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**By applying SAS congruence condition, state which of the following pairs (Fig. 28) of triangles are congruent. State the result in symbolic form**

1) We have OA = OC and OB = OD andâˆ AOB =âˆ COD which are vertically opposite angles.

Therefore by SAS condition, â–³AOC â‰… â–³BOD.

2) We have BD = DC

âˆ ADB =âˆ ADC = 90Â°

and AD = AD

Therefore by SAS condition, â–³ADB â‰… â–³ADC.

3) We have AB = DC

âˆ âˆ ABD = âˆ CDB and BD = DB

Therefore by SAS condition, â–³ABD â‰… â–³CBD.

4) We have BC = QR

âˆ ABC =âˆ PQR = 90Â°

and AB = PQ

Therefore by SAS condition, â–³ABC â‰… â–³PQR.

**State the condition by which the following pairs of triangles are congruent.**

1) AB = AD

BC = CD

and AC = CA

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

2) AC = BD

AD = BC and AB = BA

Therefore by SSS condition, â–³ABDâ‰…â–³BAC.

3) AB = AD

âˆ BAC=âˆ DACâˆ BAC=âˆ DAC

and AC = AC

Therefore by SAS condition, â–³BACâ‰…â–³DAC.

4) AD = BC

âˆ DAC = âˆ BCA

and AC = CA

Therefore by SAS condition, â–³ABCâ‰…â–³ADC.

**In Fig. 30, line segments AB and CD bisect each other at O. Which of the following statements is true?**

(i) âˆ† AOC â‰… âˆ† DOB

(ii) âˆ† AOC â‰… âˆ† BOD

(iii) âˆ† AOC â‰… âˆ† ODB.

State the three pairs of matching parts, yut have used to arive at the answer.

We have AO = OB.

And CO = OD

Also âˆ AOC = âˆ BOD

Therefore by SAS condition, â–³AOC â‰… â–³BOD.

Therefore, statement (ii) is true.

**Line-segments AB and CD bisect each other at O. AC and BD are joined forming triangles AOC and BOD. State the three equality relations between the parts of the two triangles, that are given or otherwise known. Are the two triangles congruent? State in symbolic form. Which congruence condition do you use?**

**Answer 4:**

We have AO = OB and CO = OD since AB and CD bisect each other at O.

Also âˆ AOC = âˆ BOD since they are opposite angles on the same vertex.

Therefore by SAS congruence condition,

â–³AOCâ‰…â–³BOD

**âˆ† ABC is isosceles with AB = AC. Line segment AD bisects âˆ A and meets the base BC in D.**

(i) Is âˆ† ADB â‰… âˆ† ADC?

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

(iii) Is it true to say that BD = DC?

(i) We have AB = AC (given)

âˆ BAD =âˆ CADâˆ BAD =âˆ CAD (AD bisects âˆ BAC)

and AD = AD (common)

Therefore by SAS condition of congruence, â–³ABDâ‰…â–³ACD

(ii) We have used AB, AC; âˆ BAD =âˆ CAD; AD, DA.

(iii) Nowâ–³ABDâ‰…â–³ACD therefore by c.p.c.t BD = DC.

**In Fig. 31, AB = AD and âˆ BAC = âˆ DAC.**

(i) State in symbolic form the congruence of two triangles ABC and ADC that is true.

(ii) Complete each of the following, so as to make it true:

(a) âˆ ABC = ........

(b) âˆ ACD = ........

(c) Line segment AC bisects ..... and .....

i) AB = AD (given)

âˆ BAC =âˆ DAC (given)

AC = CA (common)

Therefore by SAS conditionof congruency, â–³ABC â‰…â–³ADC

ii) âˆ ABC =âˆ ADC (c.p.c.t)

âˆ ACD =âˆ ACB (c.p.c.t)

**In Fig. 32, AB || DC and AB = DC.**

(i) Is âˆ† ACD â‰… âˆ† CAB?

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

(iii) Which angle is equal to âˆ CAD?

(iv) Does it follow from (iii) that AD || BC?

**Answer 7:**

(i) Yes â–³ACDâ‰…â–³CAB by SAS condition of congruency.

(ii) We have used AB = DC, AC = CA and âˆ DCA=âˆ BAC.

(iii) âˆ CAD =âˆ ACB since the two triangles are congruent.

(iv) Yes, this follows from ADâˆ¥âˆ¥BC as alternate angles are equal.If alternate angles are equal the lines are parallel.