## RD Sharma Solutions Class 9 Chapter 8 Ex 8.4

**Q 1: Â In below fig. AB CD and âˆ 1 and âˆ 2Â are in the ratio 3 : 2. Determine all angles from 1 to 8.**

Ans :Â Â Â Â Â Let âˆ 1 = 3Î¸ and âˆ 2Â = 2Î¸

As you can see, âˆ 1 andÂ âˆ 2Â are linear pair of angle

Therefore, we can write it as;

=> âˆ 1 + âˆ 2 = 180

=> Â 3Î¸ + 2Î¸ = 180

=> 5Î¸Â = 180

=>Â Î¸ = 180 / 5

=>Â Î¸ = 36

âˆ 1 = 3Î¸ = 108Â°, âˆ 2 = 2Î¸ = 72Â°

As we know already from the theorem, the vertically opposite angles are equal, therefore;

âˆ 1 = âˆ 3 = 108Â°

âˆ 2 = âˆ 4 = 72Â°

âˆ 6 = âˆ 7 = 108Â°

âˆ 5 = âˆ 8 = 72Â°

We also know, if a transversal intersects any parallel lines, then the corresponding angles are equal

âˆ 1 = âˆ 5 = 108Â°

âˆ 2 = âˆ 6 = 72Â°

**Q 2: In the below fig, I, m and n are parallel lines intersected by transversal p at X. Y and Z respectively. Find âˆ 1, âˆ 2 andÂ âˆ 3.**

Ans : From the given figure :

âˆ 3+ âˆ m YZ = 180Â°Â Â Â Â Â Â [From Linear pair angles theorem]

=> âˆ 3= 180 – 120

=> âˆ 3= 60Â°

From the figure, line l is parallel to line m, therefore,

âˆ 1 = âˆ 3Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Both are Corresponding angles]

âˆ 1 = 60Â°

Also, line mÂ is parallel to line n;

âˆ 2 =Â âˆ mYZÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Alternate interior angles are equal]

âˆ´ âˆ 2 = 120Â°

Hence, âˆ 1 = 60Â°,Â âˆ 2 = 120Â° andÂ âˆ 3 =Â 60Â°.

**Q.3. In the below fig, AB || CD || EF and GH || KL Find âˆ HKL.**

Ans :Â Â Â Extend LK to meet GF at P.

Now, CD and GF are parallel lines, therefore, alternate angles are equal.

âˆ CHG =âˆ HGP = 60Â°

âˆ HGP =âˆ KPF = 60Â°Â Â Â Â Â [Corresponding angles of parallel lines are equal]

Hence, âˆ KPG =180 – 60 = 120Â°

=> âˆ GPK = âˆ AKL= 120Â°Â [Corresponding angles of parallel lines are equal]

âˆ AKH = âˆ KHD = 25Â°Â Â Â Â [alternate angles of parallel lines]

Therefore, âˆ HKL = âˆ AKH + âˆ AKL = 25 + 120 = 145Â°

**Q 4 : In the below fig, show that AB || EF.**

Ans : Produce EF to intersect AC at N.

Now, âˆ DCE + âˆ CEF = 35 + 145 = 180Â°

Therefore, EF || CD Â Â Â Â Â Â (Since Sum of Co-interior angles of parallel lines is 180)Â —–(1)

Now, âˆ BAC = 57Â°

âˆ ACD = 22Â°+35Â° = 57Â°

Therefore,

=>BA || EFÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Alternative angles of parallel lines are equal]Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â —–(2)

From (1) and (2)

AB || EFÂ Â Â Â Â Â Â Â Â Â Â Â Â [Since, Lines parallel to the same line are parallel to each other]

Hence proved.

**Q 5 : If below fig.if AB || CD and CD || EF, find âˆ ACE.**

Ans : Â Since EF || CD

Therefore, âˆ Â EFC + âˆ ECD = 180Â°Â Â Â Â Â Â Â [sum of co-interior angles are supplementary to each other]

=> âˆ ECD = 180Â° – 130Â° = 50Â°

Also, BA || CD

=> âˆ BAC = âˆ ACD = 70Â° Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [alternative angles of parallel lines are equal]

But, âˆ ACE + âˆ ECD =70Â°

=> âˆ ACE = 70Â° â€” 50Â° = 20Â°

**Q 6Â : In the below fig, PQ || AB and PR || BC. If âˆ QPR = 102Â°, determine âˆ ABC.Â Give reasons.**

Ans :Â Extend line AB to meet line PR at G.

Since PQ || AB,

âˆ´ âˆ QPR = âˆ BGR =102Â°Â Â Â Â Â Â Â Â Â Â Â Â Â [corresponding angles of parallel lines are equal]

Since PR || BC,

âˆ´ âˆ RGB+ âˆ CBG =180Â°Â Â Â Â Â Â Â Â Â Â Â Â Â [ Since Corresponding angles are supplementary]

âˆ CBG = 180Â° â€“ 102Â° = 78Â°

Since, âˆ CBG =Â âˆ ABC

âˆ´ âˆ ABC = 78Â°

**Q 7 : In the below fig, state Which lines are parallel and why?**

Ans : As we know, for parallel lines, vertically opposite angles are equal, therefore;

=> âˆ EOC = âˆ DOK = 100Â°

âˆ DOK =âˆ ACO = 100Â°

Here two lines ED and CA are cut by a third line DC and the corresponding angles to it are equal.

Therefore, ED || AC.

**Q.8. In the below fig. if l||m, n || p and âˆ 1Â = 85Â°. find âˆ 2.**

Ans : Given,Â âˆ 1 = 85Â°

As we know, when a line cuts the parallel lines, the corresponding angles are equal, therefore,

=> âˆ 1 = âˆ 3 = 85Â°

Now we know,Â co-interior angles are supplementary;

âˆ 2 + âˆ 3 = 180Â°

âˆ 2 + 55Â° =180Â°

âˆ 2 = 180Â° – 85Â°

âˆ 2 = 95Â°

**Q 9 : If two straight lines are perpendicular to the same line, prove that they are parallel to each other.**

Ans :Â Â Â Â Â Given line m and line l are perpendicular to line n.

âˆ 1= âˆ 2=90Â°

Since, lines l and m are cut by a transversal n and the corresponding angles are equal, therefore,

l || m

Hence, proved.

**Q 10 : Prove that if the two arms of an angle are perpendicular to the two arms of another angle. then the angles are either equal or supplementary.**

**Ans :**Â Let the angles beÂ âˆ ACB and âˆ ABD

Given,Â CA perpendicular to AB,Â also CD is perpendicular to BD

To prove : âˆ ACD = âˆ ABD Â (or) âˆ ACD + âˆ ABD =180Â°

Proof : Â In a quadrilateral = âˆ A+ âˆ C+ âˆ D+ âˆ B = 360Â°

[ Sum of angles of quadrilateral is 360 ]=> 180Â°Â + âˆ C + âˆ B = 360Â°

=> âˆ C + âˆ B = 360Â° â€“180Â°

Hence, âˆ ACD + âˆ ABD = 180Â° Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â —–(1)

Also, âˆ D + âˆ ABD = 180Â°

=> âˆ ABD =180Â° â€“ 90Â° = âˆ ACD = 90Â° Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â —–(2)

From (i) and (ii), âˆ ACD = âˆ ABD = 90Â°

Hence, it is proved that, the angles are equal as well as supplementary.

**Q 11 : In the below fig, lines AB and CD are parallel and P is any point as shown in the figure. Show that âˆ ABP + âˆ CDP = âˆ DPB.**

Ans :

Given that AB ||CD

Let QR be the parallel line to AB and CD which passes through P

As we know, for parallel lines, alternate interior angles are equal.

Therefore, âˆ ABP = âˆ BPR …………..(1)

And,

âˆ CDP = âˆ DPR ………………..(2)

Adding equation 1 and 2, we get,

âˆ ABP +Â âˆ CDP = âˆ BPR + âˆ DPR

âˆ ABP + âˆ CDP = âˆ DPB

Hence proved.

**Q 12: In the below fig, AB || CD and P is any point shown in the figure. Prove that : âˆ angle ABP + âˆ BPD + âˆ CDP = 360Â°**

Ans:

Given, AB parallel to CD, P is any point.

To prove: âˆ ABP+ âˆ BPD+ âˆ CDP = 360Â°

Through P, draw a line PM parallel to AB or CD.

Now,

AB || PM => âˆ ABP + âˆ BPM = 180Â°

And

CD||PM = âˆ MPD +Â âˆ CDP = 180Â°

Adding (i) and (ii), we get âˆ ABP + ( âˆ BPM + âˆ MPD ) +Â âˆ CDP = 360Â°

=> âˆ ABP + âˆ BPD + âˆ CDP = 360Â°

**Q 13 : Two unequal angles of a parallelogram are in the ratio 2 : 3. Find all its angles in degrees.**

Ans :Â Let âˆ A = 2Î¸ and âˆ B = 3Î¸

Now, âˆ A +âˆ B = 180Â°Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Co-interior angles are supplementary for parallel lines]

2Î¸Â + 3Î¸ = 180Â°Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [AD // BC and AB is the transversal)

=> 5Î¸ = 180Â°

Î¸Â = 180Â°Â /5

Î¸Â = 36Â°

Therefore, âˆ A = 2Â Ã— 36 = 72Â°

âˆ B = 3Â Ã— 36 Â = 108Â°

Now, âˆ A = âˆ C = 72Â°Â Â Â Â Â Â Â Â Â [Opposite angles of a parallelogram are equal]

âˆ B = âˆ D = 108Â°

**Q 14 : Â If each of the two lines is perpendicular to the same line, what kind of lines are they to each other?**

Ans :

Let lines AB and CD be perpendicular to line mn.

Line mn intersects AB and CD st O and P respectively.

âˆ AON = Â 90Â° [AB perpendicular to mn]Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â —– (i)

âˆ CPN = 90Â° [CD perpendicular to mn] Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â —– (ii)

From (i) and (ii), we get,

âˆ AON = âˆ CPN = 90Â°

Since the corresponding angles are equal of lines AB and CD.

Therefore, AB // CD.

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

Ans : Â Â Â Â Given :

âˆ Â 1 = 60Â° and âˆ 2 =(2/3)rd of a right angle

To prove : l is parallel to m

Proof:

âˆ Â 1 = 60Â°

âˆ 2 = (2/3) ofÂ 90Â° = 60Â°

Since, the corresponding anglesÂ âˆ Â 1 andÂ âˆ 2, both are equal to 60Â°. Therefore,Â l|| m.

**16. In the below figure, if l||m||n and Â âˆ Â 1Â =60Â°. Find âˆ 2.**

Ans :

Given, l||m||n and p is a transversal.

âˆ Â 1Â =60Â°

To find:Â âˆ 2 ?

SinceÂ âˆ 1 and âˆ 3 are corresponding angles,

therefore, âˆ 1 = âˆ 3 = 60Â°

And,Â âˆ 3Â and âˆ 4Â are linear pair of angles, formed on the transversal intersecting the line m.

Therefore,

âˆ 3 + âˆ 4Â = 180Â°

60Â° + âˆ 4Â = 180Â°

âˆ 4Â = 180Â° â€” 60Â°

= 120Â°

Now, we can see, m||n and p is the transversal, therefore the alternate interior angles will be equal.

i.e. âˆ 4Â Â = âˆ 2Â = 120**Â°**

Hence,Â âˆ 2Â = 120Â°

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

Ans : Let AB and CD are two straight lines perpendicular to the line mn.

Now, as per the given question,

âˆ ABD = 90Â°Â Â [ AB is perpendicular to mn ]Â Â Â Â Â —–(i)

âˆ CDn =90Â°Â Â Â [CD is perpendicular to mn ] Â Â Â Â Â —–(ii)

On comparing eq. (i) and (ii);

âˆ ABD = âˆ CDN =90Â°Â [From (i) and (ii)]

Now, as we know, if a transversal intersects two parallel lines, then their corresponding angles are equal.

Therefore, Â AB||CD.

**Q 18 : The opposite sides of a quadrilateral are parallel. If one angle of the quadrilateral is 60Â°. Find the other angles.**

Ans : Let us draw a quadrilateral ABCD, then as per the question,

AB || CD and AC|| BD

Hence, ABCD is a parallelogram, which has opposite angles as equal.

âˆ A =Â âˆ D andÂ âˆ B =Â âˆ C

Since,Â âˆ C = 60Â°, from the above figure.

Therefore,Â âˆ B = 60Â°

Since AB || CD and AC is the transversal line intersecting the lines AB and CD.

Therefore, âˆ A + âˆ C = 180Â° (Since sum of Co-interior angles equal to 180)

âˆ A + âˆ C = 180Â°

âˆ A = 180Â° â€“ 60Â°

âˆ A = 120Â°

AndÂ âˆ D = 120Â°

Hence, âˆ CÂ = âˆ B = 60Â° and âˆ B = âˆ D = 120Â°

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

Ans :

Given, as per the question,

âˆ AOC + âˆ COB + âˆ BOD = 270Â°

To find : âˆ AOC , âˆ COB , âˆ BOD, âˆ DOA = ?

Here, âˆ AOC + âˆ COB + âˆ BOD +Â âˆ AOD = 360Â°Â (Sum of all the angles)

=>Â Â Â Â Â Â Â Â Â Â 270Â° + âˆ AOD = 360Â°Â Â (given,Â âˆ AOC + âˆ COB + âˆ BOD = 270Â°)

=> Â Â Â Â Â Â Â Â Â âˆ AOD = 360Â° â€” 270Â°

=> Â Â Â Â Â Â Â Â Â âˆ AOD Â = Â 90Â°

Now, âˆ AOD + âˆ BOD = 180Â°Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Linear pair angles]

90Â° + âˆ BOD = 180Â°

=> Â âˆ BOD = 180Â° â€“ 90Â°

=> âˆ BOD Â = 90Â°

âˆ AOD Â = âˆ BOC = 90Â°Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Vertically opposite angles are equal]

âˆ BOD Â = âˆ AOC = 90Â°Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Vertically opposite angles are equal]

Hence, all the angles are found.

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

Ans :

Given here,

âˆ 2=120Â°Â Â and âˆ 5=60Â°

To prove: âˆ 2+ âˆ 1 =180Â°Â Â Â [ As Linear pair angles are supplementary ]

120 + âˆ 1=180Â°

âˆ 1=180Â° -120Â°

âˆ 1=60Â°

Since âˆ 1 = âˆ 5 =60Â°

Since,Â pair of corresponding angles are equal, therefore,

m||n. Proved.

**Q 21: In the below fig. transversal l intersects two lines m and n, âˆ 4 =110Â°Â and âˆ 7 =65Â°.Â Is m|| n?**

Ans : Â Given,

âˆ 4 =110Â°Â and âˆ 7 =65Â°

To find: Is m||n?

Here, âˆ 7 = âˆ 5 = 65Â°Â Â Â Â Â Â [Since vertically opposite angles are equal]

Add, âˆ 4 + âˆ 5 = 110Â° + 65Â° = 175Â°

âˆ 4 andÂ âˆ 5 are the pair of co-interior angles. Then, they should be supplementary to each other. But their sums has resulted inÂ 175Â° rather than 180Â°.

Therefore, m is not parallel to n.

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

Ans : From the figure;

âˆ A+ âˆ B = 115 + 65 = 180Â°

Therefore, ADÂ || BC [ sum of co interior angles equalsÂ 180Â° orÂ are supplementary]

Similarly,

âˆ B + âˆ C =65+115=180Â°

Therefore, AB || CD (sum of co interior angles equalsÂ 180Â° orÂ are supplementary]

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

Ans :

Given, l||m, n perpendicular to l.

To prove: n is perpendicular to m

Since, l||m and n intersect l.

âˆ´ âˆ 1 = âˆ 2Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Since the Corresponding angles are equal for parallel lines]

But,Â âˆ 1 = 90Â°

=> âˆ 2Â =90Â°

Hence, it is proved, n is perpendicular to m.

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

Ans :

Given, as per question,

AB || DE and BC || EF

To prove : âˆ ABC= âˆ DEF

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

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

âˆ ABC = âˆ DMPÂ Â Â Â Â Â Â Â —–(i)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Corresponding angles are equal for parallel lines]

Also, since, BP || EF and DE is the transversal

âˆ DMP = âˆ DEF Â Â Â Â Â Â Â Â Â Â Â —–(ii)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Corresponding angles are equal for parallel lines]

From (i) and (ii), we get,

âˆ ABC= âˆ DEF

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

Ans :

Given, as per the question,

AB // DE and BC // EF

To prove: âˆ ABC + âˆ DEF = 180Â°

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

Proof :

Since AB || EP and BP is the transversal AB and ED.

âˆ ABC = âˆ EPMÂ Â Â Â —–(i) Â Â Â [Corresponding angles are equal for parallel lines]

Also,

EF || PM and EP is the transversal.

As we know, co-interior angles for parallel lines are supplementary

âˆ DEF + âˆ EPM = 180Â°Â Â Â Â ………… (ii)

From (i) and (ii) we get,

âˆ DEF + âˆ ABC = 180Â°. Hence, proved.

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

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

Ans: False

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

Ans: True

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

Ans: False.

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

Ans: True.

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

Ans: 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 180′. then the lines are _____________

Ans :

(i) Equal

(ii) Parallel

(iii) Supplementary

(iv) Parallel

(v) Parallel

(vi) Parallel