Similarity & Congruence Cheat Sheet

A printable reference covering similarity ratios, congruent figures, triangle criteria, scale factors, perimeter ratios, and area ratios for grades 7-10.

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Similarity and congruence help students compare shapes by looking at matching angles and corresponding side lengths. This cheat sheet covers how to prove figures are the same shape or exactly the same size. It is useful for solving triangle problems, working with scale drawings, and understanding transformations. Students need these rules because many geometry proofs and calculations depend on matching corresponding parts correctly. Similar figures have equal corresponding angles and proportional corresponding sides, while congruent figures have equal corresponding angles and equal corresponding sides. A scale factor $k$ compares corresponding lengths in similar figures. Perimeters scale by $k$ , but areas scale by $k^2$ . Triangle similarity and congruence shortcuts such as $AA$ , $SSS$ , $SAS$ , $SSS$ , $SAS$ , $ASA$ , $AAS$ , and $HL$ make proofs faster and more organized.

Key Facts

Similar figures have equal corresponding angles and proportional corresponding sides, so $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} = k$ .
Congruent figures have equal corresponding angles and equal corresponding sides, so the scale factor is $k = 1$ .
The perimeter ratio of similar figures equals the scale factor, so $\frac{P_2}{P_1} = k$ .
The area ratio of similar figures equals the square of the scale factor, so $\frac{A_2}{A_1} = k^2$ .
The $AA$ similarity theorem says two triangles are similar if two pairs of corresponding angles are congruent.
The $SSS$ similarity theorem says two triangles are similar if all three pairs of corresponding sides are proportional.
The $SAS$ similarity theorem says two triangles are similar if two pairs of corresponding sides are proportional and the included angles are congruent.
Triangle congruence can be proven by $SSS$ , $SAS$ , $ASA$ , $AAS$ , or $HL$ for right triangles.

Vocabulary

Similar figures: Figures that have the same shape, equal corresponding angles, and proportional corresponding side lengths.
Congruent figures: Figures that have the same shape and the same size, with all corresponding sides and angles equal.
Scale factor: The constant multiplier $k$ that compares corresponding side lengths of similar figures.
Corresponding parts: Matching sides or angles that are in the same relative positions in two figures.
Included angle: The angle formed between two given sides of a triangle.
Dilation: A transformation that enlarges or reduces a figure by a scale factor while preserving its shape.

Common Mistakes to Avoid

Matching the wrong corresponding sides is wrong because proportional ratios must compare sides in the same relative positions.
Using the scale factor for area is wrong because area changes by $k^2$ , not by $k$ .
Assuming similar figures are congruent is wrong because similar figures may have different sizes unless $k = 1$ .
Using $SSA$ to prove triangle congruence is wrong because $SSA$ does not guarantee one unique triangle in general.
Forgetting to prove the included angle in $SAS$ is wrong because the angle must be between the two proportional or congruent sides.

Practice Questions

1 Triangles $ABC$ and $DEF$ are similar with $AB = 6$ , $BC = 9$ , and $DE = 10$ . If $AB$ corresponds to $DE$ , find the scale factor from $\triangle ABC$ to $\triangle DEF$ and find the side corresponding to $BC$ .
2 Two similar rectangles have side lengths $4$ cm by $7$ cm and $12$ cm by $21$ cm. Find the scale factor, the perimeter ratio, and the area ratio.
3 A triangle has sides $5$ , $8$ , and $10$ . A second triangle has sides $15$ , $24$ , and $30$ . Determine whether the triangles are similar and name the similarity theorem.
4 Explain why two triangles with three pairs of equal corresponding angles are similar but not necessarily congruent.

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