# 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

## 3 Answers

3 and 4 are just SAS and HL except that the corresponding sides need to be proportional instead of congruent. You've already accepted the similarity version of SSS, and AAS and ASA are not on the list because they are covered by AA.

For (3), you could use the sine rule for the two triangles to show that the other two angles are equal, respectively, as well. Thus, they are similar.

As you pointed out, (4) comes from (3).

To see why (3) is true, note that two sides and the angle between them determine a triangle up to congruence. Thus, if in two triangles the angles are equal and the sides around the equal angle severally bear the same ratio to each other, then it means that you have scaled up one triangle by some constant factor to get the other. Since these three elements determine a triangle, it follows that the third sides must also bear the same ratio to each other, and thus be similar.

As for (4), this is just a special case of (3) where the included angle is right.