Lines and Angles - 80 Mark Question Paper

LINES AND ANGLES

Chapter 6 - Mathematics Question Paper

Total Marks

80

Time Limit

3 Hours

Total Questions

20

Class

IX

SECTION A: Multiple Choice Questions (1 Mark Each)

1 If a ray stands on a line, the sum of adjacent angles formed is: 1 Mark

(a) 90°

(b) 180°

(c) 270°

(d) 360°

2 Vertically opposite angles are: 1 Mark

(a) Supplementary

(b) Complementary

(c) Equal

(d) Unequal

3 Two angles whose sum is 90° are called: 1 Mark

(a) Supplementary angles

(b) Complementary angles

(c) Adjacent angles

(d) Linear pair

4 The complement of 35° is: 1 Mark

(a) 35°

(b) 55°

(c) 145°

(d) 325°

5 If two parallel lines are cut by a transversal, corresponding angles are: 1 Mark

(a) Supplementary

(b) Equal

(c) Complementary

(d) Unequal

6 A line segment has: 1 Mark

(a) One endpoint

(b) Two endpoints

(c) No endpoints

(d) Infinite endpoints

7 An obtuse angle measures: 1 Mark

(a) Less than 90°

(b) Exactly 90°

(c) Between 90° and 180°

(d) Greater than 180°

8 If ∠A and ∠B are supplementary and ∠A = 75°, then ∠B = 1 Mark

(a) 15°

(b) 75°

(c) 105°

(d) 165°

9 Which of the following is a ray? 1 Mark

(a) A line with two endpoints

(b) A line with one endpoint extending infinitely

(c) A line extending infinitely both ways

(d) A curved path

10 Lines which are parallel to the same line are __________ to each other. 1 Mark

(a) Perpendicular

(b) Intersecting

(c) Parallel

(d) Coincident

SECTION B: Short Answer Type Questions (2 Marks Each)

11 In the figure, lines AB and CD intersect at O. If ∠AOC = 60°, find all other angles. 2 Marks

12 Two adjacent angles are in the ratio 3:2. If they form a linear pair, find both angles. 2 Marks

13 Find the measure of an angle which is 24° more than its complement. 2 Marks

14 If the angles of a linear pair are (2x + 5)° and (3x - 10)°, find x and the measures of both angles. 2 Marks

15 State the Linear Pair Axiom and the Converse of Linear Pair Axiom. 2 Marks

16 In the figure, PQ || RS and AB is a transversal. If ∠1 = 70°, find ∠2. 2 Marks

17 An angle is twice its complement. Find the angle and its complement. 2 Marks

18 If the supplement of an angle is three times the angle, find the angle. 2 Marks

19 In figure, ray OS stands on line PQ. Ray OR and OT are angle bisectors of ∠POS and ∠SOQ respectively. Find ∠ROT. 2 Marks

20 Explain why vertically opposite angles are equal. 2 Marks

SECTION C: Long Answer Type Questions (4 Marks Each)

21 Prove that if two lines intersect, then vertically opposite angles are equal. (Draw figure and provide complete proof) 4 Marks

22 In figure, if PQ || RS, and ∠MXQ = 135°, ∠MYR = 40°, find ∠XMY. (Hint: Draw line AB through M parallel to both PQ and RS) 4 Marks

23 In the given figure, ∠PQR = ∠PRQ and ∠PQS = ∠PRT. Prove that QS = RT. 4 Marks

24 If x + y = w + z (refer figure with four rays from point O), prove that AOB is a straight line. 4 Marks

25 State and prove Theorem 6.6: Lines which are parallel to the same line are parallel to each other. 4 Marks

SECTION D: Application Based Questions (2 Marks Each)

26 An architect is designing a building with intersecting roof lines. If one angle is 65°, what are all four angles formed at the intersection? 2 Marks

27 A ladder leans against a wall making an angle of 60° with the ground. What is the angle it makes with the wall? 2 Marks

Instructions:

• All questions are compulsory

• Use of calculator is NOT allowed

• Diagrams should be drawn neatly

ANSWER KEY & SOLUTIONS

Lines and Angles - Complete Solutions with Explanations

SECTION A: Answer Key (10 × 1 = 10 Marks)

Question 1

Correct Answer: (b) 180°

📌 Explanation:

This is the Linear Pair Axiom (Axiom 6.1). When a ray stands on a line, the two adjacent angles formed always sum to 180° because a straight line measures exactly 180°.

Question 2

Correct Answer: (c) Equal

📌 Explanation:

Theorem 6.1: When two lines intersect, the vertically opposite angles are always equal. This is one of the most important properties in geometry.

Question 3

Correct Answer: (b) Complementary angles

📌 Explanation:

By definition, two angles whose sum is 90° are called complementary angles. Example: 30° + 60° = 90°

Question 4

Correct Answer: (b) 55°

📌 Explanation:

The complement of an angle = 90° - the angle
Complement of 35° = 90° - 35° = 55°
Verification: 35° + 55° = 90° ✓

Question 5

Correct Answer: (b) Equal

📌 Explanation:

Corresponding Angles Axiom: When a transversal cuts two parallel lines, the corresponding angles (angles in the same relative position at each intersection) are equal.

Question 6

Correct Answer: (b) Two endpoints

📌 Explanation:

A line segment is a part of a line with two fixed endpoints. It's finite and has a definite length. Example: A pencil is like a line segment - it has two ends.

Question 7

Correct Answer: (c) Between 90° and 180°

📌 Explanation:

By definition, an obtuse angle measures more than 90° but less than 180°. Example: 120°, 135°, 150° are all obtuse angles.

Question 8

Correct Answer: (c) 105°

📌 Explanation:

If two angles are supplementary, their sum = 180°
∠A + ∠B = 180°
75° + ∠B = 180°
∠B = 180° - 75° = 105°

Question 9

Correct Answer: (b) A line with one endpoint extending infinitely

📌 Explanation:

A ray has one fixed endpoint and extends infinitely in one direction. Example: A light beam from a flashlight starts from the flashlight (endpoint) and goes on infinitely.

Question 10

Correct Answer: (c) Parallel

📌 Explanation:

Theorem 6.6: If m || l and n || l, then m || n. This is the parallel transitivity property. If both lines are parallel to the same reference line, they must be parallel to each other.

SECTION B: Short Answer Solutions (10 × 2 = 20 Marks)

Question 11

Given: Lines AB and CD intersect at O, ∠AOC = 60°

To Find: All other angles

1 ∠AOC = 60° (Given)

2 ∠BOD = ∠AOC = 60° (Vertically opposite angles are equal)

3 ∠AOD = 180° - 60° = 120° (Linear pair: ∠AOC + ∠AOD = 180°)

4 ∠BOC = ∠AOD = 120° (Vertically opposite angles)

Answer: ∠AOC = 60°, ∠BOD = 60°, ∠AOD = 120°, ∠BOC = 120°

Question 12

Given: Two adjacent angles in ratio 3:2 form a linear pair

To Find: Both angles

1 Let the angles be 3x and 2x

2 Since they form a linear pair: 3x + 2x = 180°

3 5x = 180°

4 x = 36°

5 First angle = 3x = 3(36°) = 108°
Second angle = 2x = 2(36°) = 72°

Answer: The two angles are 108° and 72°

Question 13

Given: Angle is 24° more than its complement

To Find: The angle

1 Let the angle = x

2 Its complement = 90° - x

3 According to problem: x = (90° - x) + 24°

4 x = 90° - x + 24°
x + x = 114°
2x = 114°
x = 57°

5 Verification: Complement = 90° - 57° = 33°
57° - 33° = 24° ✓

Answer: The angle is 57°

Question 14

Given: Linear pair angles are (2x + 5)° and (3x - 10)°

To Find: x and both angle measures

1 Since they form a linear pair:
(2x + 5) + (3x - 10) = 180°

2 2x + 5 + 3x - 10 = 180°
5x - 5 = 180°
5x = 185°
x = 37°

3 First angle = 2(37°) + 5 = 74° + 5 = 79°
Second angle = 3(37°) - 10 = 111° - 10 = 101°

4 Verification: 79° + 101° = 180° ✓

Answer: x = 37°; Angles are 79° and 101°

Question 15

Linear Pair Axiom (Axiom 6.1):

If a ray stands on a line, then the sum of the two adjacent angles formed is 180°.

Converse of Linear Pair Axiom (Axiom 6.2):

If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a straight line.

📌 Explanation:

These two axioms are reverses of each other. Axiom 6.1 says: If line then sum = 180°. Axiom 6.2 says: If sum = 180° then line.

Question 16

Given: PQ || RS, AB is transversal, ∠1 = 70°

To Find: ∠2

1 Since PQ || RS and AB is a transversal

2 ∠1 and ∠2 are alternate interior angles (or corresponding angles, depending on their positions)

3 If they are alternate interior angles: ∠2 = ∠1 = 70° (Alternate angles are equal)

4 If they are interior angles on same side: ∠1 + ∠2 = 180° (Co-interior angles)
70° + ∠2 = 180°
∠2 = 110°

Answer: ∠2 = 70° (if alternate) or 110° (if co-interior)

Note: The exact answer depends on the positions of ∠1 and ∠2 in the figure.

Question 17

Given: Angle is twice its complement

To Find: The angle and its complement

1 Let the angle = x

2 Its complement = 90° - x

3 According to problem: x = 2(90° - x)

4 x = 180° - 2x
x + 2x = 180°
3x = 180°
x = 60°

5 Complement = 90° - 60° = 30°
Verification: 60° = 2(30°) ✓

Answer: Angle = 60°, Complement = 30°

Question 18

Given: Supplement of angle is three times the angle

To Find: The angle

1 Let the angle = x

2 Its supplement = 180° - x

3 According to problem: 180° - x = 3x

4 180° = 3x + x
180° = 4x
x = 45°

5 Verification: Supplement = 180° - 45° = 135° = 3(45°) ✓

Answer: The angle is 45°

Question 19

Given: Ray OS stands on line PQ. OR bisects ∠POS, OT bisects ∠SOQ

To Find: ∠ROT

1 Since OS stands on line PQ:
∠POS + ∠SOQ = 180° (Linear pair)

2 OR bisects ∠POS, so: ∠ROS = ½∠POS

3 OT bisects ∠SOQ, so: ∠SOT = ½∠SOQ

4 ∠ROT = ∠ROS + ∠SOT
= ½∠POS + ½∠SOQ
= ½(∠POS + ∠SOQ)
= ½(180°)
= 90°

Answer: ∠ROT = 90°

Question 20

Explanation of Vertically Opposite Angles being Equal:

1 When two lines intersect, they form 4 angles at the point of intersection.

2 Consider lines AB and CD intersecting at O.

3 ∠AOC and ∠BOD are vertically opposite angles.

4 ∠AOC and ∠AOD are a linear pair, so: ∠AOC + ∠AOD = 180°

5 ∠AOD and ∠BOD are a linear pair, so: ∠AOD + ∠BOD = 180°

6 From (4) and (5): ∠AOC + ∠AOD = ∠AOD + ∠BOD

7 Therefore: ∠AOC = ∠BOD (Subtracting ∠AOD from both sides)

This proves that vertically opposite angles are equal.

SECTION C: Long Answer Solutions (5 × 4 = 20 Marks)

Question 21: Prove Vertically Opposite Angles are Equal

Given: Two lines AB and CD intersect at point O

To Prove: ∠AOC = ∠BOD and ∠AOD = ∠BOC

1 Statement: Ray OA stands on line CD
Reason: Given (A, O, B are collinear)

2 Statement: ∠AOC + ∠AOD = 180°
Reason: Linear Pair Axiom (Axiom 6.1)

3 Statement: Ray OB stands on line CD
Reason: A, O, B are collinear (given)

4 Statement: ∠AOD + ∠BOD = 180°
Reason: Linear Pair Axiom (Axiom 6.1)

5 Statement: ∠AOC + ∠AOD = ∠AOD + ∠BOD
Reason: From steps 2 and 4 (both equal 180°)

6 Statement: ∠AOC = ∠BOD
Reason: Subtract ∠AOD from both sides (Axiom 3)

7 Statement: Similarly, ∠AOD = ∠BOC
Reason: Same reasoning using different linear pairs

Hence proved: Vertically opposite angles are equal. ✓

Question 22: Finding Angle with Parallel Lines

Given: PQ || RS, ∠MXQ = 135°, ∠MYR = 40°

To Find: ∠XMY

1 Draw line AB through M parallel to both PQ and RS
By Theorem 6.6: If m || l and n || l, then m || n
So: AB || PQ and AB || RS

2 Using AB || PQ with transversal XM:
∠QXM and ∠XMB are co-interior angles
∠QXM + ∠XMB = 180° (Co-interior angles with parallel lines)
135° + ∠XMB = 180°
∠XMB = 45°

3 Using AB || RS with transversal YM:
∠BMY and ∠MYR are alternate angles
∠BMY = ∠MYR = 40° (Alternate angles with parallel lines)

4 Finding ∠XMY:
∠XMY = ∠XMB + ∠BMY
∠XMY = 45° + 40°
∠XMY = 85°

Answer: ∠XMY = 85°

Question 23: Geometry Proof

Given: ∠PQR = ∠PRQ and ∠PQS = ∠PRT

To Prove: QS = RT

1 Since ∠PQR = ∠PRQ, triangle PQR is isosceles with PQ = PR

2 In triangle PQS and triangle PRT:
PQ = PR (from step 1)
∠PQS = ∠PRT (given)
∠QPS = ∠RPS (same angle)

3 By ASA congruence: Triangle PQS ≅ Triangle PRT

4 Therefore: QS = RT (corresponding sides of congruent triangles)

Hence proved: QS = RT ✓

Question 24: Prove AOB is a Straight Line

Given: Four rays OA, OB, OC, OD from point O and x + y = w + z

To Prove: AOB is a straight line

1 The four rays create angles around point O.
Total angle around O = 360°
x + y + w + z = 360°

2 Given: x + y = w + z

3 From (1) and (2):
(x + y) + (w + z) = 360°
(x + y) + (x + y) = 360° (since x + y = w + z)
2(x + y) = 360°
x + y = 180°

4 x and y are adjacent angles whose sum = 180°

5 By Converse of Linear Pair Axiom (Axiom 6.2):
If the sum of two adjacent angles is 180°, then their non-common arms form a straight line.

Hence proved: AOB is a straight line ✓

Question 25: Theorem 6.6 - Lines Parallel to Same Line are Parallel

Theorem: Lines which are parallel to the same line are parallel to each other.

Given: Line m || line l and line n || line l

To Prove: Line m || line n

1 Let t be a transversal cutting lines l, m, and n

2 Since m || l and t is a transversal:
∠1 = ∠2 (corresponding angles)

3 Since n || l and t is a transversal:
∠1 = ∠3 (corresponding angles)

4 From (2) and (3): ∠2 = ∠3

5 ∠2 and ∠3 are corresponding angles formed by transversal t with lines m and n

6 Since corresponding angles are equal: m || n
(Converse of corresponding angles axiom)

Hence proved: Lines which are parallel to the same line are parallel to each other. ✓

SECTION D: Application Based Solutions (2 × 2 = 4 Marks)

Question 26: Architect's Building Design

Given: Intersecting roof lines with one angle = 65°

To Find: All four angles

1 When two lines intersect, they form 4 angles.
One angle = 65° (given)

2 Vertically opposite angle = 65° (vertically opposite angles are equal)

3 Adjacent angles to the 65° angle form a linear pair with it.
Each adjacent angle = 180° - 65° = 115°

4 Verification: 65° + 115° + 65° + 115° = 360° ✓

Answer: The four angles are: 65°, 115°, 65°, 115°

📌 Real-World Application:

Architects use these angle relationships to design buildings with proper structural integrity. The angles must be calculated precisely so that roof beams meet at correct angles for proper load distribution.

Question 27: Ladder Against Wall

Given: Ladder makes 60° angle with the ground

To Find: Angle with the wall

1 When a ladder leans against a wall, it forms a right angle with the ground and wall junction.

2 The angle between the ladder and ground = 60°
The angle between the wall (vertical) and ground = 90°

3 The angle between the ladder and wall is complementary to the angle with ground.
Angle with wall = 90° - 60° = 30°

4 Verification: The three angles in the triangle formed = 90° + 60° + 30° = 180° ✓

Answer: The ladder makes an angle of 30° with the wall

📌 Real-World Application:

Safety experts use this principle to ensure ladders are positioned correctly. An angle of 30-45° with the wall is considered safe, providing optimal stability and preventing the ladder from slipping.

📊 MARKS SUMMARY

Section	Question Type	Questions	Marks Each	Total Marks
A	Multiple Choice	10	1	10
B	Short Answer	10	2	20
C	Long Answer	5	4	20
D	Application Based	2	2	4
TOTAL				80 MARKS

Important Formulas & Concepts:

• Linear Pair Axiom: If a ray stands on a line, sum of adjacent angles = 180°

• Vertically Opposite Angles: When two lines intersect, opposite angles are equal

• Complementary Angles: Sum = 90°

• Supplementary Angles: Sum = 180°

• Parallel Lines Theorem: Lines parallel to the same line are parallel to each other

Total Marks: 80 | Time: 3 Hours