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Lines And Angles

Class 9th Mathematics AP Board Solution
Exercise 4.1
  1. In the given figure, name: i. any six points ii. any five line segments iii. any…
  2. Observe the following figures and identify the type of angles in them.…
  3. State whether the following statements are true or false : i. A ray has no end…
  4. What is the angle between two hands of a clock when the time in the clock is (a)…
Exercise 4.2
  1. In the given figure three lines bar ab , vector cd and vector square f…
  2. Find the value of x in the following figures. i. ii. iii. iv.
  3. In the given figure lines vector ab and vector cd intersect at O. If ∠AOC + ∠BOE…
  4. In the given figure lines bar xy and vector mn intersect at O. If ∠POY = 90° and…
  5. In the given figure ∠PQR = ∠PRQ, then prove that ∠PQS = ∠PRT. delta…
  6. In the given figure, if x + y = w + z, then prove that AOB is a line. phi…
  7. In the given figure vector pq is a line. Ray vector or is perpendicular to line…
  8. It is given that ∠XYZ = 64° and XY is produced to point P. A ray YQ bisects…
Exercise 4.1
Question 1.

In the given figure, name:

i. any six points

ii. any five line segments

iii. any four rays

iv. any four lines

v. any four collinear points
Answer:

i) A point specifies the exact location and looks like a small dot.


∴ The points in the given figure are A,B,C,D,E,F,G,H,M,N,P,Q,X,Y.


ii) A part of a line with two end points is a line segment.


∴ The line segment in the given figure are AX,XM,MP,AM,AP,AB,XB,XY…etc


iii) A ray has a starting point but no end point. it may go to infinity.


∴ The ray in the given figure are A,B,C,D,E,F,G,H.


iv) A line goes without end in both direction.


∴ The line segment in the given figure are AB,CD,EF,GH.


v) if three or more points lie on the same line, they are called collinear points


∴ The collinear points in the given figure are AXM,EMN,GPQ,CYN…
Question 2.

Observe the following figures and identify the type of angles in them.
Answer:

A) The given figure shows the larger angle area, so it is reflex angle. The angle is more than 180° and less than 360°.


B) The given figure shows the right angle that is B = 90°


C) The given figure shows the smaller angle area, so it is acute angle. The angle is more than 0° and less than 90° .
Question 3.

State whether the following statements are true or false :

i. A ray has no end point.

ii. Line is the same as line .

iii. A ray is same as the ray

iv. A line has a define length.

v. A plane has length and breadth but no thickness.

vi. Two distinct points always determine a unique line.

vii. Two lines may intersect in two points.

viii. Two intersecting lines cannot both be parallel to the same line.
Answer:

(i) False


A ray has a starting point but no end point but it goes to infinity.


(ii) True


A line goes without end in both direction. so, is same as


(iii) False


A ray has a starting point but no end point but it goes to infinity. so


a ray is not same as


(iv) False


A line goes without end in both direction. so, it does not have a define length.


(v) True


A line goes without end in both direction. so, is same as


(vi) True


If two points are joined together then forms a line.


(vi) False


Two lines intersected at one point.


(vii) True


A parallel line does not have any intersecting point.
Question 4.

What is the angle between two hands of a clock when the time in the clock is

(a) 9’O clock

(b) 6’O clock

(c) 7:00 PM
Answer:

(a) Let draw the 9’O clock and find the angle between the lines



∴ The angle is 90°


(b) Let draw the 6’O clock and find the angle between the lines



∴ The angle is 180°


(c) Let draw the 7.00 PM and find the angle between the lines



∴ The angle is 210°
Exercise 4.2
Question 1.

In the given figure three lines and intersecting at O. Find the values of x, y and z it is being given that x : y : z = 2 : 3 : 5
Answer:

From the given, the three angles are x,y,z


If the two lines intersect at a point then its vertically opposite angles are equal.


∴ A = x, B = z and C = y


We know that,The sum of all the angles around at a point is equal to 360°


∴ A + B + C + x + y + z = 360°


⇒ x + y + z + x + y + z = 360°


⇒ 2x + 2y + 2z = 360°


⇒ 2(x + y + z) = 360°


⇒ (x + y + z) =


⇒ x + y + z = 180° ------(1)


Given that, x : y : z = 2 : 3 : 5


Let x = 2m,y = 3m,z = 5m (∵ m = constant)


Substitute these values in equation (1) we get


2m + 3m + 5m = 180


10m = 180


m =


∴ m = 18


Substituting m = 18 in x,y,z


x = 2m,x = 2(18) = 36°


y = 3m,y = 3(18) = 54°


z = 5m,z = 5(18) = 90°


∴ x = 36°,y = 54°,z = 90°
Question 2.

Find the value of x in the following figures.

i. ii.

iii. iv.


Answer:

(i) From the given figure,


3x + 18° + 93° = 180°


⇒ 3x + 111° = 180°


⇒ 3x = 180°-111°


⇒ 3x = 69°


⇒ x =


∴ x = 24°


(ii) From the given figure,


(x-24)° + 29° + 296° = 360°


⇒ (x-24)° = 360°-325°


⇒ (x-24)° = 35°


⇒ x = 35° + 24°


∴ x = 59°


(iii) From the given figure,


(2 + 3x)° = 62°


⇒ 3x = 62°-2° = 60°


⇒ x =


∴ x = 20°


(iv) From the given figure,


40° + (6x + 2)° = 90°


⇒ (6x + 2)° = 90°-40°


⇒ (6x + 2)° = 50°


⇒ 6x = 50°-2° = 48°


⇒ x =


∴ x = 8°
Question 3.

In the given figure lines and intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.
Answer:

Given that,


The lines and intersect at O.


∠AOC + ∠BOE = 70° ----(1)


∠BOD = 40° ----(2)


If the two lines intersect at a point then its vertically opposite angles are equal.


∴ ∠AOC = ∠BOD


Substitute (2) in (1)


⇒ 40° + ∠BOE = 70°


⇒∠BOE = 70°-40°


∴∠BOE = 30°


From the figure,AOB is a straight line and its angle is 180°


So, ∠AOC + ∠BOE + ∠COE = 180°


From equation (1)


⇒ 70° + ∠COE = 180°


⇒∠COE = 180°-70°


∴∠COE = 110°


Reflex ∠COE = 360° - 110° = 250°


∴∠BOE = 30° and Reflex ∠COE = 250°
Question 4.

In the given figure lines and intersect at O. If ∠POY = 90° and a: b = 2 : 3, find c.
Answer:

Given that,


The lines and intersect at O.


From the figure, XOY is a straight line and its angle is 180°


So, ∠XOM + ∠MOP + ∠POY = 180°


From the given, Let ∠a = 2x and ∠b = 3x


⇒∠b + ∠a + ∠POY = 180°----(1)


Given that ∠POY = 90°


Substitute the values in equation (1),


⇒2x + 3x + 90° = 180°


⇒5x + 90° = 180°


⇒5x = 180°-90° = 90°


⇒x =


∴ x = 18°


⇒ ∠a = 2x = 2×18° = 36°


⇒ ∠b = 3x = 3×18° = 54°


From the figure, MON is a straight line and its angle is 180°


⇒∠b + ∠c = 180°


⇒54° + ∠c = 180°


⇒∠c = 180°-54°


∴ ∠c = 126°
Question 5.

In the given figure ∠PQR = ∠PRQ, then prove that ∠PQS = ∠PRT.
Answer:

In the figure, ST is a straight line and its angle is 180°


So, ∠PQS + ∠PQR = 180° ----(1)


And ∠PRT + ∠PRQ = 180°-----(2)


From the two equations, we get


∠PQS + ∠PQR = ∠PRT + ∠PRQ


Given that,


∠PQR = ∠PRQ


⇒ ∠PQS + ∠PRQ = ∠PRT + ∠PRQ


⇒ ∠PQS = ∠PRT + ∠PRQ -∠PRQ


⇒ ∠PQS = ∠PRT


So, ∠PQS = ∠PRT is proved.
Question 6.

In the given figure, if x + y = w + z, then prove that AOB is a line.
Answer:

In a circle, the sum of all angles is 360°


∴ ∠AOC + ∠BOC + ∠DOB + ∠AOD = 360°


⇒ x + y + w + z = 360°


Given that, x + y = w + z


⇒ w + z + w + z = 360°


⇒ 2w + 2z = 360°


⇒ 2(w + z) = 360°


⇒ w + z = 180° or ∠DOB + ∠AOD = 180°


If the sum of two adjacent angles is 180° then it forms a line.


So AOB is a line.
Question 7.

In the given figure is a line. Ray is perpendicular to line is another ray lying between rays and



Prove that


Answer:

Given that, is a line and is perpendicular to line .


⇒ ∠POR = 90°


The sum of linear pair is always equal to 180°


∴ ∠POS + ∠ROS + ∠POR = 180°


Substitute ∠POR = 90°


⇒90° + ∠POS + ∠ROS = 180°


⇒∠POS + ∠ROS = 90°


∴ ∠ROS = 90°-∠POS----(1)


⇒ ∠QOR = 90°


Given that OS is another ray lying between OP and OR


⇒∠QOS-∠ROS = 90°


∴∠ROS = ∠QOS-90°----(2)


On adding two equations (1) and (2) we get


2∠ROS = ∠QOS-∠POS


⇒ ∠ROS = (∠QOS-∠POS)


So, ∠ROS = (∠QOS-∠POS) is proved.
Question 8.

It is given that ∠XYZ = 64° and XY is produced to point P. A ray YQ bisects ∠ZYP. Draw a figure from the given information. Find ∠XYQ and reflex ∠QYP.
Answer:

Let us draw a figure from the given,



Given that, a ray YQ bisects ∠ZYP


So, ∠QYP = ∠ZYQ


Here, PX is a straight line, so the sum of the angles is equal to 180°


∠XYZ + ∠ZYQ + ∠QYP = 180°


Given that, ∠XYZ = 64° and ∠QYP = ∠ZYQ


⇒ 64° + 2∠QYP = 180°


⇒ 2∠QYP = 180°-64° = 116°


∴ ∠QYP = = 58°


Also, ∠QYP = ∠ZYQ = 58°


Using the angle of reflection,


∠QYP = 360°-58° = 302°


∠XYQ = ∠XYZ + ∠ZYQ


⇒∠XYQ = 64° + 58° = 122°


∴ Reflex ∠QYP = 302° and ∠XYQ = 122°