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
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)
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.
![](data:image/png;base64,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)
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 ![](data:image/png;base64,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)
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 ![](data:image/png;base64,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)
(iii) False
A ray has a starting point but no end point but it goes to infinity. so
a ray
is not same as ![](data:image/png;base64,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)
(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 ![](data:image/png;base64,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)
(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
![](data:image/png;base64,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)
∴ The angle is 90°
(b) Let draw the 6’O clock and find the angle between the lines
![](data:image/png;base64,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)
∴ The angle is 180°
(c) Let draw the 7.00 PM and find the angle between the lines
![](data:image/png;base64,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)
∴ 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
![](data:image/png;base64,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)
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) = ![](data:image/png;base64,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)
⇒ 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 = ![](data:image/png;base64,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)
∴ 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. ![](data:image/png;base64,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)
iii.
iv. ![](data:image/png;base64,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)
Answer:(i) From the given figure,
3x + 18° + 93° = 180°
⇒ 3x + 111° = 180°
⇒ 3x = 180°-111°
⇒ 3x = 69°
⇒ x = ![](data:image/png;base64,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)
∴ 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 = ![](data:image/png;base64,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)
∴ x = 20°
(iv) From the given figure,
40° + (6x + 2)° = 90°
⇒ (6x + 2)° = 90°-40°
⇒ (6x + 2)° = 50°
⇒ 6x = 50°-2° = 48°
⇒ x = ![](data:image/png;base64,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)
∴ 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.
![](data:image/jpeg;base64,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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.
![](data:image/png;base64,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)
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 = ![](data:image/png;base64,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)
∴ 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.
![](data:image/png;base64,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)
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.
![](data:image/png;base64,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)
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 ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Prove that ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAAA1CAYAAABspyZhAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAXGSURBVHja7V09btswGOUBeoAGMLt165LSuYIjFD1AHaHoUnQKCASdAzmbgSIX6JYxQ+cM7dqlN8iQG/QObShFEiXzV6RMyXnDW1rLpt77Hr+PHymFXF1dEQAADhMgAQBgcAAAYHAAAGBwAABgcAAAYPDpgfP18WrFLikl93R1/hWcADD4gWCTs/eE0nthbkLIPxgcgMEP1egwOACDw+AAAIPD4AAAg8PgAACDw+AADA7A4AAMDsDgAAwOgwMADA6Dk4Jnx1nO34FHQAeeZ+8yXhzD4DMzuLiOrfhncAjYcMboNcs372HwhAYn7OzGJ3NTmt1+2W5fgEPAhu32y4uM0ltTJg/+kSoo3/x8S8mfMqA1oHR551p2ioc1lpTe+Vxf8PVreQyv2On3NS9e54wVH4viaF+kl3wQ8nfn/i2ZvBRrQX7ZZmR/bZZ3/fsfwm/otVPRJ04807/1+FPzVCaSRfZLlxQiBDO9r4OyKD4eVYNjD/JgxHqhDnrKzq5NQS7KDkHgSZZf1IMW/55nJxfldyhupjHVY7as/4/nJx+q3+yOZdIZ3yBUqDau/Ap9VGMYqs2c9OlzVk66S/JD3HOdJav7pUVt9H723DdPwnOMkAddYogeQCILqQbTZrZdUto1BbkRxOkG25S9PYKq63Z/s/r89A3e8OZRzvtq48WvYhxDtZmLPmbOdmP2fEW/6mNxvzyV12kmjcHZRpChmr20Bn+aaXQGbwizBHlNYF3ytr+pnk1PX7HvUzd4PfnFKM912rj0BFouuwE6VJu56KPjzGRwVcJKxVOlrdpXUckwGbwJMMX/6QLLdQasCVPNYlNb46lQBYa+sok38dr57WemUG2mro8pnn0MnpKneiyqPo8XGWI9YApElcE3nL9s1h6aa13Kd9Nnu42tx7VPzj/MqRuqK81iaRPCb6g2U9bHFs9OJfqTbil5ktfugw1ekcEeTJ1DeRbrgL79Y+oettndh5zuTLnTvX78TdNYd4WyY4xTaaaqJ5Y2w4Kv4jeGNiH6jGtuczzrDN42vtr7TMlTs/xVZH6PEtJORj9YXW86FjltB3NcU0ZdfxvEiaVNCL+xtJmSPq7xLJfNi8Xit2y8/pZXSp7a/tZuknAj43EWcckuqmzUL2XiZRj9Z/t7iTH3lsdqsA0xuKs2qUv0ofqMUV35xLOpRB87hn14MjXoopKhMninbNd0F+XP2IRSNSh0DYhmFoy4v6yCaxDKCDW4jzbtDO/Pb6g2U9BnaDz7GDwlT4MMXn6pJ/G69aStJFF1b20lk/w9YqzK5pJD+Zt6DT7E4EO08eVXvtcQbUL1idfI9OfMx+ApefIyeFP/DyDD1DCSS3XV2sdlm0G3l1sSpqgOGmIiHSAZdQ3u0GSLo43/IYwQbVLrE8KZr8FT8WTynYIM/akYl2Ct1gxqQpp9vqctgP5vdMqcLC9cj1LW3yvO7XK+edm5lwj7y2PCtYseqk1XHz2/9izhp01KfUI5k4+quvZxUvDk1EXXbnE5la6fLtXX9kp1aS1oahyIs+uqg/przpU3WK9dxHVyU6K6Jt1WTKwsEaZNd1nRMXPvAQr57LQOvtqk0meceHafjPbJkynDTzrwnwtinWQLMQDePDNfRDvJBowrUIrtPLnJCJPPE1HPogMjZtNEzcDOfmt5copPtmcBaJZ4MZ8mAxzN4nBMtzsLp310sn5uQJzamnpzEugmh+jPgwP6UlvVuLGV32O80QV4JuX5WG90AXYNKm+NiGzeNrLs2RnvZAO8Y27sd7IBrTlVb0M1nTLSfg/LC3AK2MBX7DPeqjoB+D7vjfeiA1Zz473o0yrdp3xcFjhcgIQ9lO6U+D3BBAAw+GzKc/tfnwAAGHyGEKfEcDoMgMEPEPgbYwAMfsDrbmx1ATD4gWZu1Z9nEodeGMu+4RALAIPPFPLrdPf1yicAgMH3ZG7ziwTw8AYAgwMAAIMDAACDA8Azx3/VPftxq0cmkwAAAABJRU5ErkJggg==)
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,
![](data:image/png;base64,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)
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°