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**Q.1. State which pairs of triangles in Fig. are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form: ****Sol.** (i) In Î” ABC and Î” PQR

We have:

âˆ A = âˆ P [Each 60Â°]

âˆ B = âˆ Q [Each 80Â°]

âˆ C = âˆ R [Each 40Â°]

Î”ABC ~ Î”PQR [AAA criterion]

(ii) In Î”ABC and Î”QRP

Hence, Î”ABC ~ Î”QRP [SSS criterion]

(iii) In Î”LMP and Î”EFD

âˆ´ Î”LMP is not similar to Î”EFD.

Since the three ratios are not same.

(iv) In Î”MNL and Î”QPR

(v) In Î”ABC and Î”DEF

Î”ABC is not similar to Î”DEF

âˆµ Angles between two sides are not same

(vi) In Î”DEF and Î”PQR

âˆ E = âˆ Q = 80Â°

âˆ F = âˆ R = 30Â°

[âˆ´ âˆ F = 180Â° - (80Â° + 70Â°) = 30Â°]

âˆ´ Î”DEF ~ Î”PQR [AA]**Q.2. In the figure, Î”ODC ~ Î”OBA, âˆ BOC = 125Â° and âˆ CDO = 70Â°. Find âˆ DOC, âˆ DCO and âˆ OAB.**

**Sol.**

â‡’

In Î”DOC,

â‡’ 125Â° + âˆ DCO = 180Â°

â‡’ âˆ DCO = 180Â° - 125Â° = 55Â°

Î”ODC ~ Î”OBA

âˆ OAB = âˆ DCO = 55Â°

â‡’ âˆ DOC = âˆ OAB = 55Â°**Q.3. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that ****Sol.** Given: Diagonals AC and BD intersect at O.

AB **||** DC

Proof: In Î”AOB and Î”COD

âˆ 1 = âˆ 2

âˆ 3 = âˆ 4 [Alternate angles]

âˆ´ Î”AOB ~ Î”COD [AA]

â‡’

[Corresponding sides of similar triangles]**Q.4. In the figure, **** and âˆ 1 = âˆ 2. Show that Î”PQS ~ Î”TQR.****So**l. Given

To Prove: Î”POS âˆ¼ Î”TOR

To Prove: Î”PQS

Proof: In Î”PQR,

âˆ 1 = âˆ 2 [Given]

PQ = PR [Sides opposite to equal angles]

OR/QS = QT/PR [Given]

Or

In Î”PQS and Î”TQR,

(Proved)

â‡’

âˆ 1 = âˆ 1 [Common]

âˆ´ âˆ PQS ~ âˆ TQR [SAS]**Q.5. S and T are points on sides PR and QR of Î”PQR such that âˆ P = âˆ RTS. Show that **Î”**RPQ ~ **Î”**RTS.****Sol. **In Î” RPQ ~ Î”RTS

âˆ P = âˆ RTS [Given]

âˆ R = âˆ R [Common]

âˆ´ Î”RPQ ~ Î”RTS [AA]**Q.6. In the figure, if **Î”**ABE â‰Œ Î”ACD, show that **Î”**ADE ~ **Î”**ABC.****Sol.**

Given: Î”ABE â‰Œ Î”ACD

To Prove: Î”ADE **~** Î”ABC

Proof: Î”ABE **~** Î”ACD

AB = AC and AE = AD

â‡’

In Î”ADE and Î”ABC,

[Proved above]

âˆ A - âˆ A [Common]

âˆ´ Î”ADE ~ Î”ABC [SAS]**Q.7. In the figure, altitudes AD and CE of **Î”**ABC intersect each other at the point P. Show that:****(i) **Î”**AEP ~ **Î”**CDP ****(ii) **Î”**ABD ~ **Î”**CBE ****(iii) **Î”**AEP ~ **Î”**ADB ****(iv) **Î”**PDC ~ **Î”**BEC****Sol.** Given: AD and CE are altitudes of the Î”ABC

(i) To Prove: Î”AEP ~ Î”CDP

Proof: In Î”AEP and Î”CDP,

âˆ AEP = âˆ CDP [Each'90Â°]

âˆ APE = âˆ CPD [Vertically opposite angles]

Î”AEP **~** Î”CDP [AA]

(ii) In Î”ABD and Î”CBE,

âˆ ADB = âˆ CEB [Each 90Â°]

âˆ ABD = âˆ CBE [Common]

Î”ABD ~ Î”CBE [AA]

(iii) In Î” AEP and Î”ADB,

âˆ AEP = âˆ ADB [Each 90Â°]

âˆ A = âˆ A [Common]

âˆ´ Î”AEP ~ Î”ADB [AA]

(iv) In Î”PDC and Î”BEC,

âˆ PDC = âˆ BEC [Each 90Â°]

âˆ PCD = âˆ BCE [Common]

Î”PDC ~ Î”BEC [AA]**Q.8. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that **Î”** ABE ~ **Î”**CFB.****Sol.** In Î”ABE and Î”CFB,

âˆ 1 = âˆ 2

âˆ 4 = âˆ 3 [Alternate angles]

â‡’ Î”ABE âˆ¼ Î”CFB [AA]**Q.9. In the figure, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that:****(i) **Î”** ABC ~ **Î”** AMP ****(ii) ****Sol.** (i) In Î”ABC and Î”AMP,

âˆ B = âˆ AMP [Each 90Â°]

âˆ A = âˆ A [Common]

â‡’ Î”ABC âˆ¼ Î”AMP [AA]

(ii) Î”ABC ~ Î”AMP [proved above]

[Ratio of the Corresponding sides of similar Î”s]**Q.10. CD and GH are respectively the bisectors of âˆ ACB and âˆ EGF such that D and H lie on sides AB and FE of **Î”** ABC and **Î”** EFG respectively. If **Î”**ABC ~ **Î”**FEG, show that: (i)**

â‡’ âˆ A = âˆ F

(i) In Î”ACD and Î”FGH

âˆ A = âˆ F [Given]

âˆ´ Î”ACD ~ Î”FGH

[Corresponding sides of similar triangles]

(ii) [Proved above]

In Î”DCB and Î”HGE,

âˆ´ Î”DCB ~ Î”HGE [SAS]

(iii) In Î”DCA and Î”HGF,

âˆ 1 = âˆ 2 [Bisectors]

â‡’ Î”DCA ~ Î”HGF [SAS]**Q.11. In the figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD âŠ¥BC and EF âŠ¥AC, prove that **Î”**ABD ~ **Î”**ECF.****Sol.** In Î”ABD and Î”ECF,

âˆ ADB = âˆ EFC [Each 90Â°]

âˆ B = âˆ C

[angles opposite to equal sides are equal]

â‡’ Î”ABD ~ Î”ECF [AA]**Q.12. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of Î” PQR (see figure). Show that **Î”** ABC ~ **Î”**PQR.****Sol. **In Î”ABC and Î”PQR

â‡’ Î”ABD **~** Î”PQM [SAS]

âˆ´ âˆ B = âˆ Q

[Corresponding angles of similar triangles]

In Î”ABC and Î”PQR,

[Given]

âˆ B = âˆ Q [As provecd]

âˆ´ Î”ABC **~** Î”PQR [SAS]**Q.13. D is a point on the side BC of a triangle ABC such that âˆ ADC = âˆ BAC. Show that CA ^{2} = CB.CD.**

âˆ BAC = âˆ ADC [Given]

âˆ C = âˆ C [Common]

Î”ACB

â‡’

[Corresponding sides of similar triangles]

â‡’ CA

âˆ´ E is mid-point of AB (by converse of mid-point theorem

âˆ´ Î”ADE ~ Î”PMS [SSS similarly]

Now, in Î”ABC and Î”PQR,

[Given]

âˆ A = âˆ P [Proved above]

âˆ´ Î”ABC ~ Î”PQR (SAS)

Length of shadow = 4 m

Let height of tower = h m

Length of shadow = 28 m

âˆ´

In Î”ABD and Î”PQM

âˆ´ Î”ABD

[Corresponding sides of similar triangles]

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