Understanding Congruence Rules of Triangles
Two triangles are said to be congruent if they are identical in shape and size. This means that if you place one triangle over the other, they will cover each other exactly. In congruent triangles, all corresponding sides and angles are equal.
Triangle Congruence Criteria
🏗️ The 5 Major Congruence Criteria
To prove two triangles are congruent, you don't need to check all 6 parts (3 sides and 3 angles). You only need to verify one of the following five rules:
1. Side-Side-Side (SSS)
If the three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.
- Condition: Side = Side, Side = Side, Side = Side.
2. Side-Angle-Side (SAS)
If two sides and the included angle (the angle between them) of one triangle are equal to the corresponding parts of another triangle, they are congruent.
- Condition: Side = Side, Included Angle = Included Angle, Side = Side.
3. Angle-Side-Angle (ASA)
If two angles and the included side (the side between them) of one triangle are equal to the corresponding parts of another triangle, they are congruent.
- Condition: Angle = Angle, Included Side = Included Side, Angle = Angle.
4. Angle-Angle-Side (AAS)
If two angles and any one side (not necessarily included) of one triangle are equal to the corresponding parts of another triangle, they are congruent.
- Condition: Angle = Angle, Angle = Angle, Side = Side.
5. Right-Hypotenuse-Side (RHS)
This rule applies only to Right-Angled Triangles. If the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another right triangle, they are congruent.
- Condition: Right Angle (90°), Hypotenuse = Hypotenuse, One Side = One Side.
📊 Summary Table
| Rule | Full Form | Requirements |
|---|---|---|
| SSS | Side-Side-Side | 3 Corresponding Sides |
| SAS | Side-Angle-Side | 2 Sides + 1 Included Angle |
| ASA | Angle-Side-Angle | 2 Angles + 1 Included Side |
| AAS | Angle-Angle-Side | 2 Angles + 1 Non-included Side |
| RHS | Right-Hypotenuse-Side | 1 Right Angle + Hypotenuse + 1 Side |
CPCT (Corresponding Parts of Congruent Triangles): Once you prove two triangles are congruent using any of the rules above, you can conclude that all their other corresponding parts are also equal by CPCT.