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**In the following pairs of triangles (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 form.**

1) In Î”ABC and Î”DEFâˆ†ABC and âˆ†DEF

AB = DE = 4.5 cm (Side)

BC = EF = 6 cm (Side)

and AC = DF = 4 cm (Side)

Therefore, by SSS criterion of congruence, â–³ABCâ‰…â–³DEFâ–³ABCâ‰…â–³DEF.

2)

In â–³ACB and â–³ADB

AC =AD (Side)

BC = BD (Side)

and AB=AB (Side)

Therefore, by SSS criterion of congruence, â–³ACBâ‰…â–³ADB

3)

In â–³ABD and â–³FEC,

AB = FE (Side)

AD = FC (Side)

BD = CE (Side)

Therefore, by SSS criterion of congruence, â–³ABDâ‰…â–³FECâ–³ABDâ‰…â–³FEC.

**In Fig. 16, AD = DC and AB = BC.**

(i) Is âˆ† ABD â‰… âˆ† CBD?

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

Yes â–³ABDâ‰…â–³CBDâ–³ABDâ‰…â–³CBD by the SSS criterion.

We have used the three conditions in the SSS criterion as follows:

AD = DC

AB = BC

and DB = BD

**In Fig. 17, AB = DC and BC = AD.**

**(i) Is âˆ† ABC â‰… âˆ† CDA?**

(ii) What congruence condition have you used?

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

We have AB = DC

BC = AD

and AC = AC

Therefore by SSS â–³ABCâ‰…â–³CDA.

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

Yes, we have used the fact that AC = CA.

**If âˆ† PQR â‰… âˆ† EFD, (i) Which side of âˆ† PQR equals ED? (ii) Which angle of âˆ† PQR equals âˆ E?**

**Answer 4:**

â–³PQR â‰… â–³EDF

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

2) âˆ QPR = âˆ FED since the corresponding angles of congruent triangles are equal.

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

If âˆ B = 50Â°, what is the measure of âˆ R?

We have AB = AC in isosceles â–³ABC

and PQ = PR in isosceles â–³PQR.

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

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

Hence, â–³ABCâ‰…â–³PQR.

Now

âˆ ABC = âˆ PQR (Since triangles are congruent)

However, â–³PQR is isosceles.

Therefore, âˆ PRQ =âˆ PQR =âˆ ABC = 50Â°

*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 âˆ *BAC* = 40Â° andâˆ *BDC* = 100Â°; then find âˆ *ADB*.

YES â–³ADB â‰…â–³ADC (By SSS)

AB = AC , DB = DC AND AD= DA

âˆ BAD=âˆ CAD (c.p.c.t)

âˆ BAD+âˆ CAD=40Â°

2âˆ BAD=40Â°

âˆ BAD=40Â°/2=20Â°

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

Since Î”ABC is an isosceles triangle,

âˆ ABC=âˆ BCA

âˆ ABC+âˆ ABC+40Â°=180Â°

2âˆ ABC=180Â°âˆ’40Â°=140Â°

âˆ ABC=140Â°/2=70Â°

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

Since Î”ABC is an isosceles triangle,

âˆ DBC=âˆ BCD

âˆ DBC+âˆ DBC+100Â°=180Â°

2âˆ DBC=180Â°âˆ’100Â°=80Â°

âˆ DBC=80Â°/2=40Â°

In Î”BAD,

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

30Â°+20Â°+âˆ ADB=180Â° (âˆ ABD=âˆ ABC-âˆ DBC)

âˆ ADB=180Â°âˆ’20Â°âˆ’30Â°

âˆ ADB=130Â°

âˆ ADB =130Â°

**âˆ† ABC and âˆ† 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) âˆ† ABC â‰… âˆ† ABD

(ii) âˆ† ABC â‰… âˆ† ADB

(iii) âˆ† ABC â‰… âˆ† BAD

In â–³ABC and â–³BAD we have,

AC = BD (given)

BC = AD (given)

and AB = BA (common)

Therefore by SSS criterion of congruency, â–³ABC â‰…â‰…â–³BAD.

There option (iii) is true.

**In Fig. 19, âˆ† ABC is isosceles with AB = AC, D is the mid-point of base BC.**

(i) Is âˆ† ADB â‰… âˆ† ADC?

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

We have AB = AC.

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

And AD = DA.

Therefore by SSS condition, â–³ABD â‰…â–³ADCâ–³ABD â‰…â–³ADC.

We have used AB, AC : BD, DC and AD, DA.

**In Fig. 20, âˆ† ABC is isosceles with AB = AC. State if âˆ† ABC â‰… âˆ† ACB. If yes, state three relations that you use to arrive at your answer.**

Yes â–³ABC â‰…â–³ACB by SSS condition.

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

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

Yes.

In Î”ABC and Î”DB

CAB=DB (Given)

AC=DC (Given)

BC=BC (Common)

By SSS criterion of congruency, Î”ABCâ‰…Î”DBC

No, âˆ ABD andâˆ ACD are not equal

because AB â‰ AC.