Congruent means same shape and same size. Similar means same shape but any size. That single word, size, is the entire difference between them, and every question in this unit is really asking you to notice which of the two you are looking at.
In Unit 2 you found that three angles do not fix a triangle, because a tiny 60-60-60 triangle and an enormous one both have those angles. At the time that looked like a failure. It was not. It was the discovery of similarity: those two triangles are the same shape and different sizes, and that whole family has a name.
What stays the same and what changes
| Congruent | Similar | |
|---|---|---|
| Angles | All equal | All equal |
| Sides | All equal | In the same ratio, not equal |
| Size | Identical | Can be any size |
| Symbol | ≅ | ~ |
| Everyday example | Two copies from the same photocopier at 100% | The same photo printed small and printed large |
Similar shapes have sides in the same ratio. If two shapes are congruent, that ratio is 1, which is still a ratio. So congruent is a special case of similar, in the same way that a square is a special case of a rectangle.
The scale factor
For similar shapes, the number you multiply by to get from the small one to the big one is called the scale factor. Find it by dividing one pair of matching sides. If a 3 cm side matches a 12 cm side, the scale factor is 4, and every other side must also be 4 times bigger.
The most common mistake is dividing the wrong pair. A side only matches the side in the same position in the other shape, and position is decided by the angles at its ends, not by looking. Before you divide anything, label the corners of both shapes so that matching corners have matching letters.
Similarity is how you measure things you cannot reach. Stand a metre stick upright and measure its shadow, then measure the shadow of a tall tree at the same moment. The two triangles are similar because the sun is in the same place for both. A one metre stick with a two metre shadow, next to a tree with a fourteen metre shadow, means a seven metre tree.
The full lesson is waiting for you
Create a free account to unlock activities, practice tests, presentations and videos - and to track your progress and confidence score for every subject.
Triangle ABC has sides 4, 6, 8 cm. Triangle PQR is similar, with PQ = 10 cm matching AB = 4 cm. Find QR and RP.
Measure a flagpole without climbing it
That's a peek at activity 1 of 5 - sign in for all of them, full length.
Congruent: same shape AND same size. The symbol is a congruence sign.
11 more points to remember - sign in to see the rest.
Question 1 of 10
Sign in to watch the video for this topic.