Geometry HL
If you have the right three pairs of corresponding congruent parts from two triangles, then that is all you need to prove that two triangles are congruent. If the two triangles you are working with are right triangles, you already have one of those three parts. To prove that two right triangles are congruent, consider these combinations HL – Hypotenuse Leg HA – Hypotenuse Acute angle LL – Leg Leg LA – Leg Acute angle
Of these four, only HL needs to be learned. Why? HA – The Hypotenuse-Acute Angle Theorem is a rule specially designed for use with right triangles. It states if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, the two triangles are congruent. Instead of learning the HA theorem, we will continue to use AAS.
LL – The Leg-Leg Theorem is a rule specially designed for use with right triangles. It states if the legs of one right triangle are congruent to the legs of another right triangle, the two right triangles are congruent.
Instead of learning the LL theorem, we will continue to use SAS.
LA – The Leg-Acute Angle Theorem is a rule specially designed for use with right triangles. It states if a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.
Instead of learning the LA theorem, we will continue to use ASA.
Hypotenuse Leg Theorem HL – If a leg and the hypotenuse of one right triangle are congruent to a leg and the hypotenuse of a second right triangle, then the two right triangles are congruent. M
B
F
T
N
S
If BMN and FST are right triangles and BM FS and MN ST, then BMN FST because of HL.
This is the one case where SSA is valid. The angle must be a right angle, thus the triangles must be right triangles. Here’s a reason why this works. Since we know the hypotenuse and one other side, the third side can be accurately determined, due to the Pythagorean Theorem. So this is really a version of the SSS case.