The document RD Sharma Solutions Ex-8.4, (Part -2), Lines And Angles, Class 9, Maths Class 9 Notes | EduRev is a part of the Class 9 Course RD Sharma Solutions for Class 9 Mathematics.

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**Q 15 : In the below fig, âˆ 1 = 60 ^{âˆ˜} and âˆ 2 = (2/3)rd of a right angle. Prove that l|| m.**

Ans : Given :

âˆ 1 = 60^{âˆ˜} and âˆ 2 = (2/3)rd of a right angle

To prove : parallel Drawn to m

Proof âˆ 1 = 60

âˆ 2 = (23)Ã—90 = 60

Since âˆ 1 =âˆ 1 = 60^{âˆ˜}

Therefore, Parallel to m as pair of corresponding angles are equal.

**16. In the below fig, if l||m||n and âˆ 1 = 60 ^{âˆ˜}. Find âˆ 2.**

**Ans :** Since l parallel to m and p is the transversal

Therefore, Given: l||m||n

âˆ 1 = 60^{âˆ˜}

To find âˆ 2

âˆ 1=âˆ 3 = 60^{âˆ˜ } [Corresponding angles]

Now, âˆ 3andâˆ 4 are linear pair of angles

âˆ 3+âˆ 4 = 180^{âˆ˜}

60 + âˆ 4 = 180

âˆ 4 = 180 â€” 60

â‡’ 120

Also, m||n and P is the transversal

Therefore âˆ 4 = âˆ 2 = 120 (Alternative interior angle]

Hence 2 âˆ 2 = 120

**Q 17 : Prove that the straight lines perpendicular to the same straight line are parallel to one another.**

**Ans :** Let AB and CD be drawn perpendicular to the Line MN

âˆ ABD = 90^{âˆ˜} [ AB is perpendicular to MN ] â€”â€“(i)

âˆ CON = 90^{âˆ˜} [CD is perpendicular to MN ] â€”â€“(ii)

Now,

âˆ ABD = âˆ CDN = 90^{âˆ˜} [From (i) and (ii)]

Therefore, AB||CD, Since corresponding angles are equal.

**Q 18 : The opposite sides of a quadrilateral are parallel. If one angle of the quadrilateral is 601. Find the other angles.**

**Ans :** Given AB || CD

AD|| BC

Since AB || CD and AD is the transversal

Therefore, A + D = 180 (Co-interior angles are supplementary)

60 + D = 180

D = 180 â€“ 60

D = 120

Now. AD || BC and AB is the transversal

A + B = 180 (Co-interior angles are supplementary)

60 +B = 180

B = 180 â€” 60

= 120

Hence, âˆ A = âˆ C = 60^{âˆ˜} and âˆ B =âˆ D = 120âˆ˜

**Q 19 : Two lines AB and CD intersect at O. If âˆ AOC+âˆ COB+âˆ BOD = 270 ^{âˆ˜}, find the measures of âˆ AOC,âˆ COB,âˆ BOD,âˆ DOA**

Ans :

Given : âˆ AOC+âˆ COB+âˆ BOD=270âˆ˜

To find : âˆ AOC,âˆ COB,âˆ BOD,âˆ DOA

Here, âˆ AOC+âˆ COB+âˆ BOD = 270^{âˆ˜} [ Complete angle]

â‡’ 270 + AOD = 360

â‡’ AOD = 360 â€” 270

â‡’ AOD = 90

Now, AOD + BOD = 180 [Linear pair]

90 + BOD = 180

â‡’ BOD = 180 â€“ 90

â‡’ BOD = 90

AOD = BOC = 90 [Vertically opposite angles]

BOD = AOC = 90 [Vertically opposite angles]

**Q 20. In the below figure, p is a transversal to lines m and n, âˆ 2 = 120 ^{âˆ˜} and âˆ 5 = 60^{âˆ˜}. Prove that m|| n.**

Ans :

Given that

âˆ 2 = 120^{âˆ˜} and âˆ 5 = 60^{âˆ˜}

To prove,

âˆ 2+âˆ 1 = 180^{âˆ˜} [ Linear pair ]

120+âˆ 1 = 180

âˆ 1 = 180âˆ’120

âˆ 1 = 60^{âˆ˜}

Since âˆ 1 = âˆ 5 = 60^{âˆ˜}

Therefore, m||n [As pair of corresponding angles are equal]

**Q 21 : In the below fig. transversal t intersects two lines m and n, âˆ 4 =110 ^{âˆ˜} and âˆ 7 = 65^{âˆ˜} Is m||n ?**

Ans : Given :

âˆ 4 = 110^{âˆ˜} and âˆ 7 = 65^{âˆ˜}

To find : Is m||n

Here. âˆ 7 = âˆ 5 = 65^{âˆ˜} [Vertically opposite angle]

Now. âˆ 4+âˆ 5 = 110+65 = 175^{âˆ˜}

Therefore, m is not parallel to n as the pair of co interior angles is not supplementary.

**Q 22 : Which pair of lines in the below fig. is parallel ? give reasons.**

Ans : âˆ A+âˆ B = 115+65 = 180^{âˆ˜}

Therefore, AB || BC [ As sum of co interior angles are supplementary]

âˆ B+âˆ C = 65+115 = 180^{âˆ˜}

Therefore, AB || CD (As sum of interior angles are supplementary]

**Q 23 : If I, m, n are three lines such that I|| m and n perpendicular to l, prove that n perpendicular to m.**

Ans :

Given, l||m, n perpendicular to I

To prove: n perpendicular to m

Since l||m and n intersects

âˆ´ âˆ 1 = âˆ 2 [Corresponding angles]

But, U = 90

â‡’ âˆ 2 = 90^{âˆ˜}

Hence n is perpendicular to m

**Q 24 : In the below fig, arms BA and BC of âˆ ABC are respectively parallel to arms ED and EF ofâˆ DEF. Prove that âˆ ABC = âˆ DEF.**

**Ans :**

Given

AB || DE and BC || EF

To prove : âˆ ABC=âˆ DEF

Construction: Produce BC to x such that it intersects DE at M.

Proof : Since AB || DE and BX is the transversal

ABC = DMX [Corresponding angle] â€”â€“(i)

Also, BX || EF and DE Is the transversal

DMX = DEF [Corresponding angles] â€”â€“(ii)

From (i) and (ii)

âˆ ABC =âˆ DEF

**Q 25: In the below fig, arms BA and BC of ABC are respectively parallel to arms ED and EF of DEF Prove that âˆ ABC+âˆ DEP = 180 ^{âˆ˜}**

**Ans :**

Given:

AB II DE, BC II EF

To prove: âˆ ABC+âˆ DEF = 180^{âˆ˜}

Construction: Produce BC to intersect DE at M

Proof :

Since AB || EM and BL is the transversal

âˆ ABC = âˆ EML [Corresponding angle] â€”â€“(i)

Also,

EF || ML and EM is the transversal

By the property of co-interior angles are supplementary

âˆ DEF+âˆ EML = 180^{âˆ˜} (ii)

From (i) and (ii) we have

Therefore âˆ DEF+âˆ ABC = 180^{âˆ˜}

**Q 26 : With of the following statements are true (T) and which are false (F)? Give reasons.**

**(1) If two lines are intersected by a transversal, then corresponding angles are equal.**

**(ii) If two parallel lines are intersected by a transversal, then alternate interior angles are equal.**

**(ii) Two lines perpendicular to the same line are perpendicular to each other.**

**(iv) Two lines parallel to the same line are parallel to each other.**

**(v) If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.**

**Ans :**

(i) False

(ii)True

(iii) False

(iv) True

(v) False

**Q 27: Fill in the blanks in each of the following to make the statement true:**

**(i) If two parallel lines are intersected by a transversal, then each pair of corresponding angles are ____________ **

**(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are _____________**

**(iii) Two lines perpendicular to the same line are _______ to each other**

**(Iv) Two lines parallel to the same line are __________ to each other.**

**(v) If a transversal intersects a pair of lines in such a way that a pair of alternate angles we equal. then the lines are ___________**

**(vi) If a transversal intersects a pair of lines in such a way that the sum of interior angles on the seine side of transversal is 180â€². then the lines are _____________**

**Ans :**

(i) Equal

(ii) Parallel

(iii) Supplementary

(iv) Parallel

(v) Parallel

(vi) Parallel