Geometry
Triangle Congruence
Two triangles are congruent if they have the same size and shape. Congruence postulates: SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), AAS (2 angles + non-included side), HL (right triangles: hypotenuse + leg).
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Worked Examples
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Problem
Look at the two triangles below. All three pairs of sides are marked equal. What postulate proves they are congruent?
Step-by-Step Solution
Count the tick marks: one tick on AB and DE, two ticks on AC and DF, three ticks on BC and EF.
Three pairs of equal sides → SSS (Side-Side-Side) postulate.
SSS: if all three sides match, the triangles must be congruent.
Answer
SSS (Side-Side-Side)
Problem
The diagram shows two sides and the angle between them marked equal. What postulate applies?
Step-by-Step Solution
Two sides are marked equal (tick marks on AB=DE and AC=DF).
The angle at vertex A equals the angle at vertex D — it sits between the two marked sides.
Two sides + the included angle = SAS (Side-Angle-Side).
Answer
SAS (Side-Angle-Side)
Problem
These are right triangles. The hypotenuse and one leg are marked equal. Are they congruent?
Step-by-Step Solution
Both triangles have a right angle (shown by the square corner).
The hypotenuse (longest side, opposite the right angle) is marked equal.
One leg is also marked equal.
Right triangle + equal hypotenuse + equal leg → HL (Hypotenuse-Leg) postulate.
Answer
Yes — congruent by HL (Hypotenuse-Leg)
Problem
Two angles and the side between them are marked equal. What postulate proves congruence?
Step-by-Step Solution
Angle at A = Angle at D (red arcs), Angle at B = Angle at E (red arcs).
The side AB = DE is between those two angles (tick marks).
Two angles + the included side = ASA (Angle-Side-Angle).
Answer
ASA (Angle-Side-Angle)
Problem
Only two sides are marked equal on these triangles. Is that enough to prove congruence?
Step-by-Step Solution
Two sides are marked equal — but the triangles look different.
SS (two sides only) is NOT a congruence postulate.
You need a third piece: a third side (SSS), an included angle (SAS), or a right angle + hypotenuse (HL).
Answer
No — SS alone is not sufficient to prove congruence
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
All three pairs of sides are marked equal. What postulate proves congruence?
Two sides and the included angle are marked equal. What postulate?
Two angles and the included side are marked equal. What postulate?
Two angles and a non-included side are marked equal. What postulate?
Right triangles with equal hypotenuse and one equal leg. What postulate?
Only two sides are marked equal. Can you prove congruence?
All three angles are marked equal but the triangles are different sizes. Are they congruent?
Two sides and a non-included angle are marked equal (SSA). Is this a valid postulate?
Two sides (7 cm, 5 cm) and the included 70° angle are equal. What postulate?
Three sides (3 cm, 4 cm, 5 cm) are all equal. What postulate?
These triangles are flipped but all three sides are marked equal. Are they congruent?
These triangles have the same shape but different sizes. Are they congruent or similar?
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