MathematicsChapter 9: Geometry and Polygons
Similar Figures
Two figures are similar when they have the same shape, not necessarily the same size.
Learning objectives
- Explain similar figures in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
Definition
Two figures are similar when they have the same shape, not necessarily the same size. The usual symbol is ~ (read 'similar'). Congruent figures are similar and the same size: you can lay one on the other. All circles are similar, because every circle is the same shape. A rectangle and a non-rectangular parallelogram need not be similar: the angles do not match, so the shape is different even if both have four sides.
Where / symbols
- same shape
- Two figures are similar when they have the same shape, not necessarily the same size.
- not necessarily
- Two figures are similar when they have the same shape, not necessarily the same size.
- ~
- The usual symbol is ~ (read 'similar').
- Congruent
- Congruent figures are similar and the same size: you can lay one on the other.
- All circles
- All circles are similar, because every circle is the same shape.
Overview
Match corners first, then compare side ratios:
Corresponding Parts
In similar figures, corresponding angles are equal and corresponding sides are proportional. Correspondence is about matching the same role (the two right angles, the two longest sides), not about which letter comes first in the alphabet. Write the letters of two similar triangles in matching order: triangle ABC ~ triangle DEF means angle A = angle D, and so on.
Examples A photo and its enlargement: same shape, different size, so similar · Two circles of radii 3 cm and 10 cm are similar · A long thin parallelogram is not similar to a square
Aa Sas and Scale Factor
AA similarity: if two angles of one triangle equal two angles of another, the third pair follows from 180 deg, so the triangles are similar. SAS similarity: two pairs of sides proportional and the included angles equal. SSS similarity: all three pairs of sides proportional. The scale factor is the common side ratio. The perimeter ratio equals that same side ratio.
Examples AA: two triangles each have 40 deg and 70 deg, so they are similar · SAS: AB/DE = AC/DF and angle A = angle D (the included angle) · Sides 3, 4, 5 and 6, 8, 10 have scale factor 2; perimeters 12 and 24, same ratio 2
WHAT — Core idea
Two figures are similar when they have the same shape, not necessarily the same size. The usual symbol is ~ (read 'similar'). Congruent figures are similar and the same size: you can lay one on the other. All circles are similar, because every circle is the same shape. A rectangle and a non-rectangular parallelogram need not be similar: the angles do not match, so the shape is different even if both have four sides.
HOW — Rules, method, and examples
1. CORRESPONDING PARTS: In similar figures, corresponding angles are equal and corresponding sides are proportional. Correspondence is about matching the same role (the two right angles, the two longest sides), not about which letter comes first in the alphabet. Write the letters of two similar triangles in matching order: triangle ABC ~ triangle DEF means angle A = angle D, and so on. 2. AA SAS AND SCALE FACTOR: AA similarity: if two angles of one triangle equal two angles of another, the third pair follows from 180 deg, so the triangles are similar. SAS similarity: two pairs of sides proportional and the included angles equal. SSS similarity: all three pairs of sides proportional. The scale factor is the common side ratio. The perimeter ratio equals that same side ratio.
WHY — Meaning and connections
Congruent means same shape and same size; similar allows a scale factor other than 1. SSS here is a similarity test (side ratios), not the congruent-triangle SSS you meet in constructions. If only two sides match in ratio but the equal angles are not the included pair, do not claim SAS similarity.
HOW — Answer in an exam
Similar (~) is same shape; congruent is same shape and size. All circles are similar. AA is enough for triangles because the third angle follows. SAS similarity needs the included angle. Perimeter ratio = side ratio (the scale factor).
Explanation
Congruent means same shape and same size; similar allows a scale factor other than 1. SSS here is a similarity test (side ratios), not the congruent-triangle SSS you meet in constructions. If only two sides match in ratio but the equal angles are not the included pair, do not claim SAS similarity.
In simple words
Similar (~) is same shape; congruent is same shape and size. All circles are similar. AA is enough for triangles because the third angle follows. SAS similarity needs the included angle. Perimeter ratio = side ratio (the scale factor).
Key points
- same shape
- not necessarily
- ~
- Congruent
- corresponding angles
- proportional
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Similar Figures
Sticky Notes · Quick Revision
WHAT
- Two figures are similar when they have the same shape, not necessarily the same size.
Key Points
- Two figures are similar when they have the same shape, not necessarily the same size.
- In similar figures, corresponding angles are equal and corresponding sides are proportional.
- AA similarity: if two angles of one triangle equal two angles of another, the third pair follows from 180 deg, so the triangles are similar.
- Congruent means same shape and same size; similar allows a scale factor other than 1.
HOW
- 1.
- In similar figures, corresponding angles are equal and corresponding sides are proportional.
- AA similarity: if two angles of one triangle equal two angles of another, the third pair follows from 180 deg, so the triangles are similar.
Examples
- A photo and its enlargement: same shape, different size, so similar
- Two circles of radii 3 cm and 10 cm are similar
- A long thin parallelogram is not similar to a square
- AA: two triangles each have 40 deg and 70 deg, so they are similar
- SAS: AB/DE = AC/DF and angle A = angle D (the included angle)
Common Mistakes
- Correction: Similar (~) is same shape; congruent is same shape and size.
- Correction: All circles are similar.
- Correction: AA is enough for triangles because the third angle follows.
Exam Tip
- Similar (~) is same shape; congruent is same shape and size. All circles are similar. AA is enough for triangles because the third angle follows. SAS similarity needs the included angle. Perimeter ratio = side ratio (the scale factor).
60-second revision
- Two figures are similar when they have the same shape, not necessarily the same size.
- In similar figures, corresponding angles are equal and corresponding sides are proportional.
- AA similarity: if two angles of one triangle equal two angles of another, the third pair follows from 180 deg, so the triangles are similar.
- Congruent means same shape and same size; similar allows a scale factor other than 1.
- Similar (~) is same shape; congruent is same shape and size.
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