**Triangle Congruence Postulates and Theorems :**

In this section, we are going to see, how to prove two triangles are congruent using congruence postulates and theorems.

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

5. Hypotenuse-Leg **(HL)** Theorem

6. Leg-Acute **(LA)** Angle Theorem

7. Hypotenuse-Acute **(HA)** Angle Theorem

8. Leg-Leg **(LL)** Theorem

**Be caution :**

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

**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.

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

**Explanation :**

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

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

**Explanation :**

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

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

**Explanation :**

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

**Hypotenuse-Leg (HL) Theorem**

**Explanation :**

If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.

This principle is known as Hypotenuse-Leg theorem.

**Leg-Acute (LA) Angle Theorem**

**Explanation :**

If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the two right triangles are congruent.

This principle is known as Leg-Acute Angle theorem.

**Hypotenuse-Acute (HA) Angle Theorem**

**Explanation :**

If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

This principle is known as Hypotenuse-Acute Angle theorem.

**Leg-Leg (LL) Theorem**

**Explanation :**

If the legs of one right triangle are congruent to the legs of another right triangle, then the two right triangles are congruent.

This principle is known as Leg-Leg theorem.

**Example 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 |

**Example 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

**Example 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 |

**Example 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 |

**Example 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 |

Apart from the problems given above, if you need more problems on triangle congruence postulates,

If you need problems on triangle congruence theorems,

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