This chapter builds the basic ideas of geometry — points, line segments, lines, rays and angles. These are the building blocks of plane geometry and help us understand shapes later on.
If you press a sharp pencil tip on paper, you get a tiny dot. That dot gives the idea of a point. A point shows an exact location, but it has no length, no width and no height.
Everyday models of a point: the tip of a compass, the sharpened end of a pencil, and the pointed end of a needle.
Points are named using single capital letters, such as Point P, Point Z and Point T. The real dot is imagined to be so thin that it has no size.
Three separate points, each named with a capital letter.
2.2 Line Segment
Fold a sheet of paper and open it — the crease you see is like a line segment. Mark two points A and B. Out of all the paths joining them, the shortest path from A to B is the line segment.
A line segment is the shortest path between two points, including both endpoints. It is written as AB or BA. Points A and B are called the endpoints.
Line segment AB — has two fixed endpoints.
2.3 Line
Now imagine the segment AB stretched beyond A on one side and beyond B on the other side, with no end at all. This never-ending straight path is a line.
A line through points A and B is written with a double-arrow symbol as AB. It goes on forever in both directions, so we can never draw a full picture of it — we only draw a part. A line is sometimes named with a small letter like l or m.
Imp for exams: Any two points decide exactly one line that passes through both of them.
A line goes endlessly in both directions.
2.4 Ray
A ray is a part of a line that starts at one point (called the starting point or initial point) and goes on endlessly in one direction only.
Everyday models of a ray: a beam of light from a lighthouse, light from a torch, and sunrays. If a ray starts at A and passes through P, it is written as AP.
Ray AP — starts at A, moves through P and never ends.
Figure
Ends
Named as
Line segment
Two endpoints
AB
Line
No endpoint (both sides endless)
AB (double arrow) or l
Ray
One endpoint (one side endless)
AP (single arrow)
Figure it Out Exercise (Page 15–16)
Q1. Rihan marked one point. How many lines can pass through it? Sheetal marked two points. How many lines can pass through both?
Through a single point, Rihan can draw countless (unlimited) lines. Through two fixed points, Sheetal can draw only one line.
Q2. Name the line segments in Fig. 2.4. Which points lie on exactly one segment, and which lie on two?
Segments: LM, MP, PQ, QR. Points L and R lie on exactly one segment. Points M, P and Q lie on two segments each.
Q3. Name the rays in Fig. 2.5. Is T the starting point of each ray?
Rays: TA, TB, TN and NB. No — T is the starting point of TA, TB and TN, but not of NB (that one starts at N).
Q4. Draw rough figures with correct labels for: (a) lines OP and OQ meeting at O, (b) ray XY and line PQ crossing at M, (c) line l passing through E and F but not D, (d) point P lying on AB.
Q5. In Fig. 2.6, name five points, a line, four rays and five line segments.
Points: D, E, O, B, C. Line: DB (the line through D, E, O, B). Rays: OC, OB, OE, OD (other rays are also possible). Segments: DE, DO, DB, EO, EB (OB and OC are also acceptable).
Q6. Ray OA starts at O and passes through A and B. (a) Can we also call it OB? (b) Can OA be written as AO?
(a) Yes. O is the starting point and B lies on the same ray, so OA and OB are the same ray.
(b) No. OA starts at O, while AO starts at A. Their starting points are different, so they are not the same ray.
2.5 Angle
An angle is formed by two rays that share the same starting point. The shared point is the vertex, and the two rays are the arms.
If rays BD and BE start from B, the angle can be named as Angle B, or more clearly as ∠DBE or ∠EBD.
Imp for exams: When naming an angle with three letters, the vertex is always the middle letter. So the vertex of ∠DBE is B.
Angle ∠DBE — vertex B, arms BD and BE.
Size of an angle = amount of turn
Just as a line has a size called length, an angle has a size too. The size of an angle is the amount of rotation (turn) needed to move the first arm onto the second arm around the vertex. More turn means a bigger angle. The lengths of the arms do not change the angle.
Figure it Out Exercise (Page 19–20)
Q1. Find angles in the given pictures. Draw the rays of any one angle and name its vertex.
In the bicycle picture, one angle is ∠BDC. Its vertex is D, and its arms are rays DC and DB. Other pictures also contain many such angles.
Q2. Draw and label an angle with arms ST and SR.
The vertex is S, and the two arms are rays ST and SR.
Q3. Explain why ∠APB cannot be labelled as ∠P.
At vertex P there are several rays (towards A, B and C), so many different angles meet at P. Writing only ∠P would not tell us which two arms we mean. So the three-letter name ∠APB is needed to point out the exact angle.
Q4. Name the angles marked in the given figure (arms towards P, Q, R from vertex T).
The two marked angles are ∠RTQ and ∠RTP.
Q5. Mark three points A, B, C not on one line. Draw all lines through pairs. How many lines and angles?
You get 3 lines: AB, BC and CA. Using A, B, C you can name 3 angles: ∠ABC (or ∠CBA), ∠BCA (or ∠ACB), and ∠CAB (or ∠BAC).
Q6. Mark four points A, B, C, D with no three on one line. How many lines and angles?
You get 6 lines: AB, BC, CD, DA, AC and BD. You can name 12 angles: ∠BAC, ∠CAD, ∠BAD, ∠ADB, ∠BDC, ∠ADC, ∠DCA, ∠ACB, ∠DCB, ∠CBD, ∠DBA, ∠CBA.
2.6 Comparing Angles
When animals open their mouths, the wider the jaws turn, the bigger the angle. To arrange angles from smallest to largest, we compare their amount of turn.
Comparing by superimposition
To compare two angles, we place one over the other so their vertices sit exactly on top of each other. After overlapping, it becomes clear which angle is bigger.
If one arm matches but the second arm falls outside, that angle is larger.
If both arms line up perfectly, the two angles are equal in size.
Imp for exams: Two angles are equal when their vertices and both arms overlap completely. The length of the arms never affects the size of an angle.
With one common arm, the blue angle (PQR) is larger than the red angle (ABC).
Comparing without superimposition (using a circle)
Sometimes we cannot move an angle to overlap. Instead, place a transparent circle so its centre sits on the vertex and mark where the arms cross the circle. Doing this for both angles lets us compare the gap between the marks — the wider gap belongs to the bigger angle.
Figure it Out Exercise (Page 20–23)
Is it always easy to compare two angles?
No. Angles that are very close in size, such as 89° and 91°, cannot be compared just by looking. We must measure them or overlap them. Angles that differ clearly are easy to compare by eye.
Where else do we use superimposition to compare?
We overlap to compare line segments, squares and circles as well, and many other shapes.
Q1. Fold a rectangular sheet and draw a line on the fold. Compare the angles it makes with the sides.
The fold creates four angles with the sides. Some folds give larger angles and some smaller. By making several folds and overlapping, you can decide the largest and smallest angle you formed.
Q2. In each case, decide which angle is greater and why. (a) ∠AOB or ∠XOY (b) ∠AOB or ∠XOB (c) ∠XOB or ∠XOC
(a) ∠AOB is greater, because ∠AOB is made up of ∠AOX + ∠XOY + ∠YOB.
(b) ∠AOB is greater.
(c) Neither — ∠XOB and ∠XOC are equal.
Q3. Which angle is greater: ∠XOY or ∠AOB?
We cannot tell just by looking at the figure. We must overlap them or measure them to be sure.
2.8 Special Types of Angles
Straight angle
When a book cover is opened flat until both arms lie in one straight line, a straight angle is formed. It is half of a full turn.
Straight angle ∠AOB — the two arms make one straight line.
Right angle and perpendicular lines
If a ray OC divides a straight angle into two equal parts, each part is a right angle. So a straight angle contains two right angles. A right angle looks like the letter “L”. When two lines meet at right angles, they are called perpendicular lines.
Imp for exams: A right angle is exactly half of a straight angle, which is one quarter (¼) of a full turn.
A right angle — shaped like an “L”.
Figure it Out Exercise (Page 29–30)
Q1. How many right angles do your classroom windows contain? Any others?
A rectangular window has 4 right angles at its corners. Other right angles are found at door corners, tables, blackboards and books.
Q4. Make a slanting crease, then a second crease perpendicular to it. (a) How many right angles? (b) Describe the folding.
(a) Four right angles form where the two creases cross — each is one-fourth of the full turn around the point. (b) First fold to make a slanting crease. Then fold again so that one part of the slanting crease falls exactly on the other part; this new crease is perpendicular and gives four equal right angles.
Classifying Angles
Angles are sorted into groups based on how much they turn compared with a right angle:
Acute angle — smaller than a right angle (less than a quarter turn).
Right angle — exactly a quarter turn.
Obtuse angle — bigger than a right angle but smaller than a straight angle.
Figure it Out Exercise (Page 31–32)
Q3. “Acute” means sharp and “obtuse” means blunt. Why were these words chosen?
In an acute angle the two arms open only a little, giving a sharp, pointed look. In an obtuse angle the arms open much wider, giving a blunt, spread-out look. So the names match their shapes.
Q4. Count the acute angles in the triangle figures. What is the pattern?
The counts are 3, 12, 21. The next figure will have 30 acute angles.
The pattern is: 3×0 + 1, 3×1 + 1, 3×2 + 1, 3×3 + 1, … where 0, 1, 2, 3, … is the number of inner triangles. (Actually the given sequence 3, 12, 21, 30 increases by 9 each time.)
2.9 Measuring Angles
To give an exact number to an angle, mathematicians divided a full turn into 360 equal parts. Each part is 1 degree, written as 1°. The measure of an angle is simply how many 1° parts fit inside it.
Angle
Fraction of full turn
Degree measure
Full turn
1 whole
360°
Straight angle
½ turn
180°
Right angle
¼ turn
90°
A pinch of history: Why 360? The Rigveda speaks of a wheel with 360 spokes, and many old calendars used 360 days. Most usefully, 360 is the smallest number that divides evenly by every number up to 10 except 7, so a circle can be split into 1, 2, 3, 4, 5, 6, 8, 9 or 10 equal whole-degree parts. It is also divisible by 12 (months) and 24 (hours).
Circle divided into equal parts
Number of parts
Degree of each angle (360 ÷ parts)
1
360°
2
180°
3
120°
4
90°
5
72°
6
60°
8
45°
9
40°
10
36°
12
30°
The protractor
A protractor is the tool used to measure angles. It is a half-circle divided into 180 equal parts (1° each). It has two number scales — one running left-to-right and one right-to-left — so you can measure from either side.
Imp for exams — how to measure: Place the protractor centre on the vertex, line one arm along the 0° mark, then read where the second arm crosses the scale. Using the 0° line lets you read the answer directly, without subtracting.
The vertex sits at the centre; one arm on 0° lets you read ∠AOB directly.
Figure it Out Exercise (Page 35–37)
Q1. Write the measures of ∠KAL, ∠WAL and ∠TAK (unlabelled protractor).
∠KAL = 30°, ∠WAL = 50°, ∠TAK = 120°. Yes — using the medium and long marks you can count in 5s and 10s.
Name the angles in the labelled-protractor figure and write their measures.
Angle
Measure
Angle
Measure
∠POQ
35°
∠QOT
125°
∠POR
95°
∠QOU
145°
∠POS
125°
∠ROS
30°
∠POT
160°
∠ROT
65°
∠QOR
60°
∠ROU
85°
∠QOS
90°
∠SOT
35°
∠SOU
55°
∠TOU
20°
Angle Bisector
Cutting an angle into two equal halves is called bisecting the angle. The line that does this is the angle bisector.
Each angle = 22.5°. A straight angle of 180° is folded into 8 equal parts, so each part = 180° ÷ 8 = 22.5°.
Figure it Out Exercise (Page 40–43)
Q1. Find the degree measures of the given angles (using a protractor).
∠IHJ = 47°, ∠GHK = 23°, and the third ∠IHJ = 108°.
Q3. Find the measures for the given angles. Can your paper protractor be used?
∠IHJ = 42° and ∠IHJ = 116°. No, a paper protractor made only for special angles cannot read these odd values accurately.
Q4. How can you find the degree measure of the reflex angle shown?
Measure the smaller (unmarked) angle first, then subtract it from 360°. Marked angle = 360° − unmarked angle = 360° − 100° = 260°.
Q5. Measure each angle (a to f).
a = 80°, b = 120°, c = 60°, d = 130°, e = 130°, f = 60°.
Q6. Find ∠BXE, ∠CXE, ∠AXB and ∠BXC.
∠BXE = 115°, ∠CXE = 85°, ∠AXB = 65°, ∠BXC = 30°.
Q7. Find ∠PQR, ∠PQS and ∠PQT.
∠PQR = 45°, ∠PQS = 100°, ∠PQT = 150°.
Figure it Out Where are the angles? (Page 45–46)
Q1. Angles in a clock.
(a) The centre of the clock is a full turn (360°) split into 12 equal gaps, so each gap = 360° ÷ 12 = 30°; at 1 o'clock the hands are one gap apart.
(b) At 2 o'clock = 60°, at 4 o'clock = 120°, at 6 o'clock = 180°.
(c) At 3 o'clock = 90°, at 9 o'clock = 270°, and so on.
Q2. The angle of a door.
Yes. The vertex is the hinge line where the door meets the wall. The arms are the edge of the door and the edge of the wall. The wider the door opens, the larger the angle.
Q3. Where is the angle in Vidya's swing?
The angle is between the swing's resting (vertical) position and the position it is pulled back to at the start. A bigger starting angle gives a faster swing.
Q4. The toy with slanting slabs.
Yes, angles describe the slopes — a bigger angle means a steeper slope and faster rolling balls. For each angle, one arm is the horizontal base (not always visible) and the other arm is the slanting edge of the slab (visible).
Q5. The insect and its rotated version.
Yes, the amount of turning is an angle. The vertex is the point on the horizontal line, one arm is the original direction and the other arm is the rotated direction of the insect.
2.10 Drawing Angles
To draw an angle such as ∠TIN = 30° with a protractor, follow these steps:
Step 1: Draw the base ray IN.
Step 2: Put the protractor's centre on I and line up IN with the 0° mark.
Step 3: Count from 0 up to 30 and mark point T at 30°.
Step 4: Join I and T with a ruler. Now ∠TIN = 30°.
∠TIN = 30° drawn using a protractor.
Figure it Out Exercise (Page 49–50)
Q1. In Fig. 2.23, list all possible angles, guess and then measure them.
Angles you can name include: ∠CAP, ∠ACD, ∠APL, ∠DLP, ∠RPL, ∠SLP, ∠PRS, ∠LSR, ∠BRS, ∠CLP and more. Guess each first, then check with a protractor and note down how close your guesses were.
Q2. Draw angles of 110°, 40°, 75°, 112° and 134°.
For each, draw a base ray, place the protractor centre on the vertex, line the ray on 0°, count up to the required number and mark the point, then join. This gives all five angles.
Q3. Draw an angle equal to the given ∠IHJ and note the steps.
First measure the given angle with a protractor. Then draw a base ray, set the same measure on the protractor and mark the second arm, and join to get an equal angle.
2.11 Types of Angles and their Measures
We can now describe every type of angle by its degree measure:
Type of angle
Degree measure
Turn
Acute
More than 0° and less than 90°
Less than ¼ turn
Right
Exactly 90°
¼ turn
Obtuse
More than 90° and less than 180°
Between ¼ and ½ turn
Straight
Exactly 180°
½ turn
Reflex
More than 180° and less than 360°
Between ½ and 1 full turn
Imp for exams: Learn this table well — most objective questions ask you to classify an angle from its degree measure.
A reflex angle — bigger than a straight angle (180°).
