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Congruence Exercise 16.2

Question: 1

In the following pairs of triangle (Fig. 12 to 15), the lengths of the sides are indicated along sides. By applying SSS condition, determine which are congruent. State the result in symbolic.

Congruence Exercise 16.2 Question: 1A

Congruence Exercise 16.2 Question: 1B

Solution:

Congruence Exercise 16.2 Solution: 1A

1) In Δ ABC and Δ DEF

AB = DE = 4.5 cm (Side)

BC = EF = 6 cm (Side) and

AC = DF = 4 cm (Side)

Therefore, by SSS criterion of congruence, ΔABCΔDEF

2)


Congruence Exercise 16.2 Solution: 1B

In Δ ACB and Δ ADB

AC = AD (Side)

BC = BD (Side) and

AB = AB (Side)

Therefore, by SSS criterion of congruence, ΔACBΔADB

3) In Δ ABD and Δ FEC,

AB = FE (Side)

AD = FC (Side)

BD = CE (Side)

Therefore, by SSS criterion of congruence, ΔABDΔFEC

4) In Δ ABO and Δ DOC,

AB = DC (Side)

AO = OC (Side)

BO = OD (Side)

Therefore, by SSS criterion of congruence, ΔABOΔODC

Question: 2

In figure, AD = DC and AB = BC

(i) Is ΔABDΔCBD?

(ii) State the three parts of matching pairs you have used to answer (i).

Congruence Exercise 16.2 Question: 2

Solution:

Yes Δ ABD = Δ CBD by the SSS criterion. We have used the three conditions in the SSS criterion as follows:

AD = DC

AB = BC and

DB = BD

Question: 3

In Figure, AB = DC and BC = AD.

(i) Is ΔABC ΔCDA?

(ii) What congruence condition have you used?

(iii) You have used some fact, not given in the question, what is that?

Congruence Exercise 16.2 Question: 3

Solution:

We have AB = DC

BC = AD

and AC = AC

Therefore by SSS ΔABC ΔCDA

We have used Side congruence condition with one side common in both the triangles.

Yes, have used the fact that AC = CA.

Question: 4

In ΔPQR ΔEFD,

(i) Which side of ΔPQR equals ED?

(ii) Which angle of ΔPQR equals angle E?

Solution:

ΔPQR ΔEFD

(i) Therefore PR = ED since the corresponding sides of congruent triangles are equal.

(ii) QPR = FED since the corresponding angles of congruent triangles are equal.

Congruence Exercise 16.2 Question: 4

Question: 5

Triangles ABC and PQR are both isosceles with AB = AC and PO = PR respectively. If also, AB = PQ and BC = QR, are the two triangles congruent? Which condition do you use?

It ∠B = 50°, what is the measure of ∠R?

Solution:

We have AB = AC in isosceles Δ ABC

And PQ = PR in isosceles Δ PQR.

Also, we are given that AB = PQ and QR = BC.

Therefore, AC = PR (AB = AC, PQ = PR and AB = PQ)

Hence, ΔABCΔPQR

Now

ABC = PQR (Since triangles are congruent)However, Δ PQR is isosceles.

Therefore, PRQ = PQR = ∠ABC = 50°

Question: 6

ABC and DBC are both isosceles triangles on a common base BC such that A and D lie on the same side of BC. Are triangles ADB and ADC congruent? Which condition do you use? If ∠BAC = 40° and BDC = 100°, then find ADB.

Solution:

BAD = CAD (c.p.c.t)

BAD + CAD = 40°/ 2 BAD = 40°

BAD = 40°/2 =20°

ABC + BCA + BAC = 180° (Angle sum property)

Since Δ ABC is an isosceles triangle,

ABC = BCA ABC +ABC + 40°= 180°

2 ABC = 180° – 40° = 140° ABC = 140°/2 = 70°

DBC +  BCD + BDC = 180° (Angle sum property)

Since Δ ABC is an isosceles triangle, DBC = BCD DBC + DBC + 100o = 180°

2 DBC = 180° – 100o = 80°

DBC=80°/2=40°

In Δ BAD,

ABD + BAD + ADB = 180°(Angle sum property)

30° + 20° + ADB = 180° (ADB = ABC – DBC), ADB =180°- 20° – 30°

ADB = 130°

ADB =130°

Question: 7

Δ ABC and Δ ABD are on a common base AB, and AC = BD and BC = AD as shown in Fig. 18. Which of the following statements is true?

(i) ΔABCΔABD

(ii) ΔABCΔADB

(iii) ΔABCΔBAD

Congruence Exercise 16.2 Question: 7

Solution:

In Δ ABC and Δ BAD we have,

AC = BD (given)

BC = AD (given)

and AB = BA (common)

Therefore by SSS criterion of congruency, ΔABCΔBAD

There option (iii) is true.

Question: 8

In Figure, Δ ABC is isosceles with AB = AC, D is the mid-point of base BC.

(i) Is ΔADBΔADC?

(ii) State the three pairs of matching parts you use to arrive at your answer.

Congruence Exercise 16.2 Question: 8

Solution:

We have AB = AC.

Also since D is the midpoint of BC, BD = DC

Also, AD = DA

Therefore by SSS condition,

ΔADBΔADC

We have used AB, AC : BD, DC AND AD, DA

Question: 9

In figure, Δ ABC is isosceles with AB = AC. State if ΔABCΔACB. If yes, state three relations that you use to arrive at your answer.

Congruence Exercise 16.2 Question: 9

Solution:

Yes, ΔABCΔACB by SSS condition.

Since, ABC is an isosceles triangle, AB = BC, BC = CB and AC = AB

Question: 10

Triangles ABC and DBC have side BC common, AB = BD and AC = CD. Are the two triangles congruent? State in symbolic form, which congruence do you use? Does ABD equal ACD? Why or why not?

Solution:

Yes,

Given,

Δ ABC and Δ DBC have side BC common, AB = BD and AC = CD

By SSS criterion of congruency, ΔABCΔDBC

No, ABD and ACD are not equal because AB AC

Congruence Exercise 16.2 Question: 10

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