Congruence
Explore what it means for shapes to be congruent, and learn the four tests for proving triangles are congruent: SSS, SAS, AAS, and RHS.
What is Congruence?
Two shapes are congruent if they have exactly the same size and shape. One shape can be mapped onto the other by a combination of translations, rotations, and reflections — but not resizing.
▵ABC ≅ ▵DEF
The symbol ≅ means "is congruent to".
Corresponding sides are equal and corresponding angles are equal.
Congruent Triangles — Same Shape, Same Size
These triangles are congruent. All sides and angles match exactly.
The Four Congruence Tests for Triangles
You don't need to know all six measurements. If any one of these four conditions is met, the triangles are guaranteed to be congruent.
All three sides of one triangle are equal to the corresponding three sides of the other triangle.
Two sides and the included angle (between those two sides) are equal.
Two angles and one corresponding side are equal. (Also written as ASA: Angle-Side-Angle.)
Both triangles have a right angle, and the hypotenuse and one other side are equal.
Corresponding Parts and Notation
When writing congruence statements, the order of letters matters. Corresponding vertices must be listed in matching order.
If ▵ABC ≅ ▵DEF, then:
Corresponding Sides:
Corresponding Angles:
Key Vocabulary
Congruent
Having exactly the same shape and size. Congruent figures can be superimposed on each other.
Corresponding Parts
Parts (sides or angles) that are in the same position in two congruent figures.
Included Angle
The angle formed between two sides. Used in the SAS congruence test.
Congruence Test
A minimum set of measurements (SSS, SAS, AAS, RHS) sufficient to guarantee congruence.
Worked Examples
In ▵PQR and ▵XYZ: PQ = XY, QR = YZ, PR = XZ. Which test applies?
All three pairs of sides are equal.
Answer: SSS — ▵PQR ≅ ▵XYZ (SSS)
In ▵ABC and ▵DEF: AB = DE, ∠B = ∠E, BC = EF. Which test applies?
Two sides and the angle between them (angle B is between AB and BC; angle E is between DE and EF).
Answer: SAS — ▵ABC ≅ ▵DEF (SAS)
Both triangles have a right angle. They share the same hypotenuse and one leg is equal. Which test?
Right angle + equal hypotenuse + one equal side = RHS.
Answer: RHS — the triangles are congruent by RHS.
Knowledge Check
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Key Concepts Summary
- ●Congruent shapes have the same shape and size; one fits exactly on top of the other.
- ●SSS: Three sides equal.
- ●SAS: Two sides and the included angle equal.
- ●AAS: Two angles and one corresponding side equal.
- ●RHS: Right angle, equal hypotenuse and one other equal side (right-angled triangles only).
- ●The order of letters in a congruence statement shows which parts correspond.