Geometry and measures - Properties and constructions: use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)

The idea

Two triangles are congruent if they are identical in size and shape. To prove this, you need to show that one of four minimum conditions is met, rather than all six corresponding parts being equal. The criteria are: * **SSS** (Side-Side-Side): Three pairs of corresponding sides are equal. * **SAS** (Side-Angle-Side): Two pairs of corresponding sides and the *included* angle (the angle between them) are equal. * **ASA** (Angle-Side-Angle): Two pairs of corresponding angles and the *included* side (the side between them) are equal. * **RHS** (Right-angle-Hypotenuse-Side): Both triangles have a right angle, their hypotenuses are equal, and one other pair of corresponding sides are equal. A proof involves listing three matching facts with reasons (e.g., 'given', 'common side'), then stating the criterion.

What earns the marks

• Stating three pairs of equal corresponding parts (sides or angles) with a valid reason for each.
• Correctly identifying the congruence criterion (SSS, SAS, ASA, or RHS) that applies.
• Writing a concluding statement of congruence with vertices in the correct corresponding order, e.g. ΔABC ≅ ΔXYZ.
• Using the established congruence to deduce that another pair of sides or angles must be equal.

Common mistakes and how to put them right

• Using SAS when the angle is not between the two sides.

• Confusing ASA (included side) with AAS (a non-included side).

  Put it right: While AAS also proves congruence, you must use the correct label for the facts you have established.

• Assuming AAA proves congruence.

  Put it right: It only proves similarity (same shape, different size).

Exam tips

• Carefully mark equal sides and angles on the diagram as you identify them. This helps you see which criterion to use.
• Look for 'hidden' equal parts: a common side shared by both triangles, or vertically opposite angles where two lines cross.