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