How do you determine if two triangles are congruent or similar?

Triangle Congruence

Two triangles are congruent if all their sides and angles are exactly the same. You can prove congruence using these shortcuts:

• SSS (Side-Side-Side): All three sides of one triangle equal the three sides of the other.
• SAS (Side-Angle-Side): Two sides and the angle between them are equal.
• ASA (Angle-Side-Angle): Two angles and the side between them are equal.
• AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
• RHS (Right angle-Hypotenuse-Side): For right triangles, the hypotenuse and one other side are equal.

Example:
If ∆ABC and ∆DEF both have sides 4, 5, 6 units long, then by SSS, they are congruent.

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Triangle Similarity

Triangles are similar if their corresponding angles are equal, and their sides are in proportion. Ways to prove similarity:

• AA (Angle-Angle): If two angles are the same, so is the third; triangles are similar.
• SSS~ (Side-Side-Side in proportion): All sides have the same ratio.
• SAS~ (Side-Angle-Side, where sides are proportional and the angle is equal).

Example:
If in triangles ∆PQR and ∆XYZ, angle P = angle X, angle Q = angle Y, then by AA they are similar.

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Key Takeaways

• Congruent: triangles are exactly the same size and shape; proven by SSS, SAS, ASA, AAS, RHS.
• Similar: triangles have the same shape but possibly different sizes; proven by AA, SSS~, SAS~.