Chapter 9: Geometric Twins
When two figures match perfectly , same shape, same size , we call them congruent. They might be in different positions or orientations, but if you could pick one up and place it on the other, every point would line up. This chapter explores when two figures are guaranteed to be congruent.
Triangles are the simplest non-trivial figure, so most of the chapter focuses on them. Surprisingly, you do not need to compare all three sides and all three angles to be sure two triangles are congruent. A small handful of pieces , three carefully chosen , is enough.
These rules , SSS, SAS, ASA, RHS , are the foundation of geometric proofs and are used by carpenters, designers and engineers every day. Once you master them, you can show that two distant figures are exact twins with just a few measurements.
What's inside
- Congruent figures , what "same shape and size" means.
- SSS criterion , three matching sides force a match.
- SAS criterion , two sides and the angle between them.
- ASA criterion , two angles and the included side.
- RHS criterion , for right triangles.
Key results
| Criterion | What you must match | Conclusion |
|---|---|---|
| SSS | All three sides | Triangles are congruent |
| SAS | Two sides + included angle | Congruent |
| ASA | Two angles + included side | Congruent |
| AAS | Two angles + a non-included side | Congruent |
| RHS | Right angle + hypotenuse + one leg | Congruent (for right triangles) |
Important non-criteria:
- AAA (all three angles) alone is not enough , it gives similar triangles, not necessarily congruent.
- SSA (two sides + a non-included angle) is ambiguous , two different triangles can have the same SSA.
How to read this chapter
Try the "test" yourself: cut out two paper triangles and label corresponding parts. If your given pieces match, you should be able to flip and slide one onto the other to coincide with the second.