Page 8.12

### Exercise 8.1

#### Question 1

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

**(i) Two distinct points in a plane determine __ line .**

Sol : unique

Page 8.13

**(ii) A line separates a plane into __ parts namely the __ and the itself.**

Sol : three , two half planes , line

**(iii) Two distinct __ in a plane cannot have more than one point in common.**

Sol : lines

**(iv) If any ray stands on a line , sum of the two adjacent angles are __**

Sol : 180°

**(v) If two lines intersect each other, vertically opposite angles are ___**

Sol : equal

#### Question 2

**Which of the following statements are true (T) and which are false (F) . Give reasons.**

**(i) Angles forming a linear pair can both be acute angles**

Sol : F

**(ii) Angles forming a linear pair are supplementary**

Sol : T

**(iii) Two distinct lines in a plane can have two points in common**

Sol : F

**(iv) If two lines intersect and one of the angles so formed is of measure 90° ****then each of the other three angles is of measure 90°**

Sol : T

**(v) If angles forming a linear pair are equal, then each of these angles is of measure 90° ?**

Sol : T

#### Question 3

**Give answer to the following questions :**

**(i) If a ray stand on a line , what will be the sum of the two adjacent angles ?**

Sol : 180°

**(ii) If sum of two adjacent angles is two right angles , what type of angles will these be ?**

Sol : supplementary

**(iii) If two lines intersect , what is a the relation between vertically opposite angles ?**

Sol : They are equal

#### Question 4

**Write the sum of all angles (in right angles) formed at any point in a plane.**

Sol :

Four right angles or 360°

#### Question 5

**Lines AB and CD intersect each other at a point O such that ∠AOC=∠COB. What is the relation between these lines ?**

Sol :

AB⊥CD

#### Question 6

**If lines AB and CD intersect each other at a point O and ∠AOC=135° , then**

(a) ∠AOD = __

Sol : 45°

(b) ∠BOD = __

Sol : 135°

(c) ∠COB = __

Sol : 45°

#### Question 7

**If a ray stands on a line, then angles formed between the bisectors of adjacent angles is __**

Sol :

Right angle

#### Question 8

**Lines AB and CD intersect at point O, Write in degree the measure of the angles between bisectors of ∠AOC and ∠BOC .**

Sol :

90°

Type 1

#### Question 9

**In the given figure, find the value of x**

#### Question 10

**In the following figure, find the value of y.**

**(i)**

**(ii)**

Sol :

#### Question 11

**In the given figure, find the value of y**

<fig to be added>

Sol :

30°

#### Question 12

**In the given figure, find ∠AOB in degree**

<fig to be added>

#### Question 13

**In the given figure, a is greater than b by one third of a right angles. Find the values of a and b .**

<fig to be added>

[Hint: Given,

⇒a-b=30°..(i)

Also ⇒a+b=180°..(ii)

On solving (i) and (ii) , we get

⇒2a=210°

⇒a=105° , b=180°-105° = 75° ]

#### Question 14

**If a ray stands on a line such that difference of adjacent angles so formed is 30° , then find the measure of each adjacent angle in degree.**

[Hint: Let a and b be the two adjacent angle. Then , a-b=30° and a+b=180°

Solving , we get a=105° , b=75°]

Sol :

#### Question 15

**In the given figure , what value of x will make POQ a straight line ?**

<fig to be added>

[Hint : For POQ to be a line , we must have

2x+3x+10°=180°

Hence ]

#### Question 16

In the given figures (i) and (ii) , find the values of x in each case

(i)

<fig to be added>

(ii)

<fig to be added>

#### Question 17

**What is the measure of the angle (in degree) which is twice of its supplementary angles ?**

Sol :

Type 2

#### Question 18

**Ray OE bisects ∠AOB and ray OF is opposite to ray OE. Show that ∠FOB=∠FOA**

Sol :

#### Question 19

**If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ, such that ∠POR=∠QOS , then prove that OR and OS lie in a line.**

Sol :

#### Question 20

**From any point O , four lines AO, OB , OC and OD are drawn respectively such that ∠AOB=∠COD and ∠BOC=∠DOA , prove that AOC and BOD are straight lines.**

Sol :

#### Question 21

**O is a point on line AB , OC and OD are perpendiculars drawn on AB in opposite directions. Prove that OC and OD lie in a straight line .**

Sol :

#### Question 22

**Two lines AB and CD intersect each other at point O. If line OP bisects ∠BOD , prove that if OP is produced backwards , then it bisects ∠AOC. If OP and OQ are respectively bisectors of ∠BOD and ∠AOC . Show that the rays OP and OQ are in the same line**

[Hint : Since OP is the bisector of ∠BOD

∴ ∠1=∠6;

If OP is produced,

then ∠1=∠4

and ∠6=∠3 [vertically opposite angles]

∴ ∠3=∠4

Thus, OQ is the bisector of ∠AOC

Also ∠2=∠5 [vertically opposite angles]

**Second part :**

Since sum of the angles formed at a point is 360°

∴ ∠1+∠2++∠3+∠4+∠5+∠6=360°

⇒ (∠1+∠6)+(∠3+∠4)+(∠2+∠5)=360°

⇒ 2∠1+2∠3+2∠2=360° [Using above equations]

⇒ ∠1+∠3+∠2=180° ∴∠POQ=180°

Hence, OP and OQ are in the same line ]

Sol :

#### Question 23

**(i) In the given figure, lines PQ and RS intersect at Point O. If ∠POR:∠ROQ= 5:7 , find all the angles.**

[Hint: ∵∠POR+∠ROQ=180° [By linear pair axiom]

Given , or

and

Now , ∠POS=∠ROQ=105° and ∠SOQ=∠POR=75°]

**(ii) Three coplanar lines AB ,CD and EF intersect at a point O, forming angles as shown in the figure. Find the values of x, y , z and v.**

[Hint: Clearly , ∠y=50° [Vertically opposite angles]

∠z=90° [Vertically opposite angles]

∠v=∠x [Vertically opposite angles]

Now, ∠x=40° ,∠y=50° , ∠z=90° and ∠v=40°]

(iii) In the given figure , find the value of x and then find ∠BOC, ∠ FOC , ∠COA

[Hint: ∵ ∠DOE=∠FOC=2x

Now ray OF stands in line AOB]

∴∠BOF+∠FOC+∠COA=180°

⇒5x+2x+3x=180°

⇒10x=180°

⇒x=18°

∴∠BOF=5x=5×18°=90°

∠FOC=2x=2×18°=36°

∠COA=3x=3×18°=54°

(iv) In the given figure. two straight lines PQ and RS intersect eaach other at O.

If ∠POT=70°, find the value of a,b and c

<fig to be added>

[Hint : Since ray OT stands on line RS

∴ ∠ROP+∠POT+∠TOS=180°

or 4b+70°+b=180°

5b=180°-70°=110°

⇒b=22°; since PQ and RS intersected at O, so

∠QOS=∠POR or a=4b

a=4×22°=88°

Now ∠ROQ=∠POS [Vertically opposite angles]

∴ 2c=70°+b=70°+22°=92°

or ]

**Alternatively,**

2c+a=180° [∵Ray OQ stands on line RS]

⇒ 2c+88°=180°

⇒ 2c=180°-88°

⇒ c=46°

#### Question 24

**If a ray OC stands on AB such that ∠AOC=∠COB , then show that ∠AOC=90°**

Sol :

#### Question 25

**Point O is the common end point of the rays OA , OB , OC , OD and OE. Show that ∠AOB+∠BOC+∠COD+∠DOE+∠EOA=360°**

[Hint: Draw a ray OP opposite to ray OA]

#### Question 26

**In the given figure, if each of ∠AOC and ∠AOB is 90° , show BOC is a line .**

Sol :

#### Question 27

**In the given figure, OE and OF bisect ∠AOC and ∠COB respectively and OE⊥OF . Show that points A,O,B are collinear.**

Sol :

#### Question 28

**In the given figure, ray OS stands on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS , and ∠SOQ , if ∠POS=x , then find ∠ROT.**

Sol :