Chapter 3: Geometry - The Science of Space and Shapes
🎯 Objective: This chapter aims to build a solid foundation in geometry, starting from the most basic elements (point, line) to complex 2D and 3D shapes. We will explore their properties, symmetry, and even learn how to construct them accurately, covering the entire PSTET syllabus for both Paper 1 and Paper 2 .
📐 Section 3.1: Basic Geometrical Ideas (2-D)
This section introduces the fundamental building blocks of all geometry.
⚫ 3.1.1 Point, Line, Line Segment, Ray
These are the alphabets of the language of geometry.
| Term | Icon | Definition | Visualization |
|---|---|---|---|
| Point | ⚫ | An exact location in space. It has no dimensions (no length, breadth, or height). It is denoted by a dot and a capital letter. | • A |
| Line | ↔️ | A straight collection of points extending infinitely in both directions. It has no endpoints. It is denoted by ↔ AB or by a single letter like 'l'. | <---A-----------------B---> |
| Line Segment | ➖ | A part of a line with two distinct endpoints. It has a fixed length. Denoted by — AB. | A•-------------•B |
| Ray | ➡️ | A part of a line that starts at one endpoint and extends infinitely in one direction. Denoted by —> AB, where A is the endpoint and B is another point along the ray. | A•-------------B---> |
🔀 3.1.2 Types of Lines – Intersecting, Parallel, Perpendicular
When lines interact, they form specific relationships.
| Type of Line | Icon | Definition | Example in Real Life |
|---|---|---|---|
| Intersecting Lines | 🙏 | Two lines that meet or cross each other at a single point (called the point of intersection). | Scissors open, crossroads. |
| Parallel Lines | ‖ | Lines in a plane that never meet, no matter how far they are extended. They are always the same distance apart. | Railway tracks, opposite edges of a ruler. |
| Perpendicular Lines | ┴ | Two lines that intersect at a right angle (90°) . | The letter 'T', the corner of a blackboard. |
📐 3.1.3 Angles – Types
An angle is formed when two rays originate from a common endpoint (called the vertex).
| Type of Angle | Icon | Degree Measure | Description |
|---|---|---|---|
| Acute Angle | 📐(sharp) | Less than 90° | A "sharp" angle. Example: 45° |
| Right Angle | 📐(perfect L) | Exactly 90° | The angle in a square corner. |
| Obtuse Angle | 📐(wide) | Greater than 90° but less than 180° | A "wide" angle. Example: 120° |
| Straight Angle | ― | Exactly 180° | Forms a straight line. |
| Reflex Angle | 🔄(wide) | Greater than 180° but less than 360° | The larger angle "outside" the acute/obtuse one. |
| Complete Angle | ⭕ | Exactly 360° | A full circle. |
📏 3.1.4 Measuring Angles Using a Protractor
A protractor is a semi-circular tool used to measure angles in degrees (°).
Steps:
Place the center of the protractor exactly on the vertex of the angle.
Align the baseline (0° line) of the protractor with one ray of the angle.
Read the measurement on the protractor's scale where the other ray crosses it.
🔷 3.1.5 Polygons – Sides, Vertices, Diagonals
A polygon is a closed figure made up of three or more line segments.
| Term | Definition | Example (in a Pentagon) |
|---|---|---|
| Sides | The line segments that form the polygon. | 5 sides |
| Vertices (singular: Vertex) | The points where two sides meet. | 5 vertices |
| Diagonals | A line segment connecting two non-adjacent vertices. | 5 diagonals |
🔺 3.1.6 Triangles – Classification
Triangles can be classified in two ways: by their sides and by their angles.
| Classification By | Type | Icon | Property |
|---|---|---|---|
| Sides | Equilateral | 🔺(equal sides) | All three sides are equal. All angles are 60°. |
| Isosceles | 🔺(two equal sides) | Two sides are equal. Angles opposite these sides are equal. | |
| Scalene | 🔺(no equal sides) | All three sides are of different lengths. All angles are different. | |
| Angles | Acute-angled | 🔺 | All three angles are acute (< 90°). |
| Right-angled | 📐 | One angle is exactly 90°. The side opposite this angle is the hypotenuse. | |
| Obtuse-angled | 🔺 | One angle is obtuse (> 90°). |
🔲 3.1.7 Quadrilaterals
A quadrilateral is a polygon with four sides, four vertices, and four angles.
| Quadrilateral | Icon | Properties |
|---|---|---|
| Square | ⬛ | All sides equal. All angles 90°. Diagonals are equal and bisect each other at 90°. |
| Rectangle | ▭ | Opposite sides equal and parallel. All angles 90°. Diagonals are equal and bisect each other. |
| Rhombus | ◆ | All sides equal. Opposite sides parallel. Opposite angles equal. Diagonals bisect each other at 90° (but are not equal). |
| Parallelogram | ▱ | Opposite sides are equal and parallel. Opposite angles are equal. Diagonals bisect each other. |
| Trapezium | ⏢ | One pair of opposite sides is parallel. |
⭕ 3.1.8 Circle – Centre, Radius, Diameter, Chord, Arc, Sector, Segment
A circle is a set of points that are all at the same distance from a fixed point.
| Term | Icon | Definition |
|---|---|---|
| Centre | ⚪• | The fixed point in the middle of the circle. |
| Radius | 📏➡️ | The distance from the centre to any point on the circle. |
| Diameter | 📏↔️ | A line segment passing through the centre, connecting two points on the circle. (Diameter = 2 × Radius). |
| Chord | ➖ | A line segment whose endpoints lie on the circle. (The diameter is the longest chord). |
| Arc | ⭕) | A part of the curve of the circle. |
| Sector | 🍕 | The region enclosed by two radii and the arc between them (like a pizza slice). |
| Segment | 🥔 | The region enclosed by a chord and the arc between them. |
🧩 Section 3.2: Understanding Elementary Shapes (2-D and 3-D)
This section focuses on measuring and comparing these shapes to understand their properties better.
📏 3.2.1 Measuring Line Segments
Comparison by Observation: Simply looking at them (not accurate).
Comparison by Tracing: Tracing one and comparing with the other.
Comparison using a Ruler/Divider: The most accurate method. A divider is used to measure the distance, which is then read on a ruler.
📐 3.2.2 Angles – Right Angle, Straight Angle, Complete Angle
Right Angle (90°): One-quarter of a complete revolution.
Straight Angle (180°): Half of a complete revolution.
Complete Angle (360°): One full revolution.
┴ 3.2.3 Perpendicular Lines
Two lines are perpendicular if the angle between them is 90°. This can be checked using a set square or a protractor.
🔺 3.2.4 Classification of Triangles (Revisited)
This is a recap of the classification in 3.1.6, but focusing on the internal measures that define them.
🔲 3.2.5 Quadrilaterals and Their Properties
We revisit the quadrilaterals from 3.1.7, but now we focus on proving or understanding their properties through the lens of elementary shapes (like parallel lines, transversals, and angle sums). The key properties are summarized in the table in 3.1.7.
🧊 3.2.6 Three-Dimensional Shapes – Cube, Cuboid, Cylinder, Cone, Sphere
Let's move from flat (2D) to solid (3D) shapes.
| Shape | Icon | Description | Real-Life Example |
|---|---|---|---|
| Cube | 🧊 | A 3D shape with 6 identical square faces, 12 edges, and 8 vertices. | Dice, Rubik's cube. |
| Cuboid | 🧱 | A 3D shape with 6 rectangular faces, 12 edges, and 8 vertices. Opposite faces are identical. | Brick, book, shoebox. |
| Cylinder | 🥫 | A 3D shape with two identical circular faces (top and bottom) and one curved surface. | Can of soda, gas cylinder. |
| Cone | 🎩 | A 3D shape with a circular base and a curved surface that tapers to a point (apex). | Ice cream cone, birthday hat. |
| Sphere | ⚽ | A perfectly round 3D shape where every point on its surface is equidistant from its centre. | Ball, globe, orange. |
🧮 3.2.7 Faces, Edges, and Vertices
This is the standard way to describe any polyhedron (3D shape with flat faces).
Face: Any of the individual flat surfaces of a solid object. 🟩
Edge: A line segment where two faces meet. ✏️
Vertex (plural: Vertices): A point where two or more edges meet. ⚫
| Shape | Faces (F) | Edges (E) | Vertices (V) | Relation (F + V - E) |
|---|---|---|---|---|
| Cube | 6 | 12 | 8 | 6 + 8 - 12 = 2 |
| Cuboid | 6 | 12 | 8 | 6 + 8 - 12 = 2 |
| Cylinder | 3 (2 flat, 1 curved) | 2 | 0 | (For curved solids, this formula doesn't apply directly) |
| Cone | 2 (1 flat, 1 curved) | 1 | 1 | - |
| Sphere | 1 (curved) | 0 | 0 | - |
📦 3.2.8 Nets of 3D Shapes
A net is a 2D shape that can be folded to form a 3D shape. It's like the "flattened out" version of the solid.
Cube: Has 11 distinct nets. One common net is a cross shape of 6 squares (like a 'T').
Cuboid: A net consists of 6 rectangles arranged so that opposite faces are not adjacent in the net.
Cylinder: A net consists of two circles (for the top and bottom) and one rectangle (for the curved surface).
Cone: A net consists of one circle (base) and a sector of a larger circle (the curved surface).
Section 3.3: Symmetry (Reflection)
Symmetry is all about balance and beauty.
⚖️ 3.3.1 Line of Symmetry
A line of symmetry is an imaginary line that divides a shape into two identical halves that are mirror images of each other. A shape can have one, many, or no lines of symmetry.
🦋 3.3.2 Symmetry in Common Shapes
| Shape | Number of Lines of Symmetry | Diagram Idea |
|---|---|---|
| Square | 4 | Two diagonals, one vertical, one horizontal through the middle. |
| Rectangle | 2 | One vertical, one horizontal through the middle. |
| Equilateral Triangle | 3 | From each vertex to the midpoint of the opposite side. |
| Isosceles Triangle | 1 | From the vertex between the equal sides to the midpoint of the base. |
| Circle | Infinite | Any line passing through the centre is a line of symmetry. |
| Regular Pentagon | 5 | - |
| Letter 'A' | 1 | Vertical line down the middle. |
3.3.3 Reflection Symmetry
Reflection symmetry is the same as line symmetry. It means one half of an object or shape is the mirror image of the other half. If you place a mirror on the line of symmetry, the reflection completes the shape.
🖼️ 3.3.4 Completing Symmetrical Figures
Given half a figure and a line of symmetry, you can complete the figure by drawing the mirror image of each point on the other side of the line. Every point on one side should be at an equal distance from the line of symmetry as its corresponding point on the other side.
🛠️ Section 3.4: Construction (Using Straight Edge Scale, Protractor, Compasses)
Geometric constructions are done using only a ruler (straight edge), a protractor, and a compass. No measurements are allowed after the initial step. This teaches precision and logical thinking.
✂️ 3.4.1 Constructing a Line Segment of Given Length
Method: Use a ruler. Mark a point 'A'. Place the ruler with its '0' at A. Mark point 'B' at the required length (e.g., 5 cm). Join A and B.
📄 3.4.2 Constructing a Copy of a Line Segment
Draw a line 'l' and mark a point 'P' on it.
Measure the given line segment (say, AB) using a compass by placing the compass needle at A and the pencil at B.
Without changing the compass width, place the needle at P and draw an arc on line 'l'. The point where the arc cuts the line, mark as Q.
PQ is the required copy of AB.
📏 3.4.3 Perpendicular Bisector
To construct a perpendicular bisector of a line segment AB:
With A as centre, draw an arc with a radius greater than half of AB.
With the same radius, draw another arc from B, cutting the first two arcs at points P and Q.
Join P and Q. This line PQ is the perpendicular bisector of AB. It meets AB at its midpoint and is perpendicular to it.
📐 3.4.4 Angle Construction (60°, 90°, 120°)
60° Angle:
Draw a ray OA.
With O as centre, draw an arc of any radius, cutting OA at B.
With B as centre and the same radius, draw an arc to cut the previous arc at C.
Join OC. ∠AOC = 60°.
120° Angle:
Follow steps for 60°.
With C as centre and the same radius, draw an arc to cut the first arc at D.
Join OD. ∠AOD = 120°.
90° Angle:
Draw a ray OA.
With O as centre, draw an arc cutting OA at B.
With B as centre and the same radius, draw an arc to cut the first arc at C. (We now have 60° at ∠AOC).
With C as centre and the same radius, draw an arc to cut the first arc at D. (We now have 120° at ∠AOD).
Now, with C and D as centres and a radius more than half of CD, draw arcs to intersect at E.
Join OE. This line is the angle bisector of the 120° angle (∠AOD), dividing it into 60° and 60°. However, a simpler method is to bisect the 180° straight angle. Draw a semicircle from O. With the same radius from both ends of the semicircle, draw arcs that intersect. Join O to that intersection point. This gives 90°.
🔪 3.4.5 Angle Bisector
To construct the bisector of a given angle ABC:
With B as centre, draw an arc of any radius, cutting BA at P and BC at Q.
With P as centre, draw an arc (radius more than half of PQ).
With Q as centre and the same radius, draw another arc, cutting the previous arc at R.
Join BR. Ray BR is the angle bisector of ∠ABC.
🔺 3.4.6 Constructing Triangles (Given Different Criteria)
SSS (Side-Side-Side) Criterion: Construct a triangle when all three sides are given.
SAS (Side-Angle-Side) Criterion: Construct a triangle when two sides and the included angle are given.
ASA (Angle-Side-Angle) Criterion: Construct a triangle when two angles and the included side are given.
RHS (Right angle-Hypotenuse-Side) Criterion: Construct a right-angled triangle when the hypotenuse and one side are given.
⭕ 3.4.7 Constructing Circles of Given Radius
Open the compass to the given radius (e.g., 3 cm) by measuring it on a ruler.
Mark the centre point 'O' on paper.
Place the compass needle at O and slowly rotate the pencil arm to draw a complete circle.
📝 Chapter Summary: Quick Revision Table for PSTET
| Section | Key Concepts | PSTET Focus |
|---|---|---|
| 3.1 Basic Geometrical Ideas | Points, Lines, Angles, Polygons, Triangles, Quadrilaterals, Circles. | Definitions, classifications (by sides & angles), identification of parts, and basic properties of all 2D shapes. |
| 3.2 Understanding Elementary Shapes | Measuring line segments, angle types, properties of 2D & 3D shapes (Faces, Edges, Vertices), nets. | Distinguishing between 2D and 3D, applying properties to classify shapes, and visualizing 3D shapes from their nets. |
| 3.3 Symmetry (Reflection) | Line of symmetry, reflection symmetry, completing symmetrical figures. | Identifying lines of symmetry in common shapes and completing a given symmetrical figure. |
| 3.4 Construction | Constructing line segments, angles (60°, 90°, 120°), bisectors, triangles, circles. | Knowing the correct steps and sequence for basic geometric constructions using ruler, compass, and protractor. |
I hope this comprehensive chapter equips you with a deep understanding of geometry. Remember, for constructions, practice is key. The more you do it, the more precise you'll become. Best of luck with your PSTET preparation and your future teaching endeavors! 👍
