Angle, Lines and Bearing – Basic 6 Mathematics Lesson Note (Third Term)

What is an Angle?

An angle is formed when two lines meet at a point. The point where they meet is called the vertex.

Parts of an Angle:

• Vertex: The point where two lines meet
• Arms: The two lines that form the angle
• Symbol: ∠ (angle symbol)

Types of Angles

Acute Angle

• Size: Less than 90°
• Example: 30°, 45°, 60°

Right Angle

• Size: Exactly 90°
• Symbol: Small square in corner
• Found in: Corners of books, walls, tables

Obtuse Angle

• Size: Between 90° and 180°
• Example: 120°, 150°

Straight Angle

• Size: Exactly 180°
• Looks like: A straight line

Reflex Angle

• Size: Between 180° and 360°
• Example: 270°, 300°

Complete Angle

• Size: Exactly 360°
• Looks like: A full turn/circle

Exercise A – Identifying Angles

Identify the type of each angle:

1. 45° = ______ angle
2. 90° = ______ angle
3. 135° = ______ angle
4. 180° = ______ angle
5. 270° = ______ angle
6. 360° = ______ angle
7. 75° = ______ angle
8. 210° = ______ angle

Measuring Angles

Using a Protractor

Steps to measure an angle:

1. Place the center of protractor on the vertex
2. Align one arm with the 0° line
3. Read where the other arm crosses the scale
4. Choose the correct scale (inner or outer)

Exercise B – Angle Measurement

Use a protractor to measure these angles (or estimate):

1. An angle that looks like a clock at 3 o’clock = ______°
2. An angle that looks like a clock at 6 o’clock = ______°
3. An angle that looks like a clock at 9 o’clock = ______°
4. The angle of your classroom door when fully open = ______°
5. The angle between the hands of a clock at 3 o’clock = ______°

Types of Lines

Parallel Lines

• Definition: Lines that never meet
• Symbol: ||
• Example: Railway tracks, opposite sides of a rectangle

Perpendicular Lines

• Definition: Lines that meet at 90°
• Symbol: ⊥
• Example: Corners of a square, walls and floor

Intersecting Lines

• Definition: Lines that cross each other
• Example: Letter X, crossroads

Exercise C – Line Types

Identify the type of lines:

1. Railway tracks: ______ lines
2. The letter X: ______ lines
3. Corner of a book: ______ lines
4. Opposite sides of a rectangle: ______ lines
5. Roads at a junction: ______ lines

Angle Relationships

Angles on a Straight Line

• Rule: Angles on a straight line add up to 180°
• Example: If one angle is 110°, the other is 70°

Angles at a Point

• Rule: Angles around a point add up to 360°
• Example: Four 90° angles around a point

Vertically Opposite Angles

• Rule: When two lines cross, opposite angles are equal
• Example: In an X shape, opposite angles are the same

Exercise D – Angle Relationships

Find the missing angles:

1. Angles on a line: 125° and ______°
2. Angles at a point: 90°, 120°, 70°, and ______°
3. Vertically opposite: If one angle is 65°, the opposite angle is ______°
4. Three angles on a line: 45°, 60°, and ______°
5. Angles around a point: 110°, 85°, 95°, and ______°

Bearing

What is Bearing?

Bearing is a way to describe direction using angles measured from North in a clockwise direction.

Key Points:

• Always measured from North (N)
• Always measured clockwise
• Written as 3-digit numbers (e.g., 045°, 270°)

The Compass

Main Directions:

• North (N): 000° or 360°
• East (E): 090°
• South (S): 180°
• West (W): 270°

Reading Bearings

Examples:

• North: 000°
• Northeast: 045°
• East: 090°
• Southeast: 135°
• South: 180°
• Southwest: 225°
• West: 270°
• Northwest: 315°

Exercise E – Basic Bearings

Give the bearing for these directions:

1. East: ______°
2. South: ______°
3. West: ______°
4. Northeast: ______°
5. Southwest: ______°
6. Northwest: ______°
7. Southeast: ______°
8. North: ______°

Back Bearings

What is a Back Bearing?

If you travel from A to B on a bearing, the back bearing is the direction from B to A.

Rule: Back bearing = Original bearing ± 180°

• If original bearing < 180°, add 180°
• If original bearing > 180°, subtract 180°

Examples:

• Bearing from A to B = 050°
• Back bearing from B to A = 050° + 180° = 230°
• Bearing from A to B = 250°
• Back bearing from B to A = 250° – 180° = 070°

Exercise F – Back Bearings

Find the back bearings:

1. Original bearing: 060°, Back bearing: ______°
2. Original bearing: 200°, Back bearing: ______°
3. Original bearing: 135°, Back bearing: ______°
4. Original bearing: 300°, Back bearing: ______°
5. Original bearing: 045°, Back bearing: ______°

Practical Applications

Exercise G – Real Life Bearings

Solve these bearing problems:

1. Navigation: A ship sails from port on bearing 120°. What bearing should it take to return directly to port? Answer: ______°
2. School Direction: From your house, the school is on bearing 075°. What bearing is your house from the school? Answer: ______°
3. Market Direction: The market is southeast of your home. What is the approximate bearing? Answer: ______°
4. Playground: If the playground is directly west of the classroom, what is its bearing from the classroom? Answer: ______°

Exercise H – Compass Navigation

Use compass directions:

1. Morning Sun: The sun rises in the east. What bearing is this? Answer: ______°
2. Evening Sun: The sun sets in the west. What bearing is this? Answer: ______°
3. Shadow Direction: At noon, your shadow points north. You are facing which direction? Answer: ______ (bearing ______°)

Angle and Bearing Problems

Exercise I – Combined Problems

Solve these angle and bearing problems:

1. Clock Problem: What angle do clock hands make at 3:00? Answer: ______°
2. Triangle Problem: In a triangle, two angles are 60° and 70°. What is the third angle? Answer: ______°
3. Navigation Problem: A plane flies north for 100 km, then turns and flies on bearing 090° for 50 km. In which direction is the plane from its starting point? Answer: ______ of starting point

Exercise J – Angle Calculations

Calculate missing angles:

1. Pentagon: Each angle in a regular pentagon is 108°. What is the sum of all angles? Answer: ______°
2. Hexagon: A regular hexagon has 6 equal angles. If the sum of angles is 720°, what is each angle? Answer: ______°
3. Star Shape: A five-pointed star has 5 sharp angles of 36° each. What is the sum of these angles? Answer: ______°

Exercise K – Practical Measurement

Apply angle knowledge:

1. Door Opening: A classroom door opens 90°. If it’s half open, what angle is it? Answer: ______°
2. Roof Slope: A roof makes a 30° angle with the horizontal. What type of angle is this? Answer: ______ angle
3. Staircase: A staircase makes a 45° angle with the ground. What type of angle is this? Answer: ______ angle

Answer Key

Exercise A:

1. Acute, 2. Right, 3. Obtuse, 4. Straight, 5. Reflex, 6. Complete, 7. Acute, 8. Reflex

Exercise B:

1. 90°, 2. 180°, 3. 270°, 4. 90°, 5. 90°

Exercise C:

1. Parallel, 2. Intersecting, 3. Perpendicular, 4. Parallel, 5. Intersecting

Exercise D:

1. 55°, 2. 80°, 3. 65°, 4. 75°, 5. 70°

Exercise E:

1. 090°, 2. 180°, 3. 270°, 4. 045°, 5. 225°, 6. 315°, 7. 135°, 8. 000°

Exercise F:

1. 240°, 2. 020°, 3. 315°, 4. 120°, 5. 225°

Exercise G:

1. 300°, 2. 255°, 3. 135°, 4. 270°

Exercise H:

1. 090°, 2. 270°, 3. South, 180°

Exercise I:

1. 90°, 2. 50°, 3. Northeast

Exercise J:

1. 540°, 2. 120°, 3. 180°

Exercise K:

1. 45°, 2. Acute, 3. Acute