Triangle congruence criteria

About this lesson

Integrated Math I › Transformations and congruence

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Triangle congruence criteria

Warm-up

Three quick questions from earlier lessons.

Discover

You know two sides of a triangle, 5 cm and 7 cm, and the 40∘40^\circ angle between them. How many different triangles can you draw?

  1. Just one (up to position)
  2. Infinitely many

Key idea

Three matching parts can be enough, if they are the right ones.

Triangles are congruent if they match by SSS, SAS (the angle between the sides), ASA or AAS. SSA (angle not between) and AAA do not guarantee congruence.

Worked example

Which criterion?

3 pairs of equal sides: SSS
All sides.
2 angles and the side between them: ASA
Side between two angles.
2 sides and a non-included angle: no conclusion
Not a criterion.

Your turn, guided

Your turn: how many of SSS, SAS, ASA, AAS, SSA, AAA are valid criteria?

44
SSA and AAA are not.

Practice ★

  1. Two triangles have two pairs of equal sides (3 cm and 5 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA
  1. Two triangles both have angles 48∘48^\circ, 60∘60^\circ and 72∘72^\circ. Are they congruent?
    1. Not necessarily: they have the same shape but may differ in size
    2. Yes, all three angles match
    3. No, never

Watch out!

Practice ★★

  1. Two triangles have two equal pairs of sides (3 and 6) and an equal angle not between them. Must they be congruent?
    1. Not necessarily: side-side-angle can give two different triangles
    2. Yes, by SSA
    3. Yes, by SAS

Exit ticket

Three questions to check you have got it.

  1. Two triangles have two pairs of equal sides (3 cm and 5 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA
  2. Two triangles have two pairs of equal sides (9 cm and 11 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA
  3. Two triangles have two pairs of equal sides (5 cm and 7 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA

Summary card

Congruence criteria

SSS, SAS, ASA, AAS. Not SSA, not AAA.

Watch out: in SAS the angle is between the sides.

End of the lesson

Back to the chapter

Answers: Triangle congruence criteria

A

  1. SAS

B

  1. Not necessarily: they have the same shape but may differ in size

C

  1. Not necessarily: side-side-angle can give two different triangles

D

  1. SAS
  2. SAS
  3. SAS

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