SSS, SAS and ASA Theorems

key notes :

Triangle Congruence Theorems

Congruent triangles are triangles that have the same shape and size. The following theorems help determine triangle congruence:

SSS (Side-Side-Side) Theorem

  • If three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.
  • Example: If △ABC has sides 5 cm, 6 cm, and 7 cm, and △DEF also has sides 5 cm, 6 cm, and 7 cm, then △ABC ≅ △DEF.

SAS (Side-Angle-Side) Theorem

  • If two sides and the included angle (the angle between the two sides) of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent.
  • Example: If △ABC has AB = DE, BC = EF, and ∠B = ∠E, then △ABC ≅ △DEF.

ASA (Angle-Side-Angle) Theorem

  • If two angles and the included side (the side between the two angles) of one triangle are equal to the corresponding two angles and included side of another triangle, the triangles are congruent.
  • Example: If △ABC has ∠A = ∠D, ∠B = ∠E, and AB = DE, then △ABC ≅ △DEF.

Importance of These Theorems

  • These theorems help in proving triangle congruence without measuring all sides and angles.
  • They are useful in geometric constructions, proofs, and real-life applications like engineering and architecture.

Learn with an example

Which rule explains why these triangles are congruent?

  • ASA
  • SSS
  • SAS
  • These triangles cannot be proven congruent.

First, look for congruent sides and angles.

CF ≅ GH and BF ≅ BG .

Next, notice that ∠CBF and ∠GBH are vertical angles. Since vertical angles are congruent, ∠CBF≅∠GBH.

Finally, put the three congruency statements in order. BF is between CF and ∠CBF, and BG is between GH and ∠GBH in the diagram.

CF ≅ GH Side

BF ≅ BG Side

∠CBF ≅ ∠GBH Angle

In order, the congruent sides and angles form SSA. This is not one of the three ways to show that triangles are congruent. There is not enough information to prove that the triangles are congruent.

Which rule explains why these triangles are congruent?

  • SSS
  • SAS
  • ASA
  • These triangles cannot be proven congruent.

First, look for congruent sides and angles.

Notice that there are no pairs of congruent sides. Since all of the congruency theorems call for at least one pair of congruent sides, there is not enough information to prove that the triangles are congruent.

Which rule explains why these triangles are congruent?

  • ASA
  • SAS
  • SSS
  • These triangles cannot be proven congruent.

First, look for congruent sides and angles.

Notice that there are no pairs of congruent sides. Since all of the congruency theorems call for at least one pair of congruent sides, there is not enough information to prove that the triangles are congruent.

Let’s try some problems!✍️