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All questions of Lines and Angles for Class 9 Exam

What is the measure of an angle which is equal to 5 times its supplement?
  • a)
    150°
  • b)
    120°
  • c)
    90°
  • d)
    135°
Correct answer is option 'A'. Can you explain this answer?

Sameer Menon answered
The measure of an angle and its supplement add up to 180 degrees. Let's call the measure of the angle x. The measure of its supplement would then be 180 - x. According to the problem, x = 5(180 - x).

Solving this equation, we get:
x = 5(180 - x)
x = 900 - 5x
6x = 900
x = 150

Therefore, the measure of the angle is 150 degrees.
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Two complementary angles are in the ratio 2 : 7. The measure of a smaller angle is:
  • a)
    70°
  • b)
    45°
  • c)
    20°
  • d)
    40°
Correct answer is option 'C'. Can you explain this answer?

Shalini Nair answered
Let the smaller angle be 2x. Then, the larger angle is 7x.

Since the angles are complementary, we have 2x + 7x = 90.

Combining like terms, we get 9x = 90.

Dividing both sides by 9, we get x = 10.

Therefore, the smaller angle is 2x = 2(10) = 20.

So, the measure of the smaller angle is 20.

The correct answer is a) 20.

Two complementary angles are such that two times the measure of one is equal to three times the measure of the other. The measure of the larger angle is:
  • a)
    60°
  • b)
    45°
  • c)
    54°
  • d)
    36°
Correct answer is option 'C'. Can you explain this answer?

Prashansa Singh answered
Let the measure of one angle be x. Then the measure of the other angle is 90-x.

According to the given information, 2x = 3(90-x).

Simplifying this equation, we get 2x = 270 - 3x.

Adding 3x to both sides, we get 5x = 270.

Dividing both sides by 5, we get x = 54.

Therefore, the measure of the larger angle is 90 - 54 = 36 degrees.

So, the answer is: b) 36.

AB || CD || EF and GH || KL. The measure of ∠HKL is
  • a)
    85°
  • b)
    135°
  • c)
    215°
  • d)
    145°
Correct answer is option 'D'. Can you explain this answer?

Shilpa Choudhury answered
Extending GH to M, we have,

∠CHG = ∠APH = 60° [Corresponding ∠S]
∠APH = ∠MPD = 60°  [Vertically opposite ∠S]
∠APH = ∠MPD, then,
∠MPD + ∠LKP = 180° [Sum of interior ∠S]
⇒ ∠LKP = 180° - 60° = 120°, also
∠KHD = ∠PKH = 25° (Alternate ∠S)
∴ ∠HKL = ∠LKP + ∠PKH
= 120° + 25° = 145° 

AB || CD and PQ || RS, then x - y =
  • a)
    30°
  • b)
    60°
  • c)
    75°
  • d)
    90°
Correct answer is option 'C'. Can you explain this answer?

Shilpa Choudhury answered
∠DD1S = ∠B1D1C1 = 2x   [Vertically opposite ∠S]
∠B1D1C1 + ∠A1 B1D1 = 180° [Interior ∠S]
⇒  2x + 2y = 180°
⇒ x + y = 90° ...(i)
Also,
∠AA1C1 + ∠A1B1D1  [Corresponding ∠S]
⇒ 30° = 2y
⇒ y = 15° ...(ii)
Using (ii) in (i), we get
x = 90° - y = 90° - 15° = 75° 

POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR, then ∠POS is equal to:
  • a)
    ∠ROS - ∠QOS
  • b)
    ∠QOS - 2∠ROS
  • c)
    ∠QOS + 2∠ROS
  • d)
    2∠ROS - ∠QOS
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
Let the measure of ∠POS be x°
∵ ∠ POQ = 180°
⇒ ∠POS +∠ROS + ∠QOR = 180°
⇒ ∠POS +∠ROS + 90° = 180°
⇒ ∠ROS =(90° - x).
⇒ ∠QOS = 90° + (90° - x) = 180° - x ∴ ∠POS = x = ∠QOS - 2∠ROS.

x = 3y = 6/7 Z, then, find the value of y.
  • a)
    36°
  • b)
    24°
  • c)
    72°
  • d)
    84°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
x = 3y, z = 21/6 y = 7/2y.
∴ x + y + z = 180°
⇒ 3y + y + 7/2 y = 180° 
⇒ 4y + 7/2 y = 180°
⇒ 15y = 180° × 2
⇒ y = 24°

If q/p = 5, r/p = 3, then r + p =
  • a)
    80°
  • b)
    120°
  • c)
    160°
  • d)
    100°
Correct answer is option 'A'. Can you explain this answer?

Shilpa Choudhury answered
q = 5p , r = 3p and 
∵ ∠POQ = 180°
⇒ p + q + r =  180°
⇒ p + 5p +3p = 180° 
⇒ 9p = 180° ⇒ p = 20°
r = 3p = 3 × 20° = 60°
∴ r + p = 60° + 20° = 80°

OP bisects ∠BOC and OQ, ∠AOC. Find the measure of ∠POQ.
  • a)
    85°
  • b)
    45°
  • c)
    90°
  • d)
    135°
Correct answer is option 'C'. Can you explain this answer?

Shilpa Choudhury answered
∵ AOB is a straight line
∴ ∠AOB = 180°
∠AOC +∠BOC = 180°

⇒ ∠COQ +∠POC = 90°

⇒ ∠POQ = 90°

In the figure, OA and OB are opposite rays ∠AOC +∠BOD = 63°. The measure of angle ∠COD is;
  • a)
    63°
  • b)
    127°
  • c)
    117°
  • d)
    27°
Correct answer is option 'C'. Can you explain this answer?

Shilpa Choudhury answered
∵ OA and OB are opposite rays.
∴ ∠AOB is a straight angle.
⇒ ∠AOB = 180°
⇒ (∠AOC + ∠BOD) + ∠COD = 180°
⇒ 63° + ∠COD = 180°
⇒  ∠COD = 117°

Find x from the figure.
  • a)
    20°
  • b)
    25°
  • c)
    10°
  • d)
    15°
Correct answer is option 'A'. Can you explain this answer?

Shilpa Choudhury answered
Here ∠COQ = ∠POD   [vertically opposite ∠s]
∵ ∠AOB = 180°    (AOB is a straight line)
⇒ ∠POA + ∠POD + ∠BOD = 180°
⇒ 2x° +3x° + 20° +3x° = 180°
⇒ 8x = 160°
⇒ x = 20°

b = a + 20°, then a =
  • a)
    145°
  • b)
    125°
  • c)
    130°
  • d)
    135°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
∵ Sum of angles around a point = 360°
∠AOB + ∠BOC = 360°
⇒ 90° + a + b = 360°
⇒ a + b = 270° ...(i)  and 
b = a + 20° ...(ii)
Using (ii) in (i)
a + (a + 20°) = 270°
⇒ 2a = 250°
⇒ a = 125°

In the adjoining figure, ∠AOC + ∠BOE = 70° and ∠BOD = 40°, then measure of reflex ∠BOE is 
  • a)
    320°
  • b)
    330°
  • c)
    290°
  • d)
    250°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
∠BOD = ∠AOC = 40°
[vertically opposite ∠s]
∵ ∠SOB is a straight angle 
∴ ∠AOB = 180°
⇒ ∠AOC +∠COE +∠BOE = 180°
⇒ ∠COE = 180° - (∠AOC + ∠BOC)
= 180° - 70° = 110°
⇒ ∠AOC + ∠BOE = 70°
∠AOE = 70° - 40° = 30°
∴ reflex (∠BOC) = 360°- 30 = 330°

The value of m is
  • a)
    60°
  • b)
    30°
  • c)
    45°
  • d)
    20°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
∠m = ∠x  [Vertically opposite ∠s]
∵ ∠AOB = 180°
⇒ ∠BOF + ∠COF + ∠AOC = 180°
⇒ ∠BOF + ∠DOE +∠AOC = 180°
⇒ x° + 2x° + 3x° = 180°
⇒  6x° = 180°
⇒ x = 30° 
∴ m = 30°

If y  - x = 10°, then y =
  • a)
     25°
  • b)
    20°
  • c)
    15°
  • d)
    10°
Correct answer is option 'A'. Can you explain this answer?

Shilpa Choudhury answered
∠AOC +∠ BOC = 180°
⇒ 7x + 3y = 180° ...(i), and
y - x = 10° 
⇒ x = y -10° ...(ii)
Using (ii) and (i)
7(y - 10°) + 3y = 180°
⇒ 10y = 250°
⇒ y = 25°

The value of y, if AB || PQ is
  • a)
    9°
  • b)
    29°
  • c)
    27°
  • d)
    7°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
From the figure,

(2y + y + y) + 35° = 180° (Interior ∠S)
⇒ 5y = 180° - 35°
⇒ y = 36° - 7
= 29°

AB || CD, and ∠RQB = 115°, and ∠PRQ = 30°. The measure of ∠APC is:
  • a)
    115°
  • b)
    45°
  • c)
    85°
  • d)
    30°
Correct answer is option 'C'. Can you explain this answer?

Shilpa Choudhury answered
Here ∠RQB + ∠RQP =180°   (∵ AB is a straight line)
⇒ ∠RQP = 180° - 115° = 65°
Now ∠PRQ = 30°
∵ ∠PRQ, ∠RQP and ∠APQ are the ∠S of D
∴ ∠PRQ + ∠RQP + ∠APQ = 180°
⇒ ∠APQ = 180°- 65°- 30° = 85°
∠APC = ∠APQ [Vertically opposite ∠S]
∴ ∠APC = 85°

If x : y = 2 : 3, then the value of y is equal to:
  • a)
    72°
  • b)
    36°
  • c)
    108°
  • d)
    144°
Correct answer is option 'C'. Can you explain this answer?

Shilpa Choudhury answered
∵  X : Y = 2 : 3 

∴ Let the angles x and y be 2k and 3k respectively.
∴ 2k + 3k = 180° 
[Sum of ∠S in the interior of transversal]
⇒ 5k = 180° ⇒ k = 36°
∴ y = 3k = 3 × 36° = 108°

Determine the value of x from the given figure;
  • a)
    45°
  • b)
    22.5°
  • c)
    25.5°
  • d)
    25°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
∠AOB = 180°
⇒ ∠AOP + ∠POB = 180°
⇒ ∠AOP + ∠POQ + ∠BOQ = 180°
⇒ 90° + 3x + x = 180°
⇒  4x = 90° 
⇒ x = 90°/4 = 22.5°

In the adjoining figure, ∠AOQ :∠AOP = 5 : 7, then the measure of ∠BOQ is:
  • a)
    75°
  • b)
    105°
  • c)
    60°
  • d)
    120°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
∵ ∠POQ = 180°     [PQ is a straight line]
⇒ ∠AOQ +∠AOP = 180°
⇒ 5k + 7k = 180°
⇒ 12k = 180°
⇒ k = 15°
∴ ∠BOQ = ∠AOP = 7 × 15° = 105°
[vertically opposite ∠s]

Find x from the adjoining figure:
  • a)
    30°
  • b)
    20°
  • c)
    40°
  • d)
    80°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
∠AOD = ∠BOC [vertically opposite ∠s]
⇒ y = 60° + x   ...(i) and,
∵ ∠DOC = 180°
⇒ 60° + x + 100° = 180°
⇒ x = 20°

Find x from the given figure (CP || DQ):
  • a)
    105°
  • b)
    130°
  • c)
    125°
  • d)
    175°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
Construct a line l || AB,

∠a = 25° [ Alternate ∠S]
∠c = 105°
∴ ∠b = 105° [Vertically opposite ∠S]
∴ x = a + b = 25° + 105° = 130°

In the adjoining figure, AB || CD and, PQ ⊥ AB, find the measure of ∠PCM.
  • a)
    120°
  • b)
    60°
  • c)
    30°
  • d)
    90°
Correct answer is option 'B'. Can you explain this answer?

Shilpa Choudhury answered
∠ANP +∠PNO = 180° [Straight angle]
⇒ ∠PNO = 180° - 120° = 60° 
∴ ∠PNO = ∠PCM = 60° [corresponding ∠S]

Find the value of x - y + z
  • a)
    77°
  • b)
    85°
  • c)
    127°
  • d)
    137°
Correct answer is option 'A'. Can you explain this answer?

Shilpa Choudhury answered
∵ ∠COD = 180° 
⇒ ∠BOC + ∠BOQ +∠DOQ = 180°
⇒ 55° + 60° + z = 180°
⇒ z = 65°
Similarly ∠BOC = ∠AOD
⇒ 2x + 3° = 55°
⇒ x = 26°
and,
∠AOP = ∠BOQ  [vertically opposite ∠s]
⇒ 5y - 10° = 60° 
⇒ y = 70°/5 = 14°
∴ x - y + z = 26° - 14° + 65°  = 77°

Chapter doubts & questions for Lines and Angles - Mathematics Olympiad for Class 9 2025 is part of Class 9 exam preparation. The chapters have been prepared according to the Class 9 exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Class 9 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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