Geometry begins with simple ideas. A point indicates a precise location and has no length, breadth or height. A tiny pencil dot, compass tip or needle point is only a model: the mathematical point is imagined to be invisibly thin. Points are usually labelled with capital letters such as A, B, P or Z.
A line segment is the shortest path between two points, including the endpoints. The segment joining A and B is written AB or BA. Its endpoints are A and B. Unlike a segment, a line extends without end in both directions. A line through A and B may be denoted by AB or by a lowercase letter such as l or m. A complete line cannot be drawn on paper because it continues endlessly. Any two distinct points determine one unique line through them.
When reading a geometry diagram, distinguish the endpoints of a segment from points that simply lie on a line. A line may be represented only by a finite drawing, but the drawing stands for an endless object. The same idea applies to rays: the visible picture is a model, not the whole mathematical object.
A ray starts at one point, called its starting or initial point, and continues endlessly in one direction. A beam of light from a torch or lighthouse is a familiar model. If a ray starts at A and passes through P, it is named ray AP. The first letter tells you the starting point; the second identifies the direction.
The order of letters matters. Ray AP starts at A and passes through P, while ray PA starts at P and travels in the opposite direction through A. Even when two points lie on the same drawn path, reversing the letters can describe a different ray. If a ray starts at O and passes through A and then B, it can be named OA or OB because both names begin at O and point along the same direction. It cannot be named AO, because AO starts at A.
When a question asks you to name line segments or rays in a figure, examine the endpoints and starting points carefully. A point may lie on more than one segment, and a ray may contain many points beyond its starting point.
An angle is formed by two rays that share a common starting point. The shared point is the vertex, and the two rays are the arms. For example, rays BD and BE form an angle with vertex B. It can be named ∠DBE or ∠EBD. The vertex must be the middle letter when three letters are used. If only one angle is formed at B in the figure, it may be called ∠B; when several angles share B, three letters avoid confusion.
The size of an angle is the amount of rotation or turn needed to move one arm to the other around the vertex. Opening a book cover, moving scissors, turning the arms of a compass or divider, and opening a jaw all make angles. A wider turn means a larger angle. The lengths of the arms do not determine the angle size: making the arms longer without changing their direction leaves the angle unchanged.
To compare angles by superimposition, place one angle over the other so their vertices coincide. If the arms also coincide, the angles are equal. If one angle opens farther than the other, it is larger. If direct tracing is inconvenient, the chapter describes using a transparent circular sheet: place its centre at the vertex and compare how the arms cross the circle.
The chapter suggests making rotating arms from two paper straws joined with a paper clip. By turning one straw while keeping the other fixed, you can create different angles and arrange them from smallest to largest. A cardboard slit cut to match an angle provides another comparison method. An angle that exactly matches the slit passes through; one that is too large or too small does not. This activity reinforces that the opening between the arms, rather than their lengths, matters.
Angles can be compared without placing them directly on top of one another by using a clear circular sheet or by reasoning about the amount of turn. When comparing, keep the vertex aligned and compare the direction of the arms. A picture can look larger simply because it is drawn at a different scale, so focus on the opening, not the arm length or the overall size of the drawing.
A full turn is one complete rotation. A straight angle is half a full turn, with its arms lying in a straight line. The textbook describes folding paper so that one arm overlaps the other; the crease divides the straight angle into two equal angles called right angles. A right angle is exactly half of a straight angle. Two lines meeting at right angles are called perpendicular lines.
Angles are classified by comparing their turn with a right angle and a straight angle. An acute angle is smaller than a right angle. A right angle is exactly half a straight angle. An obtuse angle is larger than a right angle but smaller than a straight angle. A straight angle measures a half turn. These names describe the size of the turn, not the orientation of the drawing.
To assign numerical measures, a full turn is divided into 360 equal parts. Each part is one degree, written 1°. Therefore a full turn is 360°, a straight angle is 180°, and a right angle is 90°. A protractor is used to measure angles; a common semicircular protractor divides a straight angle into 180 degree units. To measure an angle, align the protractor’s centre with the vertex, place its base along one arm, and read the degree mark where the other arm crosses the scale.
The chapter notes that the historical reason for using 360 degrees is not fully known. Ancient traditions used divisions of 360, and this number is convenient because it can be divided evenly into many parts, including 2, 3, 4, 5, 6, 8, 9 and 10 equal parts. It is also divisible by 12 and 24, numbers familiar from months and hours.
| Object or angle | How it is identified | Important point |
|---|---|---|
| Point | Capital letter such as A or P | Shows a location; has no dimensions. |
| Line segment AB | Two endpoints A and B | Finite; AB and BA name the same segment. |
| Line | Continues in both directions | Two distinct points determine one line. |
| Ray AP | Starts at A and passes through P | The first letter names the starting point. |
| Acute angle | Smaller than a right angle | Less than 90°. |
| Right angle | Half a straight angle | 90°. |
| Obtuse angle | Between a right and straight angle | More than 90° and less than 180°. |
| Straight angle | Arms lie along a straight line | 180°. |