Congruent shapes are exactly the same shape and size. Similar shapes have the same shape but can be different sizes. Congruent is similar with scale factor 1, so congruence is the stricter idea.
This lesson pulls together corresponding sides and scale factors, because both ideas depend on pairing the parts of two shapes.
How do you tell them apart?
For similar shapes, corresponding angles are equal and corresponding sides are in the same ratio. For triangles, three equal angles are enough. Equal ratios on all three sides are also enough.
For congruent triangles you need more: the triangles must match in size too. The usual sets of conditions are SSS, SAS, ASA or AAS, and RHS. Each has to include at least one side.
Some shapes are always similar to each other. Any two squares are similar, and so are any two circles.
Worked example
Decide whether each pair of triangles is congruent, similar but not congruent, or neither.
(a) Sides 6, 8, 10 and 9, 12, 15.
Ratios: 9 ÷ 6 = 1.5, 12 ÷ 8 = 1.5, 15 ÷ 10 = 1.5. Equal ratios, so they are similar. The scale factor is not 1, so they are similar but not congruent.
(b) Sides 6, 8, 10 and 8, 6, 10.
The same three lengths appear in each triangle, so they are congruent by SSS. The second triangle may be a reflection or rotation of the first, which is still congruent. Congruent.
(c) Sides 6, 8, 10 and 6, 8, 11.
Two pairs match, but 10 and 11 do not, and the ratios 6/6, 8/8 and 11/10 differ. Neither. As a further check, 6² + 8² = 100 = 10², so the first triangle has a right angle, while the second does not.
The mistake to watch for
A common slip is to claim congruence from two sides and an angle that is not between them.
Mistaken working: In triangle ABC, AB = 8 cm, BC = 11 cm and angle A = 40°. In triangle DEF, DE = 8 cm, EF = 11 cm and angle D = 40°. “Congruent by SAS.”
Angle A lies at the end of AB, between AB and AC, not between AB and BC. The given angle is not the included angle, so SAS does not apply.
The correction is to check where the angle sits. For SAS the angle must be between the two given sides. Two sides and a non-included angle is not a proof, because the triangle is not always fixed in shape.
Check yourself
Try these, then open each answer.
1. Two triangles each have angles 40°, 60° and 80°. Each has one side of length 7 cm, but the diagram does not say which angle it is opposite. Are they congruent?
Show answer
They are similar, because all angles match. They are congruent only if the 7 cm sides are opposite the same angle in both triangles. The information given does not say that, so congruence is not proved.
2. Rectangle P is 6 cm by 4 cm and rectangle Q is 9 cm by 6 cm. Are they similar? Are they congruent?
Show answer
9 ÷ 6 = 1.5 and 6 ÷ 4 = 1.5. The ratios match, so the rectangles are similar. They are different sizes, so they are not congruent.
3. Rectangle R is 6 cm by 4 cm and rectangle S is 9 cm by 5 cm. Are they similar?
Show answer
9 ÷ 6 = 1.5 but 5 ÷ 4 = 1.25. The ratios differ, so the rectangles are not similar, even though all angles are 90°.
Where this leads next
Put everything in this module together in the similarity, congruence and scale practice set. The link between equal angles and geometric reasoning is developed in angles and geometric reasoning. The non-calculator working trainer is useful for checking ratios quickly.
If you can state the conditions but freeze when choosing which one a diagram supports, a teacher in online one-to-one Mathematics tuition can practise that decision with you on fresh diagrams.