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 , , , , and .
Important triangle similarity tests include , , and similarity.
Key Facts
- Congruent figures have equal corresponding sides and equal corresponding angles, so if , then , , , and corresponding angles are equal.
- Similar figures have equal corresponding angles and proportional corresponding sides, so if , then .
- The scale factor from a preimage to an image is .
- If two figures are similar with scale factor , then all corresponding side lengths are multiplied by .
- If two figures are similar with scale factor , then their perimeters are in the ratio and their areas are in the ratio .
- The triangle congruence shortcuts are , , , , and for right triangles.
- The triangle similarity shortcuts are , , and similarity.
- Corresponding parts of congruent triangles are congruent, often written as 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 .
Common Mistakes to Avoid
- Matching the wrong corresponding sides is wrong because ratios must compare sides in the same relative positions.
- Using as a triangle congruence shortcut is wrong because 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 .
- Using the side scale factor for area is wrong because areas scale by , not by .
- 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 and are similar with , , , and . Find .
- 2 A rectangle is dilated by a scale factor of . If its original perimeter is cm and its original area is cm, find the new perimeter and new area.
- 3 In and , , , and . 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.
Understanding Similarity & Congruence Reference
The order of letters in a geometry statement carries important information. When a problem names one triangle with letters in a certain order, each position is matched to the same position in the other triangle. This matching tells you which sides and angles belong together.
Do not match parts just because they look horizontal, vertical, or longest. A diagram may not be drawn to scale.
Mark known lengths and angles, then use the stated order or trace each vertex carefully. A wrong correspondence can produce a proportion that looks reasonable but gives the wrong answer.
Similarity is closely connected to resizing. Imagine making a photocopy larger or smaller. Every length changes by one common multiplier, while angle openings stay fixed.
A scale factor greater than one makes an enlargement. A scale factor between zero and one makes a reduction. Negative scale factors can appear in coordinate geometry.
They include a turn through the center of dilation as the figure changes size. On a coordinate plane, students often identify dilation by checking whether each point moves along a line from a common center. This is useful for explaining why a resized figure keeps its shape.
Perimeter and area do not change in the same way during resizing. If each side becomes three times as long, the distance around the figure becomes three times as large. The area becomes nine times as large because both length and width have been scaled.
This distinction matters in maps, blueprints, model buildings, photographs, and scale drawings. A small model may use a simple length scale, but the amount of paint, fabric, or land involved depends on area. When solving a word problem, decide whether it asks for a one-dimensional measurement such as fencing or a two-dimensional measurement such as flooring.
Triangle tests are efficient, but each test has conditions that must be met exactly. Side-angle-side requires the angle between the two known sides. Using an angle outside those sides does not prove congruence.
Angle-angle-angle shows that triangles have the same shape, yet it cannot show that they have the same size. Side-side-angle can be ambiguous because different triangles may fit the same information. The hypotenuse-leg rule works only when both triangles are right triangles.
In a written proof, first state the facts you know, then name the rule that justifies the triangle relationship. Only after that step can you transfer facts from one triangle to the other. This order keeps the reasoning valid.