Basic Geometrical Concepts
1. Fundamental Objects
Point
A point is an exact position or location. It has no size, only position.
Denoted by a capital letter: P, Q, A, B.
Line Segment
A part of a line with two fixed endpoints.
Denoted by AB or BA. It has a definite length.
Ray
A part of a line with one endpoint, extending infinitely in one direction.
Denoted by ray AB (starting at A, passing through B).
Line
A straight path extending infinitely in both directions.
Denoted by line AB or line l.
Comparison Table:
| Object | Endpoints | Extends infinitely? | Length |
|---|---|---|---|
| Point | None | No | 0 |
| Segment | 2 | No | Definite |
| Ray | 1 | One direction | Infinite |
| Line | 0 | Both directions | Infinite |
Common Mistake: Confusing ray AB with ray BA. Ray AB starts at A and goes through B, while ray BA starts at B and goes through A. They are different.
2. Angles
An angle is formed when two rays meet at a common endpoint (vertex).
Types of Angles
| Type | Measure | Example |
|---|---|---|
| Acute | Between 0 and 90 | 30, 45 |
| Right | Exactly 90 | 90 |
| Obtuse | Between 90 and 180 | 120, 150 |
| Straight | Exactly 180 | 180 |
| Reflex | Between 180 and 360 | 270 |
| Complete | Exactly 360 | 360 |
Exam Focus (2 marks): 'Classify: (a) 35 (b) 180 (c) 215 (d) 90.'
(a) Acute (b) Straight (c) Reflex (d) Right.
Measuring an Angle
Use a protractor:
- Place the center on the vertex.
- Align the base line with one arm.
- Read the measure where the other arm crosses the scale.
Common Mistake: Reading the wrong scale on a protractor (inner vs outer). Check if the angle is acute (inner scale gives < 90) or obtuse (outer scale gives > 90).
3. Pairs of Angles
Adjacent Angles
Two angles that share a common vertex and a common arm but do not overlap.
Complementary Angles
Two angles whose sum is 90.
Example: 30 and 60 are complementary.
Worked Example: Find the complement of 42.
Complement = 90 - 42 = 48.
Supplementary Angles
Two angles whose sum is 180.
Example: 110 and 70 are supplementary.
Worked Example: Find the supplement of 115.
Supplement = 180 - 115 = 65.
Common Mistake: Confusing complementary (sum 90) with supplementary (sum 180). Remember: 'C' for Corner (90) and 'S' for Straight (180).
Vertically Opposite Angles
When two lines intersect, the angles opposite each other are equal.
If two lines intersect, angle 1 = angle 3 and angle 2 = angle 4.
Exam Focus (4 marks): 'Two lines intersect. One angle is 75. Find the other three angles.'
Vertically opposite angle = 75.
Adjacent angles (linear pair) = 180 - 75 = 105. The opposite of that = 105.
Angles: 75, 105, 75, 105.
Linear Pair
Two adjacent angles that together form a straight line. Sum = 180.
4. Drawing Angles with a Protractor
Worked Example: Construct an angle of 55.
Step 1: Draw a base ray. Step 2: Place protractor center at the endpoint, baseline along the ray. Step 3: Mark at 55 on the inner scale. Step 4: Remove protractor and draw the second ray through the mark.
5. Self-Test
- Define: (a) Line segment (b) Ray (c) Angle.
- Classify the angles: (a) 89 (b) 179 (c) 360 (d) 91.
- Find the complement of: (a) 38 (b) 17.
- Find the supplement of: (a) 130 (b) 22.
- If two lines intersect and one angle is 110, find the other three angles.
- Draw and label an acute angle, an obtuse angle, and a right angle.
- Are 120 and 50 supplementary? Justify.
6. Answers to Self-Test
- (a) Part of a line with two endpoints. (b) Part of a line with one endpoint, extending infinitely. (c) Formed by two rays meeting at a point.
- (a) Acute (b) Obtuse (c) Complete (d) Obtuse.
- (a) 52 (b) 73.
- (a) 50 (b) 158.
- Vertically opposite = 110. Adjacent = 180 - 110 = 70. Opposite of 70 = 70. Angles: 110, 70, 110, 70.
- Draw using a protractor or freehand with labels.
- No, 120 + 50 = 170, which is not 180.
