This is a core topic in Geometry. Proving that two triangles are congruent (identical in shape and size) allows you to deduce that all their corresponding sides and angles are equal. This is a powerful tool for solving complex geometry problems.

1. Core Concepts (Short Notes)

• Congruent Figures: Two plane figures are congruent if one can be placed exactly on top of the other so that they coincide perfectly. They are identical in shape and size. The symbol for congruence is ≡.

• Elements of a Triangle: Every triangle has six elements: 3 sides and 3 angles.

• The Goal: To prove that ΔABC ≡ ΔPQR, you do not need to show all six elements are equal. You only need to prove three specific elements are equal, according to one of the four conditions below.

• Corresponding Parts: Once two triangles are proven congruent, their "corresponding" parts (angles and sides that match up) are equal. This is often abbreviated as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

2. The Four Conditions for Congruence

You must memorize these four cases. In an exam, you must state the case you used as the reason.

1. SAS (Side-Angle-Side)

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

• Included Angle: The angle between the two sides.

• Example: If AB = PQ, AC = PR, and the angle between them ∠BAC = ∠QPR, then ΔABC ≡ ΔPQR (SAS).

2. AAS (Angle-Angle-Side)

If two angles and a corresponding side of one triangle are equal to two angles and a corresponding side of another triangle, then the triangles are congruent.

• Corresponding Side: The side opposite an equal angle.

• Example: If ∠ABC = ∠PQR, ∠BCA = ∠QRP, and the side opposite ∠BCA (which is AB) is equal to the side opposite ∠QRP (which is PQ), then ΔABC ≡ ΔPQR (AAS).

3. SSS (Side-Side-Side)

If the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

• Example: If AB = PQ, BC = QR, and AC = PR, then ΔABC ≡ ΔPQR (SSS).

4. RHS (Right Angle-Hypotenuse-Side)

This case is ONLY for right-angled triangles. If the hypotenuse and one other side of a right-angled triangle are equal to the hypotenuse and one other side of another right-angled triangle, then the triangles are congruent.

• Hypotenuse: The side opposite the right angle.

• Example: If ∠ABC = ∠PQR = 90°, hypotenuse AC = PR, and side AB = PQ, then ΔABC ≡ ΔPQR (RHS).

3. Tips & Tricks for Exams

• Mark Your Diagram: As you read a problem, mark the given equal sides (with dashes |, ||, |||) and equal angles (with arcs ◡, ◡◡). This helps you visually identify which of the four conditions might apply.

• Look for "Hidden" Information:

  1. Common Side: If two triangles share a side, that side is equal in both. (Reason: "Common side").

  2. Vertically Opposite Angles: If two lines intersect, the angles opposite each other at the intersection are equal. (Reason: "Vertically opposite angles").

  3. Parallel Lines: Look for Z-angles (alternate angles) or F-angles (corresponding angles) which will be equal.

• Structure Your Proof: A formal proof is usually required. Lay it out clearly.

  1. State the two triangles you are considering: "In ΔABC and ΔPQR,"

  2. List the three pairs of equal elements, giving a reason for each in brackets.

     • AB = PQ (Given)

     • ∠ABC = ∠PQR (Vertically opposite angles)

     • BC = QR (Common side)

  3. State the conclusion and the condition used.

     • ∴ ΔABC ≡ ΔPQR (SAS)

• Use CPCTC for the Next Step: Often, a question will first ask you to prove two triangles are congruent, and then ask you to prove something else (e.g., that two other sides are equal). The second part of the proof almost always uses the fact that corresponding parts of the now-proven congruent triangles are equal.

4. Important Points to Remember

• ASS is NOT a condition: Proving Angle-Side-Side (where the angle is not included between the sides) is not a valid condition for congruence.

• AAA is NOT a condition: Proving Angle-Angle-Angle only proves that the triangles are similar (same shape), not necessarily congruent (same size).

• The order of vertices matters when writing the congruence statement. ΔABC ≡ ΔPQR implies that A corresponds to P, B to Q, and C to R. This tells you exactly which sides and angles are equal.