Similarity (Cambridge (CIE) IGCSE Maths: Extended): Revision Note
Exam code: 0580 & 0980
Similarity
What are similar shapes?
Two shapes are similar if they have the same shape and same angles and their corresponding sides are in proportion
One shape is an enlargement of the other
Similarity does not imply congruence
Two similar shapes could be different enlargements of each other
How do I prove that two shapes are similar?
To show that two non-triangular shapes are similar you need to show
that their corresponding sides are in proportion
and that their corresponding angles are equal
Divide the length of one side by the length of the corresponding side on the other shape to find the scale factor
If the scale factor is the same for all corresponding sides, then the shapes are similar if the corresponding angles are equal
How do I prove that two triangles are similar?
To show that two triangles are similar you need to show that their corresponding angles are the same
If the angles are the same then corresponding lengths of a triangle will automatically be in proportion
You can use angle properties to identify equal angles
Look out for for isosceles triangles, vertically opposite angles and angles on parallel lines
If a question asks you to prove two triangles are similar
For each pair of corresponding angles
State that they are of equal size
Give a reason for why they are equal
Examiner Tips and Tricks
A pair of similar triangles can often be opposite each other in an hourglass formation. Look out for the vertically opposite, equal angles.
It may be helpful to sketch the triangles next to each other and facing in the same direction.
Worked Example
(a) Prove that the two rectangles shown in the diagram below are similar.
Answer:
All corresponding angles are equal because they are all 90°
Use the corresponding lengths (15 cm and 6 cm) to find the scale factor
Use the corresponding width (5 cm and 2 cm) to find the scale factor for the other pair of sides
The two rectangles are similar, with a scale factor of 2.5
(b) In the diagram below, AB and CD are parallel lines.
Show that triangles ABX and CDX are similar.
Answer:
State the equal angles by name, along with clear reasons
Don't forget to state that similar triangles need to have equal corresponding angles
Angle AXB = angle CXD (vertically opposite angles are equal)
Angle ABC = angle BCD (alternate angles on parallel lines are equal)
Angle BAD = angle ADC (alternate angles on parallel lines are equal)
All three corresponding angles are equal, so the two triangles are similar
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Questions students ask about Similarity
Real questions from students revising this topic on Save My Exams
How do you prove that two shapes are similar?
For triangles, show that all the corresponding angles are equal and give a reason for each; for any other shape, show that every pair of corresponding sides has the same scale factor.
Triangles
- Name each pair of equal angles
- Give a geometric reason for each — vertically opposite angles, alternate angles on parallel lines, isosceles triangles
- Conclude: all corresponding angles are equal, so the triangles are similar
Example from this page: AB and CD are parallel; show triangles ABX and CDX are similar.
- Angle AXB = angle CXD (vertically opposite angles are equal)
- Angle ABC = angle BCD (alternate angles on parallel lines are equal)
- Angle BAD = angle ADC (alternate angles on parallel lines are equal)
- All three corresponding angles are equal, so the triangles are similar
Other shapes
- Divide each side by its corresponding side on the other shape
- If every answer is the same, the shapes are similar
Example from this page: rectangles 15 cm × 5 cm and 6 cm × 2 cm. 15 ÷ 6 = 2.5 and 5 ÷ 2 = 2.5, the same scale factor, so they are similar.
What is the difference between congruent and similar shapes?
Congruent shapes are identical in shape and size; similar shapes are the same shape but can be different sizes.
Similar shapes
- Corresponding angles are equal
- Corresponding sides are in proportion — all multiplied by the same scale factor
- One shape is an enlargement of the other
Congruent shapes
- Corresponding angles are equal
- Corresponding sides are exactly equal
- They would fit exactly on top of each other, though you may need to rotate or flip one
Congruent shapes are a special case of similar shapes, with a scale factor of 1. So congruent shapes are always similar, but similar shapes are not necessarily congruent.
Example from this page: rectangles of 15 cm × 5 cm and 6 cm × 2 cm.
- 15 ÷ 6 = 2.5 and 5 ÷ 2 = 2.5, the same scale factor, so they are similar
- The scale factor is 2.5, not 1, so they are not congruent
If the second rectangle were also 15 cm × 5 cm, they would be congruent.
What is a corresponding length in similar shapes?
A corresponding length is the side in the same position on the other shape — the one it matches up with when one shape is an enlargement of the other.
How to find them:
- In triangles, corresponding sides are the ones opposite the equal angles
- In other shapes, match by position: longest with longest, shortest with shortest
- Sketch the two shapes next to each other, facing the same way, so the pairs are obvious
Example from this page: rectangles of 15 cm × 5 cm and 6 cm × 2 cm.
- Lengths correspond: 15 cm ↔ 6 cm, and 15 ÷ 6 = 2.5
- Widths correspond: 5 cm ↔ 2 cm, and 5 ÷ 2 = 2.5
- Pairing 15 cm with 2 cm would be wrong — they are not in the same position
In the hourglass triangles ABX and CDX: the page shows angle at A equals the angle at D, and the angle at B equals the angle at C, so AB ↔ DC, AX ↔ DX and BX ↔ CX.
Dividing a length by its corresponding length gives the scale factor.
