Proofs about similar figures
Congruence proofs end with equal sides. How do you prove that sides are in proportion instead?
Where you are 4 inside
What this area is about
A congruence proof ends with equal sides. Many results in geometry are about sides that are not equal but in proportion, and those are proved by showing two triangles similar and then reading a proportion off the similarity.
How these fit together
The method. Proving two triangles similar names a correspondence, finds the pairs, and cites AA, SAS or SSS similarity.
Theorems it proves. The triangle proportionality theorem shows that in \triangle ABC a line parallel to \overline{BC}, meeting \overline{AB} at D and \overline{AC} at E strictly inside those sides, cuts them in the same ratio, \displaystyle \frac{AD}{DB} = \frac{AE}{EC}, and that a cut in the same ratio at such points is parallel. The Pythagorean theorem with similar triangles drops the altitude to the hypotenuse, which cuts the right triangle into two smaller similar copies, turns two proportions from them into products, and adds those to get a^2 + b^2 = c^2.
A proof by area. The rearrangement proof reaches the same a^2 + b^2 = c^2 with no similarity at all, by filling one square two ways with four copies of a right triangle.
A path through it
Start with proving triangles similar: the other two similarity proofs are built on it. Then take the triangle proportionality theorem and the Pythagorean proof in either order. The rearrangement proof needs nothing from the other three, so it can come at any point.
Finish here and you can prove length facts from angle facts. The same move proves the power of a point relationships for chords, secants and tangents of a circle.