Congruence & Transformations
Two figures are congruent when they have the same size and the same shape. The symbol is . Position doesn't matter — a triangle that has been slid across the page, flipped over a line, or spun around a point is still the exact same triangle. It just landed somewhere else.
That's why transformations are the natural way to test congruence. If you can map one figure onto the other using only slides, flips, and turns, the figures are congruent. If getting from one to the other requires changing the size, they're not.
The three transformations that keep congruence
Translations (slides), reflections (flips), and rotations (turns) all move a figure without changing it. Every side of the image is the same length as the matching side of the original, and every angle is the same measure. So the image is always congruent to the original — no measuring required.
In the figure below, the triangle on the right is a reflection of the triangle on the left across the vertical axis. It faces the other way, but every side and every angle matches, so the two triangles are congruent.
The one transformation that doesn't
A dilation multiplies every side of a figure by a scale factor . The image keeps the same shape — same angles, same proportions — but the size changes. Congruent figures need the same size and the same shape, so a dilated image is not congruent to the original (unless , which leaves the figure unchanged).
This is the whole test in one sentence: slides, flips, and turns give a congruent image; a dilation does not.
How to answer congruence questions
When a problem shows two figures and names the transformation, you don't need to measure anything. Identify the transformation, then apply the rule. If it's a translation, reflection, or rotation — or any combination of those — the answer is congruent. If a dilation is involved anywhere, the answer is not congruent.
When no transformation is named, compare the figures directly: same shape but different size means similar, not congruent.
Worked examples
Example 1: a reflection
Triangle is a reflection of triangle . Are the triangles congruent?
Answer: Yes — the triangles are congruent.
Example 2: a dilation
Figure is a dilation of figure with scale factor . Are the figures congruent?
Answer: No — the figures are not congruent.
Example 3: a translation
A figure is translated units to the right. Is the image congruent to the original?
Answer: Yes — the image is congruent to the original.
Try one yourself
Common questions
What does congruent mean, exactly?
Same size and same shape. If you could cut one figure out and lay it perfectly on top of the other — flipping or turning it if needed — the figures are congruent. The symbol is .
Why doesn't a dilation produce a congruent image?
Because a dilation changes size. It multiplies every side by the scale factor , so unless , the image is bigger or smaller than the original. The shape survives, but congruence requires the size to survive too.
Does it matter where the figure ends up?
No. Congruence ignores position and orientation completely. A figure that has been slid, flipped, or turned is congruent to the original no matter where it lands or which way it faces.
What if several transformations are applied in a row?
Check whether a dilation is in the list. Any combination of translations, reflections, and rotations keeps the image congruent. The moment a dilation appears, the sizes stop matching and congruence is lost.
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