13. Congruence of Triangles

Explanation

Two triangles are congruent if they are exactly the same size and shape. This means all their corresponding sides and angles are equal. There are four rules to prove that two triangles are congruent without checking all sides and angles: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and RHS (Right angle-Hypotenuse-Side).

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Problems Example

1. Side-Side-Side (SSS)

Two triangles are congruent if all three sides of one triangle are equal to the corresponding sides of the other triangle.

Example:
If in ΔABC and ΔDEF:

AB = DE
BC = EF
CA = FD
Then ΔABC ≅ ΔDEF

2. Side-Angle-Side (SAS)

Two triangles are congruent if two sides and the angle between them (included angle) in one triangle are equal to the corresponding parts of the other triangle.

Example:
If in ΔPQR and ΔXYZ:

PQ = XY
∠PQR = ∠XYZ
QR = YZ
Then ΔPQR ≅ ΔXYZ

3. Angle-Side-Angle (ASA)

Two triangles are congruent if two angles and the side between them (included side) in one triangle are equal to the corresponding parts of the other triangle.

Example:
If in ΔLMN and ΔPQR:

∠LMN = ∠PQR
MN = QR
∠MNL = ∠QRP
Then ΔLMN ≅ ΔPQR

4. Angle-Angle-Side (AAS)

Two triangles are congruent if two angles and a non-included side in one triangle are equal to the corresponding parts of the other triangle.

Example:
If in ΔABC and ΔXYZ:

∠A = ∠X
∠B = ∠Y
BC = YZ
Then ΔABC ≅ ΔXYZ

5. Right-Angle-Hypotenuse-Side (RHS)

Two right-angled triangles are congruent if the hypotenuse and one other side in one triangle are equal to the corresponding parts of the other triangle.

Example:
If in ΔABC and ΔDEF (both right-angled triangles):

Hypotenuse AC = DF
AB = DE
Then ΔABC ≅ ΔDEF