# PROVING TRIANGLE CONGRUENCE WORKSHEET

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 :

 StatementsPQ  ≅  STPW  ≅  TWQW  ≅  SWΔPQW  ≅  ΔTSW ReasonsGivenGivenGivenSSS Congruence Postulate

Problem 2 :

In the diagram given below, prove that ΔAEB  ≅  ΔDEC

Solution :

 StatementsAE  ≅  DE, BE  ≅  CE∠1  ≅  ∠2ΔAEB  ≅  ΔDEC ReasonsGivenVertical Angles TheoremSAS Congruence Postulate

Problem 3 :

In the diagram given below, prove that ΔABD  ≅  ΔEBC

 StatementsBD  ≅  BCAD || EC∠D  ≅  ∠C∠ABD  ≅  ∠EBCΔABD  ≅  ΔEBC ReasonsGivenGivenAlternate Interior Angles TheoremVertical Angles TheoremASA Congruence Postulate

Problem 4 :

In the diagram given below, prove that ΔEFG  ≅  ΔJHG

 StatementsFE  ≅  JH∠E  ≅  ∠J∠EGF  ≅  ∠JGHΔEFG  ≅  ΔJHG ReasonsGivenGivenVertical Angles TheoremAAS 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)

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

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