1. Chapter at a Glance
Geometry begins with very simple ideas — a point, a line and an angle. Every shape you know, from a triangle to the outline of a building, is made by joining points with straight paths and turning at corners. In this chapter you will learn the correct names and symbols for these basic figures, understand what an angle really measures, and use a protractor to measure and draw angles in degrees.
- Point, line segment, line and ray — their meaning, naming and symbols.
- What an angle is: vertex, arms and the amount of turn.
- Comparing angles by superimposing (placing one over another).
- Measuring angles in degrees with a protractor; drawing angles of given measure.
- Types of angles: acute, right, obtuse, straight, reflex and complete.
- Angles in daily life: clock hands, doors, scissors and roads.
2. Point, Line Segment, Line and Ray
Point
A point shows an exact position. It has no length, width or thickness. We show it with a tiny dot and name it with a capital letter, such as point A. The tip of a sharpened pencil or the corner of a book gives the idea of a point.
Line segment
If you join two points A and B by the shortest straight path, you get a line segment, written as AB (with a bar on top in books). It has two end points and a fixed length that can be measured with a ruler. The edge of a table or a stretched string are examples.
Line
If a line segment AB is extended endlessly in both directions, we get a line AB. A line has no end points and cannot be measured. It is often drawn with arrowheads at both ends and may be named with a small letter, such as line l.
Ray
A ray starts at one point and goes on endlessly in one direction. A beam of light from a torch is a good example. Ray OP starts at O (the initial point) and passes through P. The starting point is always written first.
| Figure | End points | Can be measured? | Example |
|---|---|---|---|
| Line segment | 2 | Yes | Edge of a ruler |
| Line | 0 | No | Railway track imagined endlessly |
| Ray | 1 | No | Sunlight, torch beam |
Through one point, countless lines can be drawn. Through two different points, exactly one line can be drawn.
3. What is an Angle?
An angle is formed when two rays start from the same point. The common starting point is the vertex, and the two rays are the arms of the angle. The size of an angle tells us how much one arm has turned away from the other — it does not depend on how long the arms are drawn.
- An angle with vertex B and arms BA and BC is written ∠ABC or ∠CBA. The vertex letter is always in the middle.
- If there is only one angle at a vertex, we may simply write ∠B.
- Angles appear when you open a door, a book, scissors or the lid of a box, and between the hands of a clock.
Interior and exterior of an angle
The region between the two arms is the interior of the angle. Points outside it form the exterior. Points on the arms are neither inside nor outside.
Name the angles at vertex Q formed by rays QP, QR and QS (with QR between QP and QS).
Solution: Three angles are formed: ∠PQR, ∠RQS and ∠PQS. Note that ∠PQS is the biggest because it includes both of the others.
4. Comparing Angles
To compare two angles, place one on the other so that their vertices match and one arm of each lies along the same line. The angle whose second arm lies further out is the bigger angle. This method is called superimposition. You can do it by tracing one angle on thin paper.
A simpler way is to measure both angles with a protractor and compare the numbers. Long arms do not make an angle bigger — only the amount of opening matters.
Students often think an angle drawn with longer arms is bigger. This is wrong. Two angles with the same opening are equal even if one is drawn with arms 2 cm long and the other with arms 10 cm long.
5. Special Angles: Right and Straight
- When a rotating arm makes a half turn, the two arms point in opposite directions and lie on one straight line. This is a straight angle.
- Half of a straight angle is a right angle. The corner of a page, a set-square or a door frame shows a right angle. Lines that meet at a right angle are perpendicular to each other (written AB ⊥ CD).
- A full turn brings the arm back to where it started; this is a complete angle.
So: 2 right angles = 1 straight angle, and 2 straight angles = 1 complete turn.
6. Measuring Angles in Degrees
To give angles exact sizes, a full turn is divided into 360 equal parts. Each part is 1 degree, written 1°. This idea came from ancient astronomy, where the year was linked to about 360 days, and 360 is easy to divide into many equal parts (by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more).
| Turn | Angle name | Measure |
|---|---|---|
| Full turn | Complete angle | 360° |
| Half turn | Straight angle | 180° |
| Quarter turn | Right angle | 90° |
| Three-quarter turn | Reflex angle | 270° |
Using a protractor
A protractor is a semicircular tool marked from 0° to 180°. It has two scales — an inner and an outer — running in opposite directions.
- Place the centre point of the protractor exactly on the vertex of the angle.
- Line up the base line (0° line) of the protractor with one arm of the angle.
- Find the scale that starts at 0° on that arm.
- Read the number where the other arm crosses that same scale. That number is the measure.
Drawing an angle of given measure (say 65°)
- Draw a ray OA.
- Place the protractor's centre on O and its base line along OA.
- Using the scale that starts at 0° on OA, mark a dot at 65°.
- Remove the protractor and join O to the dot to get ray OB. ∠AOB = 65°.
One arm of an angle passes through 40° on the inner scale and 140° on the outer scale. The other arm is on the 0° mark of the inner scale. What is the angle?
Solution: We must read the scale that starts at 0° on the first arm — that is the inner scale. So the angle is 40°, not 140°.
7. Types of Angles
| Type | Measure | Everyday example |
|---|---|---|
| Zero angle | 0° | Closed scissors |
| Acute angle | between 0° and 90° | Slice of pizza, letter V |
| Right angle | exactly 90° | Corner of a book |
| Obtuse angle | between 90° and 180° | Reclining chair back |
| Straight angle | exactly 180° | Open book lying flat |
| Reflex angle | between 180° and 360° | Outside of a pizza slice |
| Complete angle | exactly 360° | Second hand after one minute |
Classify: 35°, 90°, 127°, 180°, 210°, 360°.
Solution: 35° acute; 90° right; 127° obtuse; 180° straight; 210° reflex; 360° complete.
8. Angles on a Clock
A clock face is a full circle of 360° divided into 12 hour-gaps. So each hour-gap measures 360° ÷ 12 = 30°, and each minute-gap measures 360° ÷ 60 = 6°.
- At 3 o'clock the hands are 3 gaps apart: 3 × 30° = 90° (a right angle).
- At 6 o'clock they are 6 gaps apart: 180° (a straight angle).
- At 2 o'clock: 60° (acute). At 5 o'clock: 150° (obtuse).
Through what angle does the minute hand turn from 1:15 to 1:40?
Solution: It moves 25 minute-gaps. Angle = 25 × 6° = 150°.
A person facing North turns clockwise to face East, then clockwise again to face West. Through how many degrees did the person turn in total?
Solution: North → East is a quarter turn = 90°. East → West (clockwise, through South) is a half turn = 180°. Total = 90° + 180° = 270°.
9. Master Formula Table
| Fact | Value / Rule |
|---|---|
| Complete turn | 360° |
| Straight angle | 180° = 2 right angles |
| Right angle | 90° = quarter turn |
| Acute angle | 0° < angle < 90° |
| Obtuse angle | 90° < angle < 180° |
| Reflex angle | 180° < angle < 360° |
| Reflex partner of an angle x | 360° − x |
| Angle between clock numbers | 30° per hour-gap |
| Minute hand per minute | 6° |
| Naming an angle | ∠ABC — vertex B in the middle |
10. Master Summary Table
| Idea | Key point | Exam focus |
|---|---|---|
| Point | Exact position, no size | Naming with capital letters |
| Line segment | Two end points, fixed length | Counting segments in figures |
| Line | No end points, endless both ways | Lines through 1 or 2 points |
| Ray | One starting point | Writing the start point first |
| Angle | Amount of turn between two rays | Vertex, arms, naming |
| Degree | 1/360 of a full turn | Protractor reading |
| Types | Acute, right, obtuse, straight, reflex, complete | Classifying given measures |
| Clock angles | 30° per hour-gap, 6° per minute | Angle between hands |
Chart: how many right angles make each special angle
11. Mind Map
- Point
- Segment, line, ray
- Perpendicular lines
- Vertex and arms
- Naming ∠ABC
- Interior, exterior
- Degrees, 360° turn
- Protractor steps
- Drawing angles
- Acute, right, obtuse
- Straight, reflex
- Clock angles
12. Tips and Tricks
- Before reading a protractor, guess whether the angle is acute or obtuse. If your reading disagrees with your guess, you have used the wrong scale.
- For clock problems, count hour-gaps and multiply by 30°.
- A reflex angle and its ordinary angle at the same vertex always add up to 360°.
- To check a right angle quickly, use the corner of a page or a set-square.
- Writing the vertex at the end, like ∠BAC for an angle at C.
- Placing the protractor's edge (not its centre point) on the vertex.
- Calling 180° an obtuse angle — it is a straight angle.
- Thinking a ray can be measured — only segments have fixed length.
13. Important One-liners
- A point has position but no size.
- A line segment has two end points; a ray has one; a line has none.
- Exactly one line passes through two given points.
- An angle is made by two rays with a common starting point called the vertex.
- The size of an angle does not depend on the length of its arms.
- In ∠ABC the middle letter B is the vertex.
- One full turn is 360°.
- A right angle is 90° and a straight angle is 180°.
- Acute angles are less than 90°; obtuse angles are between 90° and 180°.
- Reflex angles are between 180° and 360°.
- Perpendicular lines meet at a right angle.
- Each hour-gap on a clock is 30°; each minute-gap is 6°.
- A protractor measures angles from 0° to 180°.
14. Exam-pattern Questions with Detailed Answers
How many end points does a ray have?
- 0
- 1
- 2
- Countless
Show answer
A ray starts at one point and goes on endlessly in one direction, so it has exactly one end point (its initial point).
Answer: (B) 1
Classify the angles 89°, 91° and 179°.
Show answer
89° is less than 90°, so it is acute. 91° and 179° are between 90° and 180°, so both are obtuse.
Answer: 89° acute; 91° obtuse; 179° obtuse
What is the angle between the hands of a clock at 4 o'clock?
Show answer
The hands are 4 hour-gaps apart. Each gap is 30°, so the angle is 4 × 30° = 120°.
Answer: 120° (obtuse)
Four points A, B, C, D lie on a straight line in that order. How many line segments can be named using these points?
Show answer
Each pair of points gives one segment: AB, AC, AD, BC, BD, CD.
That is 3 + 2 + 1 = 6 segments.
Answer: 6
An angle measures 75°. What is the measure of the reflex angle at the same vertex?
Show answer
The angle and its reflex angle together make a full turn of 360°.
Reflex angle = 360° − 75° = 285°.
Answer: 285°
Riya stands facing East and turns anticlockwise by one right angle. Which direction does she face now? What if she turns clockwise by three right angles from East?
Show answer
Anticlockwise from East by 90° brings her to North.
Clockwise from East: 90° → South, 180° → West, 270° → North. So she again faces North.
Answer: North in both cases
Through how many degrees does the minute hand turn in 20 minutes? Is the angle acute, right or obtuse?
- 100°, obtuse
- 120°, obtuse
- 60°, acute
- 90°, right
Show answer
The minute hand turns 6° every minute. In 20 minutes it turns 20 × 6° = 120°, which is between 90° and 180°, so obtuse.
Answer: (B) 120°, obtuse
A student reads 130° for an angle that clearly looks acute. What mistake did the student make, and what is the correct measure?
Show answer
The student read the wrong scale. Both scales together always add to 180° at any mark.
The correct measure is 180° − 130° = 50°, which matches the acute look.
Answer: Wrong scale used; correct angle 50°
Five rays OA, OB, OC, OD and OE start from the same point O and all lie on one side of a straight line through O, so every angle between two of them is less than 180°. How many angles of the form ∠XOY can be named?
Show answer
Every pair of rays forms one angle named ∠XOY. The number of pairs from 5 rays is 4 + 3 + 2 + 1 = 10.
(This uses the same counting idea as line segments on a line.)
Answer: 10 angles
What is the angle between the hands of a clock at 9:00? At what other exact hour is the angle between the hands the same?
Show answer
At 9:00 the hands are 9 hour-gaps apart going clockwise from 12, but the smaller angle is measured the shorter way: 3 gaps = 3 × 30° = 90°.
At 3:00 the hands are also 3 gaps apart, so the angle is again 90°.
Answer: 90°; also at 3:00
A wheel has 12 equally spaced spokes. What is the angle between two neighbouring spokes? Between spokes that have 3 spokes in between?
Show answer
12 equal angles fill 360°, so each is 360° ÷ 12 = 30°.
With 3 spokes in between, the spokes are 4 gaps apart: 4 × 30° = 120°.
Answer: 30° and 120°
The sum of two angles is a straight angle and one of them is 40° more than the other. Find both angles and classify them.
Show answer
Together they make 180°. If the smaller is x, the larger is x + 40°.
x + x + 40° = 180°, so 2x = 140° and x = 70°. The larger is 110°.
Answer: 70° (acute) and 110° (obtuse)
15. Quick Revision Summary
- Point = position; segment = 2 end points; ray = 1 end point; line = none.
- An angle is the turn between two rays from a common vertex; name it ∠ABC with the vertex in the middle.
- Full turn 360°, straight 180°, right 90°.
- Acute < 90° < obtuse < 180° < reflex < 360°.
- Protractor: centre on vertex, base line on one arm, read the scale that starts at 0°.
- Clock: 30° per hour-gap, 6° per minute for the minute hand.