- A turn changes direction about a centre. A full turn is 360°, half-turn 180° and quarter-turn 90°.
- Clockwise follows a clock's hands; anticlockwise is the opposite. Specify direction when describing a rotation.
- An angle is formed by two rays with a common endpoint, its vertex.
- Acute angles are between 0° and 90°, obtuse between 90° and 180°, straight 180°, reflex between 180° and 360°.
- Complementary angles sum to 90°; supplementary angles sum to 180°.
- Angles completing a full turn around a point sum to 360°.
- When two lines intersect, vertically opposite angles are equal.
- Adjacent angles forming a straight line add to 180°.
- When parallel lines are cut by a transversal, corresponding and alternate interior angles are equal.
- The interior angles of a triangle add to 180°.
- An exterior angle of a triangle equals the sum of the two remote interior angles.
- A circle is the set of points at fixed distance r from its centre. Diameter is 2r.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
I'm Up and Down, and Round and Round — Class 9 Mathematics
Edition: Ganita Manjari 2026–27, Part I, Chapter 5. This chapter belongs to the 2026–27 Ganita Manjari sequence.
Direct explanation: A turn changes direction about a centre. A full turn is 360°, half-turn 180° and quarter-turn 90°.
This is an original expanded study guide. Exact NCERT exercise numbers, diagrams and activities have not been independently verified against a complete official textbook PDF. The practice below is original, not a claimed reproduction of official exercises.
Learning objectives
- Explain definitions and conditions precisely.
- Use mathematical notation and diagrams correctly.
- Show full reasoning in solved problems.
- Check answers using substitution, estimation or geometry.
Turns and orientation
A turn changes direction about a centre. A full turn is 360°, half-turn 180° and quarter-turn 90°.
Worked example
Two quarter-turns make a half-turn.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Clockwise and anticlockwise
Clockwise follows a clock's hands; anticlockwise is the opposite. Specify direction when describing a rotation.
Worked example
A clockwise quarter-turn from north points east.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Angles
An angle is formed by two rays with a common endpoint, its vertex.
Worked example
A right angle measures 90°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Angle types
Acute angles are between 0° and 90°, obtuse between 90° and 180°, straight 180°, reflex between 180° and 360°.
Worked example
135° is obtuse.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Complementary and supplementary
Complementary angles sum to 90°; supplementary angles sum to 180°.
Worked example
35° complements 55°; 110° supplements 70°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Angles around a point
Angles completing a full turn around a point sum to 360°.
Worked example
If three sectors are 80°,100°,120°, the remaining sector is 60°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Vertically opposite angles
When two lines intersect, vertically opposite angles are equal.
Worked example
If one angle is 68°, its opposite is 68°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Linear pair
Adjacent angles forming a straight line add to 180°.
Worked example
An angle of 125° has adjacent linear-pair angle 55°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Parallel lines and transversal
When parallel lines are cut by a transversal, corresponding and alternate interior angles are equal.
Worked example
A 65° corresponding angle remains 65°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Triangle angle sum
The interior angles of a triangle add to 180°.
Worked example
A triangle with angles 50°,60° has third angle 70°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Exterior angle theorem
An exterior angle of a triangle equals the sum of the two remote interior angles.
Worked example
Remote angles 40° and 65° give exterior angle 105°.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Circles and radius
A circle is the set of points at fixed distance r from its centre. Diameter is 2r.
Worked example
If r=7 cm, diameter=14 cm.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Circumference and area
For a circle, circumference C=2πr and area A=πr².
Worked example
For r=3 cm, C=6π cm and A=9π cm².
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Rotation symmetry
A figure has rotational symmetry if a non-full turn maps it onto itself.
Worked example
A square has rotational symmetry of order 4.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Coordinate rotations
About the origin, a 90° anticlockwise rotation sends (x,y) to (−y,x).
Worked example
(2,1) becomes (−1,2).
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Caution about scope
This thematic title may cover selected transformations and geometry ideas; the exact official activities and scope must be checked before presenting these as textbook exercise solutions.
Worked example
Use the original NCERT chapter to verify each activity.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Formula and concept reference
| Concept | Rule | Worked illustration |
|---|---|---|
| Turns and orientation | A turn changes direction about a centre. A full turn is 360°, half-turn 180° and quarter-turn 90°. | Two quarter-turns make a half-turn. |
| Clockwise and anticlockwise | Clockwise follows a clock's hands; anticlockwise is the opposite. Specify direction when describing a rotation. | A clockwise quarter-turn from north points east. |
| Angles | An angle is formed by two rays with a common endpoint, its vertex. | A right angle measures 90°. |
| Angle types | Acute angles are between 0° and 90°, obtuse between 90° and 180°, straight 180°, reflex between 180° and 360°. | 135° is obtuse. |
| Complementary and supplementary | Complementary angles sum to 90°; supplementary angles sum to 180°. | 35° complements 55°; 110° supplements 70°. |
| Angles around a point | Angles completing a full turn around a point sum to 360°. | If three sectors are 80°,100°,120°, the remaining sector is 60°. |
| Vertically opposite angles | When two lines intersect, vertically opposite angles are equal. | If one angle is 68°, its opposite is 68°. |
| Linear pair | Adjacent angles forming a straight line add to 180°. | An angle of 125° has adjacent linear-pair angle 55°. |
| Parallel lines and transversal | When parallel lines are cut by a transversal, corresponding and alternate interior angles are equal. | A 65° corresponding angle remains 65°. |
| Triangle angle sum | The interior angles of a triangle add to 180°. | A triangle with angles 50°,60° has third angle 70°. |
| Exterior angle theorem | An exterior angle of a triangle equals the sum of the two remote interior angles. | Remote angles 40° and 65° give exterior angle 105°. |
| Circles and radius | A circle is the set of points at fixed distance r from its centre. Diameter is 2r. | If r=7 cm, diameter=14 cm. |
| Circumference and area | For a circle, circumference C=2πr and area A=πr². | For r=3 cm, C=6π cm and A=9π cm². |
| Rotation symmetry | A figure has rotational symmetry if a non-full turn maps it onto itself. | A square has rotational symmetry of order 4. |
| Coordinate rotations | About the origin, a 90° anticlockwise rotation sends (x,y) to (−y,x). | (2,1) becomes (−1,2). |
| Caution about scope | This thematic title may cover selected transformations and geometry ideas; the exact official activities and scope must be checked before presenting these as textbook exercise solutions. | Use the original NCERT chapter to verify each activity. |
Exercise A — Definitions and foundational skills
- Define turns and orientation and give a mathematically valid example of your own.
- Define clockwise and anticlockwise and give a mathematically valid example of your own.
- Define angles and give a mathematically valid example of your own.
- Define angle types and give a mathematically valid example of your own.
- Define complementary and supplementary and give a mathematically valid example of your own.
- Define angles around a point and give a mathematically valid example of your own.
- Define vertically opposite angles and give a mathematically valid example of your own.
- Define linear pair and give a mathematically valid example of your own.
- Define parallel lines and transversal and give a mathematically valid example of your own.
- Define triangle angle sum and give a mathematically valid example of your own.
Exercise B — Worked-example practice
- Rework this example: Two quarter-turns make a half-turn. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: A clockwise quarter-turn from north points east. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: A right angle measures 90°. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: 135° is obtuse. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: 35° complements 55°; 110° supplements 70°. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: If three sectors are 80°,100°,120°, the remaining sector is 60°. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: If one angle is 68°, its opposite is 68°. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: An angle of 125° has adjacent linear-pair angle 55°. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: A 65° corresponding angle remains 65°. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: A triangle with angles 50°,60° has third angle 70°. Then change one numerical value and solve the new problem, showing every step.
Exercise C — Reasoning and challenge
- Prove or justify one central identity or rule in the chapter.
- Give a counterexample to a common false statement.
- Compare two concepts and explain how they differ.
- Create a realistic problem and explain your assumptions.
- Find and correct an intentional arithmetic or reasoning error.
- Describe how a diagram, table or graph supports a conclusion.
- Explain a result to a beginner in five clear sentences.
- Check a result using a different valid method.
Frequently asked questions with answers
How do you understand turns and orientation?
A turn changes direction about a centre. A full turn is 360°, half-turn 180° and quarter-turn 90°. Example: Two quarter-turns make a half-turn.
How do you understand clockwise and anticlockwise?
Clockwise follows a clock's hands; anticlockwise is the opposite. Specify direction when describing a rotation. Example: A clockwise quarter-turn from north points east.
How do you understand angles?
An angle is formed by two rays with a common endpoint, its vertex. Example: A right angle measures 90°.
How do you understand angle types?
Acute angles are between 0° and 90°, obtuse between 90° and 180°, straight 180°, reflex between 180° and 360°. Example: 135° is obtuse.
How do you understand complementary and supplementary?
Complementary angles sum to 90°; supplementary angles sum to 180°. Example: 35° complements 55°; 110° supplements 70°.
How do you understand angles around a point?
Angles completing a full turn around a point sum to 360°. Example: If three sectors are 80°,100°,120°, the remaining sector is 60°.
How do you understand vertically opposite angles?
When two lines intersect, vertically opposite angles are equal. Example: If one angle is 68°, its opposite is 68°.
How do you understand linear pair?
Adjacent angles forming a straight line add to 180°. Example: An angle of 125° has adjacent linear-pair angle 55°.
How do you understand parallel lines and transversal?
When parallel lines are cut by a transversal, corresponding and alternate interior angles are equal. Example: A 65° corresponding angle remains 65°.
How do you understand triangle angle sum?
The interior angles of a triangle add to 180°. Example: A triangle with angles 50°,60° has third angle 70°.
How do you understand exterior angle theorem?
An exterior angle of a triangle equals the sum of the two remote interior angles. Example: Remote angles 40° and 65° give exterior angle 105°.
How do you understand circles and radius?
A circle is the set of points at fixed distance r from its centre. Diameter is 2r. Example: If r=7 cm, diameter=14 cm.
How do you understand circumference and area?
For a circle, circumference C=2πr and area A=πr². Example: For r=3 cm, C=6π cm and A=9π cm².
How do you understand rotation symmetry?
A figure has rotational symmetry if a non-full turn maps it onto itself. Example: A square has rotational symmetry of order 4.
How do you understand coordinate rotations?
About the origin, a 90° anticlockwise rotation sends (x,y) to (−y,x). Example: (2,1) becomes (−1,2).
Revision checklist
- I can explain and apply turns and orientation.
- I can explain and apply clockwise and anticlockwise.
- I can explain and apply angles.
- I can explain and apply angle types.
- I can explain and apply complementary and supplementary.
- I can explain and apply angles around a point.
- I can explain and apply vertically opposite angles.
- I can explain and apply linear pair.
- I can explain and apply parallel lines and transversal.
- I can explain and apply triangle angle sum.
- I can explain and apply exterior angle theorem.
- I can explain and apply circles and radius.
- I can explain and apply circumference and area.
- I can explain and apply rotation symmetry.
- I can explain and apply coordinate rotations.
- I can explain and apply caution about scope.
Official textbook reference
NCERT official textbook portal. Supplementary notes and original practice exercises; not verified official NCERT question-by-question solutions or a Class 9 board marking scheme.
