Similarity & Congruence Reference Cheat Sheet

A printable reference covering congruent figures, similar figures, triangle congruence, triangle similarity, scale factors, and proportional sides for grades 8-10.

Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

This cheat sheet covers the main rules for proving figures congruent or similar in geometry. Students need these ideas to compare shapes, justify triangle relationships, and solve for missing side lengths or angles. It is especially useful when working with transformations, proportional reasoning, and formal proofs. The reference keeps the key tests and formulas in one place for quick review. Conguent figures have the same shape and size, so corresponding sides and angles are equal. Similar figures have the same shape but not always the same size, so corresponding angles are equal and corresponding sides are proportional. Important triangle congruence tests include $SSS$ , $SAS$ , $ASA$ , $AAS$ , and $HL$ . Important triangle similarity tests include $AA$ , $SAS$ , and $SSS$ similarity.

Key Facts

Congruent figures have equal corresponding sides and equal corresponding angles, so if $\triangle ABC \cong \triangle DEF$ , then $AB = DE$ , $BC = EF$ , $AC = DF$ , and corresponding angles are equal.
Similar figures have equal corresponding angles and proportional corresponding sides, so if $\triangle ABC \sim \triangle DEF$ , then $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ .
The scale factor from a preimage to an image is $k = \frac{\text{image length}}{\text{preimage length}}$ .
If two figures are similar with scale factor $k$ , then all corresponding side lengths are multiplied by $k$ .
If two figures are similar with scale factor $k$ , then their perimeters are in the ratio $k$ and their areas are in the ratio $k^2$ .
The triangle congruence shortcuts are $SSS$ , $SAS$ , $ASA$ , $AAS$ , and $HL$ for right triangles.
The triangle similarity shortcuts are $AA$ , $SAS$ , and $SSS$ similarity.
Corresponding parts of congruent triangles are congruent, often written as $CPCTC$ after two triangles have been proven congruent.

Vocabulary

Congruent Figures: Figures that have the same shape and the same size, with all corresponding sides and angles equal.
Similar Figures: Figures that have the same shape, equal corresponding angles, and proportional corresponding side lengths.
Corresponding Parts: Matching sides or angles in two figures that are in the same relative position.
Scale Factor: The multiplier that changes each side length of one figure to the matching side length of a similar figure.
Dilation: A transformation that enlarges or reduces a figure by a scale factor while keeping the same shape.
Proportion: An equation stating that two ratios are equal, such as $\frac{a}{b} = \frac{c}{d}$ .

Common Mistakes to Avoid

Matching the wrong corresponding sides is wrong because ratios must compare sides in the same relative positions.
Using $SSA$ as a triangle congruence shortcut is wrong because $SSA$ does not always determine one unique triangle.
Assuming similar figures are congruent is wrong because similar figures can have different sizes when the scale factor is not $1$ .
Using the side scale factor for area is wrong because areas scale by $k^2$ , not by $k$ .
Writing a similarity statement in the wrong order is wrong because the order of vertices determines which angles and sides correspond.

Practice Questions

1 Triangles $\triangle ABC$ and $\triangle DEF$ are similar with $AB = 6$ , $BC = 9$ , $DE = 10$ , and $EF = x$ . Find $x$ .
2 A rectangle is dilated by a scale factor of $\frac{3}{2}$ . If its original perimeter is $24$ cm and its original area is $32$ cm $^2$ , find the new perimeter and new area.
3 In $\triangle ABC$ and $\triangle XYZ$ , $AB = XY$ , $BC = YZ$ , and $\angle B \cong \angle Y$ . Which congruence shortcut can prove the triangles congruent?
4 Two triangles have two pairs of equal angles, but no side lengths are given. Explain why this is enough to prove similarity but not necessarily congruence.

Sign in to save