Two figures are congruent if they are the same shape and the same size. They may be turned, flipped, or moved to a different location, but if all corresponding sides and angles match exactly, the figures are congruent. In this topic, we connect congruence to rigid transformations and then focus on triangle congruence conditions: SSS, SAS, and ASA.

These triangle congruence conditions help us decide when two triangles must be congruent without checking every side and every angle. SSS means all three corresponding side lengths are equal. SAS means two corresponding sides and the included angle are equal. ASA means two corresponding angles and the included side are equal. Rather than memorizing these as random rules, we will think about why certain pieces of information are enough to determine one exact triangle.

We will also discuss similarity. Two figures are similar if they have the same shape but not necessarily the same size. Similar figures have corresponding angles that are equal and corresponding side lengths that are proportional. The scale factor tells how much larger or smaller one figure is compared to another. Similarity helps us use information about one shape to make conclusions about another, such as finding missing side lengths, comparing maps and models, or understanding enlargements and reductions.

As future teachers, it is important to help students distinguish between “same shape and same size” and “same shape but different size.” Congruence and similarity both involve comparison, but they answer different questions about how figures are related.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Define congruent figures.

  • Explain how rigid transformations preserve congruence.

  • Use SSS, SAS, and ASA to determine triangle congruence.

  • Define similar figures.

  • Explain the difference between congruent and similar shapes.

  • Identify corresponding sides and corresponding angles.

  • Use scale factors to compare similar figures.

  • Use similarity to find missing side lengths.

Key Vocabulary

  • Congruent- Same shape and same size.

  • Corresponding Parts - Matching sides or angles in two figures.

  • Rigid Transformation - A movement that does not change size or shape, such as a slide, flip, or turn.

  • SSS Congruence - Two triangles are congruent if all three corresponding sides are equal.

  • SAS Congruence - Two triangles are congruent if two corresponding sides and the included angle are equal.

  • ASA Congruence - Two triangles are congruent if two corresponding angles and the included side are equal.

  • Similar - Same shape, but not necessarily the same size.

  • Scale Factor - The number used to multiply side lengths to make a similar figure larger or smaller.

  • Proportional - Having the same multiplicative relationship.

Triangle Congruence Conditions

  • SSS - Three corresponding sides - The side lengths lock in one triangle shape.

  • SAS - Two corresponding sides and the included angle - The angle between the sides fixes how the triangle opens.

  • ASA - Two corresponding angles and the included side - The side and angles determine the remaining shape.

The order of the letters matters. In SAS, the angle must be between the two sides. In ASA, the side must be between the two angles.

Teacher Connection

Congruence and similarity are important because they help students reason about shape relationships. Students often begin by relying on appearance, but geometry asks them to use precise evidence. Two figures may look similar but not actually be similar, or they may look different because one has been rotated or reflected even though they are congruent.

For future teachers, this is a good place to emphasize corresponding parts and careful language. Instead of saying “these sides match” vaguely, students should identify which side in one figure corresponds to which side in the other figure.

Helpful teacher questions include:

  • Are the figures the same shape?

  • Are they the same size?

  • Which sides correspond?

  • Which angles correspond?

  • Could one figure be moved onto the other with a slide, flip, or turn?

  • Is there a scale factor?

  • Are the side lengths proportional?

  • Does this show congruence, similarity, or neither?

Quick Reflection Question

Why might a student think two similar figures are congruent? What example could you use to show the difference between “same shape” and “same shape and same size”?