Which of the following pairs of angles are supplementary?

Correct Option: B. Solution: Two angles are supplementary if their sum is 180°.

What is the measure of an angle that is complementary to a 30° angle?

Correct Option: A. Solution: Complementary angles sum up to 90°. Therefore, 90° - 30° = 60°.

What is the measure of a straight angle?

Correct Option: B. Solution: A straight angle measures 180°.

Lines and Angles

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

Understand the basic terms and definitions related to lines and angles.
Identify and classify different types of angles (acute, right, obtuse, straight, reflex).
Apply the Linear Pair Axiom to find relationships between adjacent angles.
Prove that vertically opposite angles are equal when two lines intersect.
Recognize the properties of parallel lines and the implications of corresponding angles.
Use deductive reasoning to solve problems involving angles formed by intersecting and parallel lines.
Solve geometric problems involving angles using given conditions and theorems.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 6: Lines and Angles

6.1 Introduction

Study of angles formed by intersecting lines and parallel lines.
Applications in architecture and science (e.g., ray diagrams in light).

6.2 Basic Terms and Definitions

Line Segment: A part of a line with two endpoints.
Ray: A part of a line with one endpoint.
Collinear Points: Points lying on the same line.
Angle: Formed by two rays with a common endpoint (vertex).
- Types of angles:
  - Acute Angle: 0° < x < 90°
  - Right Angle: x = 90°
  - Obtuse Angle: 90° < x < 180°
  - Straight Angle: x = 180°
  - Reflex Angle: 180° < x < 360°

6.3 Intersecting Lines and Non-intersecting Lines

Intersecting Lines: Lines that cross each other.
Non-intersecting Lines (Parallel): Lines that do not cross.
Distance between two parallel lines is constant.

6.4 Pairs of Angles

Linear Pair Axiom: If a ray stands on a line, the sum of the two adjacent angles is 180°.
Vertically Opposite Angles: When two lines intersect, the opposite angles are equal.

6.5 Lines Parallel to the Same Line

Theorem 6.6: Lines parallel to the same line are parallel to each other.

6.6 Summary

Linear pair axiom: Sum of adjacent angles = 180°.
Vertically opposite angles are equal when two lines intersect.
Lines parallel to a given line are parallel to each other.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Angle Relationships: Students often confuse the relationships between angles formed by intersecting lines, such as assuming that adjacent angles are always supplementary.
Ignoring Axioms: Failing to apply the Linear Pair Axiom correctly can lead to incorrect conclusions about angle measures.
Incorrectly Identifying Vertically Opposite Angles: Students may mistakenly think that vertically opposite angles are not equal when they are.
Confusing Types of Angles: Misidentifying angles (acute, obtuse, right, straight, reflex) can lead to errors in calculations and proofs.

Tips for Success

Review Definitions: Make sure to understand the definitions of key terms such as complementary, supplementary, adjacent, and vertically opposite angles.
Practice Drawing Diagrams: Visualizing problems with accurate diagrams can help clarify relationships between angles and lines.
Use Axioms and Theorems: Always refer back to axioms like the Linear Pair Axiom and the properties of parallel lines when solving problems.
Check Your Work: After solving an angle problem, double-check your calculations and ensure that the relationships you used are valid.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

Two angles that sum to 90°

Two angles that sum to 180°

Two angles that sum to 270°

Two angles that sum to 360°

Correct Answer: B

Solution:

Two angles are supplementary if their sum is 180°.

Corresponding angles

Alternate interior angles

Alternate exterior angles

All of the above

Correct Answer: D

Solution:

When two parallel lines are cut by a transversal, corresponding angles, alternate interior angles, and alternate exterior angles are equal.

180°

90°

125°

55°

Correct Answer: A

Solution:

Since

\angle AOC + \angle BOC = \angle AOB

and they form a linear pair,

\angle AOB = 180°

They are parallel

They are perpendicular

They are intersecting lines

They form a linear pair

Correct Answer: C

Solution:

When two lines intersect, the sum of opposite angles is equal, i.e.,

x + y = w + z

. This confirms that the lines are intersecting.

Correct Answer: B

Solution:

For alternate interior angles,

3x + 15 = 2x + 45

. Solving for

x

, we get

3x + 15 = 2x + 45 \Rightarrow x = 30

30°

40°

50°

60°

Correct Answer: D

Solution:

The exterior angle is equal to the sum of the opposite interior angles. Let the interior angles be

x

and

2x

. Then,

x + 2x = 120° \Rightarrow 3x = 120° \Rightarrow x = 40°

. Therefore, the smaller angle is

40°

Complementary angles

Supplementary angles

Adjacent angles

Vertically opposite angles

Correct Answer: B

Solution:

The sum of

\angle AOC

and

\angle BOC

is 180°, making them supplementary angles.

They are complementary.

They are supplementary.

They are equal.

They are unequal.

Correct Answer: C

Solution:

When two lines intersect, the vertically opposite angles are equal.

They are equal.

They are complementary.

They are supplementary.

They are reflex angles.

Correct Answer: A

Solution:

When two lines intersect, the vertically opposite angles are equal.

Correct Answer: A

Solution:

Adjacent angles in a parallelogram are supplementary, so

3x + 10 + 5x - 20 = 180° \Rightarrow 8x - 10 = 180° \Rightarrow 8x = 190° \Rightarrow x = 23.75

. Therefore,

x = 15

70°

110°

90°

180°

Correct Answer: A

Solution:

Since angle AOC and angle BOD are vertically opposite angles, they are equal. Thus, angle BOD is 70°.

30°

60°

90°

45°

Correct Answer: A

Solution:

Let the angles be

x

2x

, and

3x

. The sum of angles in a triangle is 180°. So,

x + 2x + 3x = 180°

. Solving,

6x = 180°

gives

x = 30°

. Therefore, the smallest angle is 30°.

The lines are parallel.

The lines are perpendicular.

The lines form a linear pair.

The lines intersect at right angles.

Correct Answer: C

Solution:

x + y = 180^\circ

, then

x

and

y

form a linear pair. This means they are adjacent angles on a straight line formed by the intersection of the two lines.

60°

90°

120°

180°

Correct Answer: A

Solution:

According to the Linear Pair Axiom, the sum of two adjacent angles is 180°. Therefore, if one angle is 120°, the other must be 180° - 120° = 60°.

Correct Answer: A

Solution:

Since vertically opposite angles are equal, we have

3x + 15 = 5x - 5

. Solving for

x

, we get

3x + 15 = 5x - 5 \Rightarrow 20 = 2x \Rightarrow x = 10

. Therefore, the correct value of

x

is 5.

60°

70°

90°

150°

Correct Answer: A

Solution:

Complementary angles sum up to 90°. Therefore, 90° - 30° = 60°.

54°

126°

90°

36°

Correct Answer: A

Solution:

Since line EF is perpendicular to line CD,

\angle GEF = 90^\circ

. The angles

\angle GED

and

\angle AGE

are on a straight line, so they are supplementary. Therefore,

\angle AGE = 180^\circ - 126^\circ = 54^\circ

54^\circ

126^\circ

144^\circ

36^\circ

Correct Answer: A

Solution:

Since

EF \perp CD

\angle GEF = 90^\circ

. Using the fact that

\angle GED + \angle AGE = 180^\circ

(linear pair), we find

\angle AGE = 180^\circ - 126^\circ = 54^\circ

Correct Answer: A

Solution:

Since

\angle AOC = \angle BOD

, we have

3x + 10 = 5x - 30

. Solving for

x

, we get

3x + 10 = 5x - 30 \Rightarrow 2x = 40 \Rightarrow x = 20

. Therefore, the correct answer is

x = 10

40°

60°

80°

100°

Correct Answer: A

Solution:

The sum of angles in a triangle is 180°. Therefore, angle C = 180° - 60° - 80° = 40°.

The lines are perpendicular.

The lines are parallel.

The lines are intersecting.

The lines are skew.

Correct Answer: B

Solution:

When two lines are cut by a transversal, and the alternate interior angles are equal, the lines are parallel as per the properties of parallel lines and transversals.

They intersect at one point.

They never intersect and are equidistant.

They form a right angle with each other.

They converge at infinity.

Correct Answer: B

Solution:

Parallel lines never intersect and remain equidistant from each other.

90°

80°

100°

110°

Correct Answer: A

Solution:

The sum of angles in a quadrilateral is 360°. Therefore, the fourth angle is

360° - 270° = 90°

Their sum is 90°.

Their sum is 180°.

They are always equal.

They are always adjacent.

Correct Answer: B

Solution:

Supplementary angles are two angles whose sum is 180°.

70°

110°

90°

180°

Correct Answer: B

Solution:

Corresponding angles formed by a transversal with parallel lines are equal. Thus, the angle with line CD is also 110°.

72^\circ

60^\circ

90^\circ

108^\circ

Correct Answer: B

Solution:

Since

OP

stands on

QR

\angle QOP + \angle POR = 180^\circ

. Therefore,

2x + 3x = 180^\circ \Rightarrow 5x = 180^\circ \Rightarrow x = 36^\circ

. Hence,

\angle QOP = 2x = 72^\circ

30°

110°

50°

70°

Correct Answer: C

Solution:

Since

\angle AOC + \angle BOE = 70^\circ

and

\angle BOD = 40^\circ

\angle AOC = \angle BOD

due to vertically opposite angles, so

\angle AOC = 40^\circ

Solution:

Since

AB \parallel CD

and

EF

is a transversal,

\angle AGE = \angle GHF

due to corresponding angles being equal. Therefore,

\angle GHF = 50^\circ

30°

70°

110°

140°

Correct Answer: B

Solution:

Since angles AOC and BOE are vertically opposite, they are equal. Thus, angle BOE = 70°.

45^ ext{°}

90^ ext{°}

135^ ext{°}

180^ ext{°}

Correct Answer: A

Solution:

When two lines intersect, the vertically opposite angles are equal. Therefore, if one angle is

45^ ext{°}

, the other vertically opposite angle is also

45^ ext{°}

. This is based on the property of vertically opposite angles.

40°

60°

80°

100°

Correct Answer: A

Solution:

The sum of angles in a triangle is 180°. Therefore, the third angle is 180° - 60° - 80° = 40°.

40^ ext{°}

60^ ext{°}

80^ ext{°}

100^ ext{°}

Correct Answer: D

Solution:

Let the angles be

2x

3x

, and

4x

. The sum of angles in a triangle is

180^ ext{°}

. Thus,

2x + 3x + 4x = 180^ ext{°}

. Solving gives

x = 20^ ext{°}

. Therefore, the largest angle is

4x = 80^ ext{°}

90°

180°

270°

360°

Correct Answer: B

Solution:

The sum of two adjacent angles formed by a ray standing on a line is 180°, known as the Linear Pair Axiom.

Corresponding angles are equal.

Alternate interior angles are not equal.

All angles are right angles.

The sum of all angles is 180°.

Correct Answer: A

Solution:

When two parallel lines are cut by a transversal, corresponding angles are equal.

30^\circ

45^\circ

60^\circ

90^\circ

Correct Answer: A

Solution:

According to the law of reflection, the angle of incidence is equal to the angle of reflection. Therefore, the angle of reflection on the second mirror is

30^\circ

45^\circ

135^\circ

90^\circ

180^\circ

Correct Answer: A

Solution:

According to the Linear Pair Axiom, the sum of the two adjacent angles is

180^\circ

. Therefore, the measure of the other angle is

180^\circ - 135^\circ = 45^\circ

70^ ext{°}

110^ ext{°}

180^ ext{°}

90^ ext{°}

Correct Answer: B

Solution:

According to the Linear Pair Axiom, the sum of two adjacent angles formed by a ray standing on a line is

180^ ext{°}

. Therefore, the other angle is

180^ ext{°} - 70^ ext{°} = 110^ ext{°}

55°

60°

75°

80°

Correct Answer: B

Solution:

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Thus,

120° = 45° + \text{other interior angle}

. Therefore, the other interior angle is

120° - 45° = 75°

20 meters

10 meters

30 meters

40 meters

Correct Answer: A

Solution:

When the angle of elevation is

45^\circ

, the height of the tower is equal to the length of the shadow. Therefore, the height of the tower is 20 meters.

110°

100°

90°

80°

Correct Answer: A

Solution:

In a parallelogram, adjacent angles are supplementary. Therefore, the adjacent angle is

180° - 70° = 110°

The vertically opposite angles are equal.

The adjacent angles are equal.

The sum of all angles is 360°.

The angles are all right angles.

Correct Answer: A

Solution:

When two lines intersect, the vertically opposite angles are equal.

40°

60°

70°

80°

Correct Answer: A

Solution:

Since PQ is parallel to ST, angle PQR and angle QRS are corresponding angles. Therefore, angle QRS = 180° - 110° = 70°.

Correct Answer: A

Solution:

Vertically opposite angles are equal, so

2x + 30 = 4x - 10

. Solving gives

2x + 30 = 4x - 10

\Rightarrow 40 = 2x

\Rightarrow x = 20

45°

90°

135°

180°

Correct Answer: A

Solution:

In a reflection setup with parallel mirrors, the angle of incidence is equal to the angle of reflection. Therefore, the angle of reflection on the second mirror is also 45°.

90°

180°

270°

360°

Correct Answer: B

Solution:

A straight angle measures 180°.

Correct Answer: A

Solution:

Since the lines are parallel, alternate interior angles are equal. Therefore,

3x + 5 = 5x - 15

. Solving for

x

, we get

3x + 5 = 5x - 15

\Rightarrow 20 = 2x

\Rightarrow x = 10

Correct Answer: B

Solution:

Corresponding angles are equal, so

4x + 10 = 2x + 50

. Solving for

x

, we get

4x + 10 = 2x + 50 \Rightarrow 2x = 40 \Rightarrow x = 20

. Therefore,

x = 20

They are complementary.

They are supplementary.

They are equal.

They are adjacent.

Correct Answer: B

Solution:

The sum of

\angle AOC

and

\angle BOC

is 180°, which means they are supplementary.

45°

60°

90°

135°

Correct Answer: A

Solution:

The sum of angles in a triangle is 180°. If one angle is 90° and another is 45°, the third angle is 180° - 90° - 45° = 45°.

\angle AOC = \angle BOD

\angle AOD = \angle BOD

\angle AOC + \angle BOD = 90^\circ

\angle AOD + \angle BOC = 270^\circ

Correct Answer: A

Solution:

When two lines intersect, the vertically opposite angles are equal. Therefore,

\angle AOC = \angle BOD

54°

126°

90°

180°

Correct Answer: A

Solution:

Since

EF

is perpendicular to

CD

\angle GEF

is 90°.

\angle GED

is 126°, so

\angle AGE = 180° - 126° = 54°

80°

90°

100°

110°

Correct Answer: D

Solution:

The sum of angles in a quadrilateral is 360°. Let the unknown angle be

x

. Then,

200° + 70° + x = 360° \Rightarrow x = 360° - 270° = 90°

. Therefore, the fourth angle is 90°.

90°

180°

270°

360°

Correct Answer: B

Solution:

Supplementary angles are two angles whose sum is 180°.

Their sum is 90°.

Their sum is 180°.

They are always equal.

They form a linear pair.

Correct Answer: A

Solution:

Complementary angles are defined as two angles whose sum is 90°.

110^\circ

130^\circ

40^\circ

50^\circ

Correct Answer: D

Solution:

Since

PQ \parallel RS

\angle QRS

is the exterior angle to

\triangle QRS

and equals

180^\circ - 130^\circ = 50^\circ

An angle greater than 180° but less than 360° is called a reflex angle, not an obtuse angle.

Lines and Angles

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 6: Lines and Angles

6.1 Introduction

6.2 Basic Terms and Definitions

6.3 Intersecting Lines and Non-intersecting Lines

6.4 Pairs of Angles

6.5 Lines Parallel to the Same Line

6.6 Summary

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Multiple Choice Questions

Q1.Which of the following pairs of angles are supplementary?

Correct Answer: B

Q2.If two parallel lines are cut by a transversal, which pair of angles are equal?

Correct Answer: D

Q3.If angle ∠AOC\angle AOC∠AOC is 125° and angle ∠BOC\angle BOC∠BOC is 55°, what is the measure of angle ∠AOB\angle AOB∠AOB?

Correct Answer: A

Q4.In a scenario where two lines intersect, forming angles xxx, yyy, zzz, and www, with x+y=w+zx + y = w + zx+y=w+z, what can be concluded about the lines?

Correct Answer: C

Q5.Two parallel lines are cut by a transversal. If one of the alternate interior angles is 3x+153x + 153x+15 and the other is 2x+452x + 452x+45, what is the value of xxx?

Correct Answer: B

Q6.In a triangle, if one of the exterior angles is 120° and the opposite interior angles are in the ratio 1:2, what is the measure of the smaller interior angle?

Correct Answer: D

Q7.If angle ∠AOC\angle AOC∠AOC is 125° and angle ∠BOC\angle BOC∠BOC is 55°, what type of angles are ∠AOC\angle AOC∠AOC and ∠BOC\angle BOC∠BOC?

Correct Answer: B

Q8.If two lines intersect, which of the following is true about the vertically opposite angles?

Correct Answer: C

Q9.If two lines intersect each other, what can be said about the vertically opposite angles formed?

Correct Answer: A

Q10.In a parallelogram, if one angle is 3x+103x + 103x+10 and its adjacent angle is 5x−205x - 205x−20, what is the value of xxx?

Correct Answer: A

Q11.If lines AB and CD intersect at point O, and angle AOC is 70°, what is the measure of angle BOD?

Correct Answer: A

Q12.In a triangle, if the angles are in the ratio 1:2:3, what is the measure of the smallest angle?

Correct Answer: A

Q13.Two lines intersect at a point forming angles xxx, yyy, zzz, and www. If x+y=180∘x + y = 180^\circx+y=180∘, what can be concluded about the lines?

Correct Answer: C

Q14.If a ray stands on a line forming two adjacent angles, and one of the angles measures 120°, what is the measure of the other angle?

Correct Answer: A

Q15.In a geometric setup, two lines intersect at a point, forming vertically opposite angles. If one of the angles is 3x+153x + 153x+15 and the other is 5x−55x - 55x−5, what is the value of xxx?

Correct Answer: A

Q16.What is the measure of an angle that is complementary to a 30° angle?

Correct Answer: A

Q17.In a geometric setup, line AB is parallel to line CD, and line EF is perpendicular to line CD. If the angle ∠GED=126∘\angle GED = 126^\circ∠GED=126∘, what is the measure of ∠AGE\angle AGE∠AGE?

Correct Answer: A

Q18.In the figure, if AB∥CDAB \parallel CDAB∥CD and EF⊥CDEF \perp CDEF⊥CD, with ∠GED=126∘\angle GED = 126^\circ∠GED=126∘, what is ∠AGE\angle AGE∠AGE?

Correct Answer: A

Correct Answer: A

Q20.In a triangle, if angle A measures 60° and angle B measures 80°, what is the measure of angle C?

Correct Answer: A

Q21.If two lines are cut by a transversal and the alternate interior angles are equal, what can be concluded about the two lines?

Correct Answer: B

Q22.Which of the following is a property of parallel lines?

Correct Answer: B

Q23.In a quadrilateral, the sum of three angles is 270°. What is the measure of the fourth angle?

Correct Answer: A

Q24.Which of the following is true about supplementary angles?

Correct Answer: B

Q25.If lines AB and CD are parallel and a transversal intersects them forming an angle of 110° with line AB, what is the corresponding angle formed with line CD?

Correct Answer: B

Q26.Consider a scenario where a ray OPOPOP stands on a line QRQRQR. If ∠QOP=2x\angle QOP = 2x∠QOP=2x and ∠POR=3x\angle POR = 3x∠POR=3x, what is the measure of ∠QOP\angle QOP∠QOP?

Correct Answer: B

Q27.In a diagram, lines AB and CD intersect at point O. If ∠AOC+∠BOE=70∘\angle AOC + \angle BOE = 70^\circ∠AOC+∠BOE=70∘ and ∠BOD=40∘\angle BOD = 40^\circ∠BOD=40∘, find ∠BOE\angle BOE∠BOE.

Correct Answer: C

Q28.If a ray stands on a line, what is the sum of the two adjacent angles formed?

Correct Answer: B

Q29.What is the measure of an obtuse angle?

Correct Answer: C

Q30.Which of the following pairs of angles are complementary?

Correct Answer: B

Q31.In a geometric setup, if two lines intersect at a point, which of the following statements is always true?

Correct Answer: B

Q32.What is the relationship between angles formed by two parallel lines and a transversal?

Correct Answer: A