Geometry

Congruence & Similarity

Using transformations to determine whether two figures are congruent or similar.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Rigid Transformations

Same Size and Shape, or Just Same Shape?

Two triangles are drawn on a page — one clearly larger than the other, but with matching angles and proportional sides. Before reading on, think about the word you'd use to describe their relationship, and how it's different from the word you'd use for two triangles that are exact copies of each other, just moved around.

Definition — Congruent and Similar

Two figures are congruent if one can be mapped onto the other using only rigid transformations — meaning they have exactly the same size and shape. Two figures are similar if one can be mapped onto the other using rigid transformations plus a dilation (a size change) — meaning they have the same shape, but not necessarily the same size.

Since rigid transformations preserve every distance, two figures related only by rigid transformations must have identical corresponding side lengths — that's exactly what congruence means. Adding a dilation into the mix scales every distance by the same factor, so angles stay identical (defining the shared shape) while side lengths scale proportionally — exactly the same scale-factor idea from Grade 7's ratio and proportional-relationship reasoning.

Worked Example — Confirming Congruence Through Transformations

Triangle A is translated 3 units right, then reflected over the x-axis, producing Triangle B. Are the triangles congruent? Since translation and reflection are both rigid transformations that preserve every distance and angle, Triangle B is congruent to Triangle A.

Worked Example — Confirming Similarity with a Scale Factor

Triangle A has side lengths 3, 4, and 5. Triangle B has side lengths 6, 8, and 10. Are they similar? Check whether a single scale factor connects every pair of corresponding sides: 6/3 = 2, 8/4 = 2, 10/5 = 2. Since the same scale factor, 2, works for all three sides, the triangles are similar.

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

To confirm similarity, check that the same scale factor connects every pair of corresponding sides — a match on just one or two sides isn't enough, the same 'check several pairs' caution from confirming a proportional relationship.

Common Mistakes

Confusing congruent and similar, describing two figures of different sizes as congruent.
Congruent figures must match in both size and shape; similar figures only need to match in shape, with sizes related by a consistent scale factor.
Confirming similarity by checking only one pair of corresponding sides, missing that the other sides don't share the same scale factor.
Verify that every pair of corresponding sides shares the exact same scale factor before concluding two figures are similar.

Key Takeaways

Congruent figures are related by rigid transformations alone, matching in both size and shape.
Similar figures are related by rigid transformations plus a dilation, matching in shape with sizes connected by a consistent scale factor.
Confirming similarity requires checking that the same scale factor connects every pair of corresponding sides.

Summary

Congruence and similarity give precise, transformation-based meaning to 'same shape' comparisons. The next lesson looks at one of the most famous relationships in geometry, connecting the three sides of a right triangle.

Track your progress

← Rigid Transformations Back to Geometry Proving the Pythagorean Theorem→