CONGRUENCE AND TRIANGLES WORKSHEET

About "Congruence and triangles worksheet"

Congruence and triangles worksheet :

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

Congruence and triangles worksheet - Problems

Problem 1 : 

The congruent triangles represent the triangles in the diagram given below. Write a congruence statement. Identify all pairs of congruent corresponding parts. 

Problem 2 : 

In the diagram given below, NPLM ≅ EFGH. 

(i) Find the value of x. 

(ii) Find the value of y. 

Problem 3 : 

Find the value of x in the diagram given below. 

Problem 4 : 

Decide whether the triangles are congruent. Justify your reasoning. 

Problem 5 : 

In the diagram given below, prove that ΔAEB  ≅  ΔDEC.

Congruence and triangles worksheet - Solution

Problem 1 : 

The congruent triangles represent the triangles in the diagram given below. Write a congruence statement. Identify all pairs of congruent corresponding parts. 

Solution : 

The diagram indicates that ΔDEF ≅  ΔRST.

The congruent angles and sides are as follows. 

Angles : 

D ≅ R, E ≅ S and ≅ T

Sides : 

DE RS, EF ≅ ST and FD ≅ TR

Problem 2 : 

In the diagram given below, NPLM ≅ EFGH. 

(i) Find the value of x. 

(ii) Find the value of y. 

Solution (i) : 

We know that LM ≅ GH.

So, we have

LM  =  GH

8  =  2x - 3

Add to 3 to both sides.

11  =  2x

Divide both sides by 2.

5.5  =  x

Solution (ii) : 

We know that N ≅ E.

So, we have

mN  =  mE

72°  =  (7y + 9)°

72  =  7y + 9

Subtract 9 from both sides.

63  =  7y

Divide both sides by 7.

9  =  y

Problem 3 : 

Find the value of x in the diagram given below. 

Solution : 

In the diagram given above, ∠N ≅ ∠R and ∠L ≅ ∠S. From the Third angles theorem, we know that ∠M ≅ ∠T. So, m∠M  =  m∠T.

From the triangle sum theorem, we have 

m∠L + m∠M + m∠N  =  180°

65° + 55° + m∠M  =  180°

Simplify

120° + m∠M  =  180°

Subtract 120° from both sides. 

m∠M  =  60° 

By Third angles theorem, we have

m∠M  =  m∠T

Substitute 60° for m∠M and (2x + 30)° for m∠M.

60°  =  (2x + 30)°

60  =  2x + 30

Subtract 30 from both sides.

30  =  2x

Divide both sides by 2. 

15  =  x

Problem 4 : 

Decide whether the triangles are congruent. Justify your reasoning. 

Solution :

From the diagram, we are given that all three pairs of corresponding sides are congruent. 

RP ≅ MN, PQ ≅ NQ and Q≅ QM

Because ∠P and ∠N have the same measure, ∠P ≅ ∠N.

By the Vertical Angles Theorem, we know that 

ΔPQR  ≅  ΔNQM

By the Third Angles Theorem, 

R  ≅  M

So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent angles, 

ΔPQR  ≅  ΔNQM

Problem 5 : 

In the diagram given below, prove that ΔAEB  ≅  ΔDEC.

Solution : 

Given :

AB || DC and AB  ≅  DC

E is the midpoint of BC and AD. 

To prove :

ΔAEB  ≅  ΔDEC.

Statements

AB || DC and AB  ≅  DC

aaaaa ∠EAB  ≅  ∠EDC aaaa aaaaa ∠ABE  ≅  ∠DCE aaaa

∠ABE  ≅  ∠DCE

E is the midpoint of AD

E is the midpoint of BC

AE  ≅  DE, BE  ≅  CE

aaaa ΔAEB  ≅  ΔDEC aaaa aaaaaaaaaaaaaaaaaaaaaaaa

Reasons

Given

Alternate Interior Angles Theorem. 

Vertical Angles Theorem

Given

Given

Definition of midpoint.

Definition of congruent triangles.

After having gone through the stuff given above, we hope that the students would have understood, "Congruence and triangles worksheet". 

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