 3 , BC = 4 and ∠B = 90

^{0}

ΔDEF : DE = 3 , DF = 4 and ∠E = 90

^{0}.

**Solution :**

Here , ΔABC not ≅ Δ DEF because there is no one to one correspondence between BC and DF.

2) Δ ABC : AB = 3 , AC = 5 and BC = 6

Δ PQR : PQ = 3 , PR = 5 and QR = 6

**Solution :**

Here ΔABC ≅ ΔPQR because there is one to one correspondence between all the sides.

**Triangles**

• Introduction to Triangles

• Types of Triangles on the basis of Sides

• Types of Triangles on the basis of Angles

• Angle Sum Property of Triangles

• Exterior and Interior angles of Triangle

• Triangle Inequality Property

• Congruent Triangles

• Postulates of Congruent Triangle

• Inequality in Triangle

• Introduction to Triangles

• Types of Triangles on the basis of Sides

• Types of Triangles on the basis of Angles

• Angle Sum Property of Triangles

• Exterior and Interior angles of Triangle

• Triangle Inequality Property

• Congruent Triangles

• Postulates of Congruent Triangle

• Inequality in Triangle