Geometry: Angles and Scale Drawings
Use angle relationships to calculate missing angles, then apply scale factors to maps, plans and diagrams. You will practise explaining why an angle has a particular size and converting between drawing lengths and real distances.
Estimated learning time: 25–35 minutes · Beginner to intermediate
By the end, you should be able to:
- Use angle facts on straight lines, around a point and at an intersection.
- Find angles formed when a transversal crosses parallel lines.
- Interpret a scale such as 1 : 100 and calculate real or drawing lengths.
- Use consistent units and scale factors in practical problems.
1. Angle facts you should know
An angle measures a turn and is measured in degrees (°). These facts are useful building blocks for solving geometry problems.
| Angle relationship | Fact |
|---|---|
| Angles on a straight line | Add up to 180° |
| Angles around a point | Add up to 360° |
| Vertically opposite angles | Are equal |
| Angles in a triangle | Add up to 180° |
| Angles in a quadrilateral | Add up to 360° |
Two adjacent angles on a straight line are 127° and x.
x + 127° = 180°
x = 180° − 127° = 53°.
Three angles around a point are 85°, 140° and x.
85° + 140° + x = 360°
x = 360° − 225° = 135°.
Two straight lines cross. If one angle is 68°, the angle directly opposite it is also 68°. The two adjacent angles are each 180° − 68° = 112°.
2. Angles made by parallel lines
When a straight line called a transversal crosses two parallel lines, special angle relationships appear.
- Corresponding angles are equal. They occupy matching positions at the two intersections.
- Alternate interior angles are equal. They lie between the parallel lines on opposite sides of the transversal.
- Co-interior angles add up to 180°. They lie between the parallel lines on the same side of the transversal.
Two parallel lines are crossed by a transversal. One interior angle is 73°. Its alternate interior angle is also 73°.
If one co-interior angle is 118°, the other is 180° − 118° = 62°.
3. What is a scale drawing?
A scale drawing represents a real object or distance using a smaller or larger drawing. The scale compares a length on the drawing with the corresponding real length, using the same units.
A scale of 1 : 100 means 1 unit on the drawing represents 100 of the same units in real life. For example, 1 cm on a floor plan represents 100 cm (1 m) in the room.
| Scale | Meaning | Example |
|---|---|---|
| 1 : 50 | 1 drawing unit = 50 real units | 2 cm represents 100 cm |
| 1 : 100 | 1 drawing unit = 100 real units | 3 cm represents 300 cm |
| 1 : 25,000 | 1 map unit = 25,000 real units | 1 cm represents 25,000 cm = 250 m |
A plan uses a scale of 1 : 100. A wall measures 4.5 cm on the plan.
Real length = 4.5 × 100 = 450 cm = 4.5 m.
A table is 180 cm long. On a drawing with scale 1 : 30, its drawing length is 180 ÷ 30 = 6 cm.
4. Scale factors and enlargements
A scale factor tells you how many times a length changes. Multiply an original length by the scale factor to find the new length. A factor greater than 1 enlarges; a positive factor between 0 and 1 reduces.
A rectangle is 3 cm wide and 5 cm long. It is enlarged by scale factor 2.
New width = 3 × 2 = 6 cm. New length = 5 × 2 = 10 cm.
Every length is doubled, so the shape keeps the same proportions.
A 12 cm line is reduced by scale factor 0.5. New length = 12 × 0.5 = 6 cm.
For a scale drawing written as 1 : n, divide the real length by n to get the drawing length, or multiply the drawing length by n to get the real length. Always check which direction the question asks you to convert.
5. Plan a room using a scale
Suppose a rectangular room is 5 m long and 3 m wide. You want to draw it using a scale of 1 : 100.
- Convert real measurements to centimetres: 5 m = 500 cm and 3 m = 300 cm.
- Divide each measurement by 100.
- Draw a rectangle 5 cm long and 3 cm wide.
The drawing is proportional to the real room. If you add a door or window, measure its real position and convert that distance using the same scale.
6. Common mistakes to avoid
- Using 360° for a straight line. Angles on a straight line total 180°; angles around a point total 360°.
- Assuming adjacent angles are equal. Only use an equality rule when the angle relationship supports it.
- Mixing up alternate and co-interior angles. Alternate angles are equal; co-interior angles total 180° when the lines are parallel.
- Multiplying when you should divide in a scale problem. Decide whether you are finding the real length or the drawing length.
- Forgetting to convert units. A scale ratio compares like units.
- Changing only one side of a shape during an enlargement. Apply the same scale factor to every length.
7. Practice questions
Try each question before opening the solution.
- Two angles lie on a straight line. One is 42°. Find the other.
Show solution
180° − 42° = 138°.
- Three angles around a point are 90°, 125° and x. Find x.
Show solution
x = 360° − 90° − 125° = 145°.
- Two lines cross. One angle is 114°. Find the vertically opposite angle and one adjacent angle.
Show solution
Vertically opposite angle = 114°. Adjacent angle = 180° − 114° = 66°.
- Parallel lines are crossed by a transversal. An alternate interior angle is 57°. Find its partner.
Show solution
Alternate interior angles are equal, so the answer is 57°.
- A map scale is 1 : 20,000. Two places are 3 cm apart on the map. Find the real distance in metres.
Show solution
3 × 20,000 = 60,000 cm. Divide by 100: 600 m.
- A 7 m wall is shown on a plan at scale 1 : 100. How long is it on the plan in centimetres?
Show solution
7 m = 700 cm. 700 ÷ 100 = 7 cm.
- A drawing of a triangle has sides 4 cm, 5 cm and 6 cm. It is enlarged by scale factor 3. Find the new side lengths.
Show solution
4 × 3 = 12 cm, 5 × 3 = 15 cm, and 6 × 3 = 18 cm. New sides: 12 cm, 15 cm and 18 cm.
- A rectangular garden is 8 m by 6 m. Draw it at scale 1 : 200. Give both drawing dimensions in centimetres.
Show solution
8 m = 800 cm; 800 ÷ 200 = 4 cm. 6 m = 600 cm; 600 ÷ 200 = 3 cm. Drawing: 4 cm by 3 cm.
8. Quick knowledge check
Choose one answer for each question, then select Check answers to see your score and explanations.
0 of 5 answered
9. Printable worksheet
Show your working. Include units in every scale-drawing answer.
Geometry: angles and scale drawings — Grade 7
Name: __________________________ Date: ______________
- Find the missing angle on a straight line if the other angle is 139°.
- Angles around a point are 75°, 105°, 90° and x. Find x.
- Two lines cross. One angle is 48°. Find the other three angles.
- Two parallel lines are crossed by a transversal. A co-interior angle is 121°. Find its partner.
- A map uses a scale of 1 : 50,000. A route is 4 cm on the map. Find its real length in kilometres.
- A real path is 1.2 km long. At scale 1 : 20,000, find its map length in centimetres.
- Enlarge a 2 cm by 7 cm rectangle by scale factor 2.5. Give its new dimensions.
- A room measures 4.8 m by 3.6 m. Draw it at scale 1 : 100. Give the drawing dimensions in centimetres.
Answer key
- 180° − 139° = 41°.
- 360° − (75° + 105° + 90°) = 90°.
- Opposite angle = 48°; each adjacent angle = 132°.
- 180° − 121° = 59°.
- 4 × 50,000 = 200,000 cm = 2,000 m = 2 km.
- 1.2 km = 120,000 cm; 120,000 ÷ 20,000 = 6 cm.
- 2 × 2.5 = 5 cm; 7 × 2.5 = 17.5 cm.
- 4.8 m = 480 cm, so 4.8 cm; 3.6 m = 360 cm, so 3.6 cm. Drawing: 4.8 cm by 3.6 cm.