# Why are the Side-Angle-Side and Hypotenuse-Leg similarity rules for triangles true?

My math book tells me that two triangles are similar if all angles are equal to each other ("Angle-Angle-Angle (AAA) Similarity"). There are 4 rules summed up for similarity, I summarised them:

1. Angle-Angle (AA) Similarity. 2 pairs of angles are equal, so the third pair is equal too.

2. Side-Side-Side (SSS) Similarity. The ratio between all corresponding sides is constant.

And now the last two, of which I want to know why these rules are true. The two above are obvious for me, but these ones not.

1. Side-Angle-Side (SAS) Similarity. One angle is the same in both triangles, and the ratio between the sides around the angle is the same.

2. Hypotenuse-Leg (HL) Similarity. In a right triangle, the ratio between the hypotenuses and between two other are equal.

Can anyone explain 3 and 4?

Lastly, if you have 2 rectangular triangles, and the ratio between the sides around the right angle is the same, they must be similar, correct? Because the third rule applies on this.

• Did you mean right angular triangles ? – The Demonix _ Hermit Oct 12 '19 at 16:48
• For 4, consider Pythagorean theorem. For 3, Thales. – Jean-Claude Arbaut Oct 12 '19 at 16:49
• Yep, with one angle of 90 degrees. – mathomato Oct 12 '19 at 16:50
• @TheDemonix_Hermit Rectangular is correct, and idiomatically better, I think. – Allawonder Oct 12 '19 at 16:50
• @Allawonder en.wikipedia.org/wiki/Right_triangle – Jean-Claude Arbaut Oct 12 '19 at 16:53