**Proving Triangle Congruence Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on congruent triangles.

Before look at the worksheet, if you would like to have the stuff related to triangle congruence,

**Problem 1 : **

In the diagram given below, prove that ΔPQW ≅ ΔTSW using two column proof.

**Problem 2 : **

In the diagram given below, prove that ΔAEB ≅ ΔDEC using two column proof.

**Problem 3 : **

In the diagram given below, prove that ΔABD ≅ ΔEBC using two column proof.

**Problem 4 : **

In the diagram given below, prove that ΔEFG ≅ ΔJHG using two column proof.

**Problem 5 : **

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

**Problem 6 :**

Check whether two triangles ABC and CDE are congruent.

**Problem 7 :**

Check whether two triangles PQR and RST are congruent.

**Problem 8 :**

Check whether two triangles ABD and ACD 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 ΔAEB ≅ ΔDEC.

**Solution : **

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

**Problem 3 : **

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 4 : **

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 |

**Problem 5 : **

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 6 :**

Check whether two triangles ABC and CDE are congruent.

**Solution :**

(i) Triangle ABC and triangle CDE are right triangles. Because they both have a right angle.

(i) AC = CE (Leg)

(ii) BC = CD (Leg)

Hence, the two triangles ABC and CDE are congruent by **Leg-Leg **theorem.

**Problem 7 :**

Check whether two triangles PQR and RST are congruent.

**Solution :**

(i) Triangle PQR and triangle RST are right triangles. Because they both have a right angle.

(ii) QR = RS (Given)

(iii) ∠PRQ = ∠SRT (Vertical Angles)

Hence, the two triangles PQR and RST are congruent by **Leg-Acute (LA) Angle** theorem.

**Problem 8 :**

Check whether two triangles ABD and ACD are congruent.

**Solution :**

(i) Triangle ABD and triangle ACD are right triangles. Because they both have a right angle.

(i) AB = AC (Hypotenuse)

(ii) AD = AD (Common side, Leg)

Hence, the two triangles ABD and ACD are congruent by **Hypotenuse-Leg (HL) **theorem.

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

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