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.

 StatementsAB || DC and AB  ≅  DCaaaaa ∠EAB  ≅  ∠EDC aaaa aaaaa ∠ABE  ≅  ∠DCE aaaa∠ABE  ≅  ∠DCEE is the midpoint of AD E is the midpoint of BCAE  ≅  DE, BE  ≅  CEaaaa ΔAEB  ≅  ΔDEC aaaa aaaaaaaaaaaaaaaaaaaaaaaa ReasonsGivenAlternate Interior Angles Theorem. Vertical Angles TheoremGivenGivenDefinition of midpoint.Definition of congruent triangles.

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