Geometry: Congruence and Similarity
Congruence and Similarity is proof-and-ratio territory. You need to select a valid congruence or similarity condition, match corresponding vertices correctly, and use side, area, and parallel-line ratios without mixing them up.
Core rules
- Congruence conditions: SSS, SAS, ASA, AAS, and RHS for right triangles. In SAS, the equal angle must lie between the two pairs of equal sides.
- Similarity conditions: AA, SAS with two sides in proportion and the included angle equal, and SSS with all three side ratios equal.
- If the similarity ratio is k, corresponding lengths are in ratio k, areas in k², and volumes in k³.
- A line parallel to one side of a triangle divides the other two sides proportionally. The midpoint theorem gives a segment half the length of the third side.
Worked example
In triangle ABC, D lies on AB, E lies on AC, and DE is parallel to BC. Given AD:DB=2:3 and BC=15, find DE. Because DE is parallel to BC, triangles ADE and ABC are similar by AA: they share angle A, and corresponding angles are equal. The ratio AD:AB=2:(2+3)=2:5. Therefore DE:BC=2:5, so DE=15×2/5=6.
Traps
- Writing a similarity statement with vertices in the wrong order.
- Using k for an area ratio instead of k².
- Treating SSA as a congruence condition.
- Using AD:DB directly as the similarity ratio instead of AD:AB.
- Forgetting that a shared angle can supply one of the AA pairs.