**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.

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