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 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 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° 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 :