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KC Sinha Mathematics Solution Class 9 Chapter 8 Lines and Angles exercise 8.2

Page 8.29

Exercise 8.2

Type 1

Question 1 

Answer the following questions :

(i) If any transversal intersects two parallel lines in such a way that the ratio of interior angles on the same side is 2 : 7 , then f‌ind the measure of the bigger angle in degrees.

(ii) If two parallel lines are intersected by a transversal, what relation will exist between alternate angles ?

(iii) If each of the two lines is parallel to a third line, what relation exists between them ?

(iv) At what angle, in degree measure, all perpendiculars drawn to a line segment are inclined to one another ?

(v) If P be a point not lying on line ‘l’ then how many lines parallel to ‘l’ through the point P, can be drawn ?

(vi) If a transversal intersects two parallel lines, then write two conditions for these two lines to be parallel.

(vii) If arms of any two angles are parallel, what relation exists between these angles ?


Question 2

Fill up the blanks :

(i) All perpendiculars drawn on a line segment are __

(ii) If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, then the two lines are __

(iii) A transversal intersects two parallel lines and one pair of alternate angles x and y are formed , then x and y are __

 


Question 3

P is a point between two parallel lines AB and CD. If ∠ABP=30° , ∠CDP=45° , then find the value of ∠DPB.

Sol :

 


Type 2

Question 4

In the given f‌igure, if AB||DE, then find the value of ∠BCD

<figure to be added>

Sol:

[Hint: Draw a line PCQ through C parallel to AB and DE ]


Question 5

In the given figure, if ∠1=130° and ∠6=60° , is AB||CD ?

<figure to be added>

Sol :

 

 


Page 8.30

Question 6

In the given figure , m||n and ∠2:∠3=2:3, then find the values of angles denoted by 1,2,…..7,8.


Sol:

 


Question 7

In figures (i) , (ii) , (iii) , (iv) if AB||CD , then find the value of x and y in each of the following cases:

(i)

 

(ii)

 

(iii)

 

(iv)

 


Question 8

Write the pair of parallel lines in each of the following cases shown in figures below:

(i)

(ii)

(iii)


Type 3

Question 9

In the given figure , p is a transversal of the lines m and n and ∠1=60° and \angle 2=\dfrac{2}{3} of a right angle. Prove that m||n.

Sol :

 


Question 10

In the given figure , m||n and p||q. If ∠1=75° , then prove that \angle 2=\angle 1+\dfrac{1}{3} of right angles


Sol :

[Hint: ∠BAC=∠1=75° [alternate angles]

Now , ∠2+∠BAC=180°  [sum of the interior angle son the same side of transversal n]

∴ ∠2=180°-∠BAC

=180°-75°

=105°=75°+30°=\angle 1+\dfrac{1}{3} of right angle ]

 


Question 11

In the given figure , if x=y and a=b , prove that r||n

Sol :

 


Question 12

lf m,n, p are three lines such that p||m and n⊥p, prove that n⊥m 

Sol :

 

 

 


Question 13

What are the conditions for two lines to be parallel ?

Sol :

 


Question 14

In the given f‌igure, l and m are two intersecting lines and p||l and q||m. Prove that p and q are also intersecting lines .


Sol:

 


Question 15

If a line is perpendicular to any one of the two parallel lines , prove that this line will also be perpendicular to the other parallel line

Sol :

 

 


Question 16

In the given f‌igure, AB||CD and AD||BC, prove that ∠DAB=∠DCB.


Sol :

[Hint: AB||DC and transversal BC meets them]

∴ ∠ABC+∠BCD=180°..(i)

Again AD||BC

∴ ∠DAB+∠ABC=180°..(ii)

From (i) and (ii) , ∠ABC+∠BCD=∠DAB+∠ABC=180°

or ∠DAB=∠BCD=∠DAB ]


Question 17

A transversal intersects two given lines m and n in such a way that interior angles on the same side of the transversal are equal , then it is always true that the given lines are parallel ? If not , state the condition under which the two lines will be parallel.

[Hint: m||n only when transversal line is perpendicular to both m and n]

 


Question 18

Prove that the two lines that are respectively perpendicular to two intersecting lines intersect each other .


[Hint: As shown in the given figure , lines MN and PQ are perpendicular to intersecting lines AB and CD respectively. If possible , suppose lines MN and PQ do not intersect , then definitely MN||PQ.

But we know that two parallel lines cannot be simultaneously perpendicular to two intersecting lines. So , it is wrong to assume that MN||PQ. Hence MN and PQ are intersecting lines ]

 


Question 19

In the given f‌igure, lines AB and CD are parallel and any point P lies between these lines . Prove that ∠ABP+∠CDO=∠DPB

[Hint: Through point P , draw a line PM||AB and proceed.]


Question 20

If a transversal intersects two lines such that the bisectors of a pair of alternate angles are parallel, then prove that the two lines are parallel

Sol :

 

[Hint: Proceed as in solved example 20 page 8.27]

 


Question 21

Prove that the two lines that are respectively perpendiculars to two parallel lines are parallel to each other.

Sol :

 


 

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