The building blocks
Geometry begins with a few simple ideas.
A point marks an exact position. It has no length or width. We name it with a capital letter, such as point A.
A line segment is the straight path between two points. It has a definite length and two end points. Segment AB is written as AB with a bar over it, or simply "segment AB".
A line extends without end in both directions. Through any two points, exactly one line can be drawn.
A ray starts at one point and goes on forever in one direction. A ray of sunlight or a torch beam is a good picture of a ray. Ray OA starts at O and passes through A.
Figure | End points | Can be measured? |
|---|---|---|
Line segment AB | two (A and B) | yes |
Ray OA | one (O) | no, it never ends |
Line AB | none | no, it never ends |
Lines that meet and lines that don't
Intersecting lines cross each other at exactly one point, called the point of intersection.
Parallel lines lie in the same flat surface and never meet, however far they are extended. The opposite edges of a ruler, or the rails of a railway track, are parallel.
Concurrent lines are three or more lines that all pass through the same point.
What is an angle?
An angle is formed when two rays start from the same point.
The common starting point is the vertex.
The two rays are the arms of the angle.
An angle with vertex O and arms OA and OB is written ∠AOB or ∠BOA. The vertex letter always goes in the middle.
You can also think of an angle as the amount of turn. The hands of a clock turn through angles as time passes.
Measuring angles
Angles are measured in degrees (°). One complete turn is 360°.
Turn | Angle |
|---|---|
Quarter turn | 90° |
Half turn | 180° |
Three-quarter turn | 270° |
Full turn | 360° |
To measure an angle with a protractor:
Place the centre point of the protractor exactly on the vertex.
Line up the zero line (the baseline) with one arm of the angle.
Read the scale that starts at 0° on that arm, where the other arm crosses it.
A protractor has two scales, one running each way. The most common mistake is reading the wrong scale. If the angle is clearly smaller than a right angle, the reading must be less than 90°.
Types of angles
Type | Size | Example |
|---|---|---|
Acute angle | more than 0° and less than 90° | 35° |
Right angle | exactly 90° | the corner of a page |
Obtuse angle | more than 90° and less than 180° | 120° |
Straight angle | exactly 180° | a straight line |
Reflex angle | more than 180° and less than 360° | 250° |
Complete angle | exactly 360° | one full turn |
Two lines that meet at a right angle are perpendicular. The symbol ⊥ means "is perpendicular to".
Angles on a clock
The 12 numbers on a clock divide a full turn of 360° into 12 equal parts, so the angle between two neighbouring numbers is 360° ÷ 12 = 30°.
Example. At 3 o'clock, the hands point to 12 and 3, which are 3 numbers apart. The angle is 3 × 30° = 90°, a right angle.
Summary
A line segment has two end points, a ray has one, and a line has none.
An angle is made by two rays with a common vertex, and is measured in degrees.
Acute < 90° = right < obtuse < 180° = straight < reflex < 360° = complete.
Perpendicular lines meet at 90°; parallel lines never meet.