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.

Detailed Answer Key

Problem 1 : 

In the diagram given below, prove that ΔPQW  ≅  ΔTSW

Solution :

Statements

PQ  ≅  ST

PW  ≅  TW

QW  ≅  SW

ΔPQW  ≅  ΔTSW

Reasons

Given

Given

Given

SSS Congruence Postulate

Problem 2 : 

In the diagram given below, prove that ΔAEB  ≅  ΔDEC

Solution : 

Statements

AE  ≅  DE, BE  ≅  CE

∠1  ≅  ∠2

ΔAEB  ≅  ΔDEC

Reasons

Given

Vertical Angles Theorem

SAS Congruence Postulate

Problem 3 : 

In the diagram given below, prove that ΔABD  ≅  ΔEBC

Statements

BD  ≅  BC

AD || EC

∠D  ≅  ∠C

∠ABD  ≅  ∠EBC

ΔABD  ≅  ΔEBC

Reasons

Given

Given

Alternate Interior Angles Theorem

Vertical Angles Theorem

ASA Congruence Postulate

Problem 4 : 

In the diagram given below, prove that ΔEFG  ≅  ΔJHG

Statements

FE  ≅  JH

∠E  ≅  ∠J

∠EGF  ≅  ∠JGH

ΔEFG  ≅  ΔJHG

Reasons

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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