Geometry Essentials
Congruency, Similarity & Key Theorems
1. Congruent Triangles ($$ \cong $$)
Same Shape & Same Size. All corresponding sides and angles are equal.
| Criteria | Description |
|---|---|
| SSS | Side-Side-Side: All 3 sides of one triangle equal to 3 sides of the other. |
| SAS | Side-Angle-Side: 2 sides and the included angle are equal. |
| ASA / AAS | Angle-Side-Angle: 2 angles and any side are equal. |
| RHS | Right-Angle Hypotenuse Side: Hypotenuse and one side of a right triangle are equal. |
Result: CPCT (Corresponding Parts of Congruent Triangles) are equal.
2. Similar Triangles ($$ \sim $$)
Same Shape, Different Size. Angles are equal, Sides are proportional.
AA (or AAA)
If two angles of one triangle are equal to two angles of another.
SAS Similarity
Ratio of 2 sides is same and included angle is equal.
SSS Similarity
Ratio of all 3 corresponding sides is equal.
Important Area Property:
$$ \frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta PQR)} = \left( \frac{AB}{PQ} \right)^2 = \left( \frac{BC}{QR} \right)^2 $$
3. Key Theorems (Tools for Class 12)
Pythagoras Theorem
In a right $$ \Delta $$:
$$ H^2 = P^2 + B^2 $$
BPT (Thales Theorem)
Line parallel to one side divides other two sides in same ratio.
$$ \frac{AD}{DB} = \frac{AE}{EC} $$
Centroid (G)
Point of intersection of Medians.
• Divides median in ratio 2:1.
• Coords: $$ (\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}) $$