Shape and scale
Similarity preserves shape while allowing a scale change. Congruence preserves both shape and size.
Learning goals
- Distinguish congruence from similarity.
- Identify figure families whose members are always similar.
- State both conditions for polygon similarity.
Concept lesson
Congruent means exact match
Congruent figures superpose exactly: corresponding lengths and angles are equal.
Congruent implies similar, not conversely
Every congruent pair is similar with scale factor 1. Similar figures may have a different scale factor and therefore need not be congruent.
Two polygon conditions
Polygons with the same number of sides are similar when corresponding angles are equal and corresponding sides are proportional. Both conditions matter for general polygons.
Always, sometimes or never similar
Any two circles, squares or equilateral triangles are similar. Arbitrary rectangles, rhombi, isosceles triangles and trapeziums need not be similar because their angles or side ratios may differ.
Scale without congruence
Compare two squares of sides 2 cm and 5 cm. Record equal angles, common side ratio 2:5 and unequal sizes; conclude similar but not congruent.
Test both polygon conditions
For supplied rectangle and rhombus pairs, check corresponding angles and every side ratio before deciding similarity.
Photograph enlargement
Explain why a uniformly enlarged photograph is similar to the original, while stretching only its width usually destroys similarity.
One-mark similarity facts
For 'always similar', test whether shape can vary within the family. For polygons, state equal corresponding angles and proportional corresponding sides.
Exit ticket
Distinguish similar and congruent figures; decide whether any two equilateral triangles and any two rectangles are always similar; justify using both polygon conditions.