**Proving Triangles are Congruent :**

How do we have to know about two triangles to prove that they are congruent ?

we can use the following postulates to prove that the two triangles are congruent.

1. Side - Side - Side (SSS) Congruence Postulate.

2. Side - Angle - Side (SAS) Congruence Postulate.

3. Angle - Side - Angle (ASA) Congruence Postulate

4. Angle - Angle - Side (AAS) Congruence Postulate

**Be caution :**

**SSA** and **AAA** can not be used to test congruent triangles.

**1. Side - Side - Side (SSS) Congruence Postulate**

**Explanation :**

If three sides of one triangle is congruent to three sides of another triangle, then the two triangles are congruent.

**2. Side - Angle - Side (SAS) Congruence Postulate**

**Explanation :**

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

**3. Angle - Side - Angle (ASA) Congruence Postulate**

**Explanation :**

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

**4. Angle - Angle - Side (AAS) Congruence Postulate**

**Explanation :**

If two angles and non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.

**Problem 1 : **

In the diagram given below, prove that ΔPQW ≅ ΔTSW.

**Solution :**

PQ ≅ ST PW ≅ TW QW ≅ SW ΔPQW ≅ ΔTSW |
Given Given Given SSS Congruence Postulate |

**Problem 2 : **

In the diagram given below, prove that ΔABC ≅ ΔFGH.

**Solution :**

Because AB = 5 in triangle ABC and FG = 5 in triangle FGH,

AB ≅ FG.

Because AC = 3 in triangle ABC and FH = 3 in triangle FGH,

AC ≅ FH.

Use the distance formula to find the lengths of BC and GH.

**Length of BC : **

**BC = √[(x₂ - x₁)² + (y₂ - y₁)²]**

**Here (**x₁, y₁) = B(-7, 0) and (x₂, y₂) = C(-4, 5)

**BC = √[(-4 + 7)² + (5 - 0)²]**

**BC = √[3² + 5²]**

**BC = √[9 + 25]**

**BC = √34**

**Length of GH : **

**GH = √[(x₂ - x₁)² + (y₂ - y₁)²]**

**Here (**x₁, y₁) = G(1, 2) and (x₂, y₂) = H(6, 5)

**GH = √[(6 - 1)² + (5 - 2)²]**

**GH = √[5² + 3²]**

**GH = √[25 + 9]**

**GH = √34**

**Conclusion :**

Because BC = √34 and GH = √34,

BC ≅ GH

All the three pairs of corresponding sides are congruent. By SSS congruence postulate,

ΔABC ≅ ΔFGH

**Problem 3 : **

In the diagram given below, prove that ΔAEB ≅ ΔDEC.

**Solution : **

AE ≅ DE, BE ≅ CE ∠1 ≅ ∠2 ΔAEB ≅ ΔDEC |
Given Vertical Angles Theorem SAS Congruence Postulate |

**Problem 4 : **

In the diagram given below, prove that ΔABD ≅ ΔEBC.

BD ≅ BC AD || EC ∠D ≅ ∠C ∠ABD ≅ ∠EBC ΔABD ≅ ΔEBC |
Given Given Alternate Interior Angles Theorem Vertical Angles Theorem ASA Congruence Postulate |

**Problem 5 : **

In the diagram given below, prove that ΔEFG ≅ ΔJHG.

FE ≅ JH ∠E ≅ ∠J ∠EGF ≅ ∠JGH ΔEFG ≅ ΔJHG |
Given Given Vertical Angles Theorem AAS Congruence Postulate |

After having gone through the stuff given above, we hope that the students would have understood, "Proving triangles are congruent".

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