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Geometry

Triangle Congruence

Lesson

Two triangles are congruentif all corresponding sides and angles are equal. You don’t need to check all six — there are five shortcuts that guarantee it.

The five postulates

  • SSS — three sides equal.
  • SAS — two sides and the included angle (the angle BETWEEN them) equal.
  • ASA — two angles and the included side (the side BETWEEN them) equal.
  • AAS — two angles and a non-included side equal.
  • HL (right triangles only) — hypotenuse and a leg equal.

Not enough

  • AAA — three angles equal proves SIMILARITY, not congruence (the triangles could be different sizes).
  • SSA(the “donkey theorem”) — two sides and a non-included angle is not enough to guarantee congruence in general.

How to type your answer

Type the code: 1 SSS, 2 SAS, 3 ASA, 4 AAS, 5 HL, 0 if the information is insufficient (AAA or SSA).

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

All three sides equal in pairs\text{All three sides equal in pairs}

Problem 2

Two sides and the included angle equal\text{Two sides and the included angle equal}

Problem 3

Two angles and the included side equal\text{Two angles and the included side equal}

Problem 4

Two angles and a non-included side equal\text{Two angles and a non-included side equal}

Practice

Standard problems matching the lesson.

Problem 5

Right triangles: hypotenuse and one leg equal\text{Right triangles: hypotenuse and one leg equal}

Problem 6

All three angles equal (no side info)\text{All three angles equal (no side info)}

Problem 7

Two sides and a non-included angle equal\text{Two sides and a non-included angle equal}

Problem 8

△ABC≅△DEF with AB=DE,BC=EF,AC=DF\triangle ABC \cong \triangle DEF \text{ with } AB=DE, BC=EF, AC=DF

Problem 9

AB=DE,∠A=∠D,AC=DFAB=DE, \angle A = \angle D, AC=DF

Problem 10

∠A=∠D,AB=DE,∠B=∠E\angle A = \angle D, AB=DE, \angle B = \angle E

Problem 11

∠A=∠D,∠B=∠E,BC=EF\angle A = \angle D, \angle B = \angle E, BC=EF

Problem 12

Right triangles share hypotenuse and one leg\text{Right triangles share hypotenuse and one leg}

Problem 13

Two sides equal but angle is not between them\text{Two sides equal but angle is not between them}

Problem 14

All three pairs of corresponding sides equal\text{All three pairs of corresponding sides equal}

Problem 15

Triangle: ∠A=70∘,∠B=50∘,AB=5. Same in other.\text{Triangle: } \angle A = 70^\circ, \angle B = 50^\circ, AB = 5. \text{ Same in other.}

Problem 16

AB=AC and DE=DF,AB=DE,∠A=∠DAB=AC \text{ and } DE=DF, AB=DE, \angle A = \angle D

Problem 17

Right triangles with ∠A=∠D=90∘,AB=DE,BC=EF\text{Right triangles with } \angle A = \angle D = 90^\circ, AB=DE, BC=EF

Problem 18

AB=DE,∠A=∠D,BC=EF (BC not adjacent to ∠A)AB=DE, \angle A = \angle D, BC=EF \text{ (BC not adjacent to } \angle A\text{)}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Only three pairs of equal corresponding angles\text{Only three pairs of equal corresponding angles}

Problem 20

Two pairs of sides equal, no angle info\text{Two pairs of sides equal, no angle info}

Problem 21

Right triangles with both legs equal in pairs\text{Right triangles with both legs equal in pairs}

Problem 22

∠A=∠D,∠B=∠E, side BC (opp ∠A) = EF (opp ∠D)\angle A = \angle D, \angle B = \angle E, \text{ side BC (opp } \angle A\text{) = EF (opp } \angle D\text{)}

Problem 23

Three corresponding sides equal in two triangles\text{Three corresponding sides equal in two triangles}

Problem 24

Right triangles with ∠A=∠D=90∘,AB=DE(legs),BC=EF(hyp)\text{Right triangles with } \angle A = \angle D = 90^\circ, AB=DE \text{(legs)}, BC=EF \text{(hyp)}

Problem 25

AB=DE,∠B=∠E,BC=EFAB=DE, \angle B = \angle E, BC=EF

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