1. Introduction: When Are Two Triangles "the Same Shape"?
Look at two photos of the same building — one small, one blown up big. The big one is not a different building. It is the same building, just scaled up. Every corner sits at the same slant. Every wall keeps the same proportion to the next.
Triangles can do this too. A small triangle and a big triangle can be the same shape even when they are not the same size. We call such triangles similar.
Here are two triangles. The second is just the first, enlarged.
- Triangle 1: angles 50°, 60°, 70°. Shortest side 3 cm.
- Triangle 2: angles 50°, 60°, 70°. Shortest side 6 cm.
Look at them. What stayed the same and what changed?
Stop scrolling. Try it in your head before reading on.
The angles stayed exactly the same — 50°, 60°, 70° in both. The sides got bigger, but each side of Triangle 2 is exactly twice the matching side of Triangle 1. So the slant is the same; only the scale changed.
That is the whole idea of similarity. Two triangles are similar when:
- their corresponding angles are equal, and
- their corresponding sides are in the same ratio (proportional).
We write it as ΔABC ~ ΔDEF. The squiggle "~" means "is similar to". The order of letters matters: A matches D, B matches E, C matches F.
But here is the real question of this lesson. To check if two triangles are similar, do you have to measure all three angles AND all three sides — six things in total? That is a lot of measuring. The answer, happily, is no. You only need to check a few things. Those shortcuts are called the criteria for similarity.
You can now say what "similar triangles" means and why we want shortcuts to test for it.