HL Theorem (Hypotenuse Leg)
Hypotenuse leg (HL) congruence theorem
This lesson will introduce a very long phrase abbreviated CPCTC. It's easy to remember because every other letter is "C," you see? The Hypotenuse Leg or HL Theorem, is not as funny as the Hypotenuse Angle or HA Theorem, but it is useful.
This theorem is really a derivation of the Side Angle Side Postulate, just as the HA Theorem is a derivation of the Angle Side Angle Postulate.
What are right triangles?
Right triangles have exactly one interior angle measuring 90°, and the other two interior angles are acute (because they can only add up to 90°).
CPCTC is an acronym for corresponding parts of congruent triangles are congruent. It is shortened to CPCTC, which is easy to recall because you use three Cs to write it.
Here are two congruent, right triangles, △PAT and △JOG. Notice the hash marks for the two acute interior angles. Notice the hash marks for the three sides of each triangle. Notice the squares in the right angles.
Every part of one triangle is congruent to every matching, or corresponding, part of the other triangle. Usually you need only three (or sometimes just two!) parts to be congruent to prove that the triangles are congruent, which saves you a lot of time.
Here are all the congruences:
∠P ≅ ∠J
∠A ≅ ∠O
∠T ≅ ∠G
Side PA ≅ JO
Side AT ≅ OG
Side TP ≅ GJ
CPCTC reminds us that, if two triangles are congruent, then every corresponding part of one triangle is congruent to the other.
The converse of this, of course, is that if every corresponding part of two triangles are congruent, then the triangles are congruent. The HL Theorem helps you prove that.
Recall the SAS Postulate used to prove congruence of two triangles if you know congruent sides, an included congruent angle, and another congruent pair of sides. The included angle has to be sandwiched between the sides.
The Hypotenuse Leg Theorem, or HL Theorem, states; If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
Hold on, you say, that so-called theorem only spoke about two legs, and didn't even mention an angle?
Aha, have you forgotten about our given right angle? Every right triangle has one, and if we can somehow manage to squeeze that right angle between the hypotenuse and another leg...
Of course you can't, because the hypotenuse of a right triangle is always (always!) opposite the right angle. So we have to be very mathematically clever. We have to enlist the aid of a different type of triangle.
Proving the HL theorem
We must first prove the HL Theorem. Once proven, it can be used as much as you need. To prove that two right triangles are congruent if their corresponding hypotenuses and one leg are congruent, we start with an isosceles triangle.
Here we have isosceles △JAK. We know by definition that JA ≅ JK, because they are legs. We are about to turn those legs into hypotenuses of two right triangles. Can you guess how?
Construct an altitude from side AK. Recall that the altitude of a triangle is a line perpendicular to the base, passing through the opposite angle. Label its point on AK as Point C.
That altitude, JC, complies with the Isosceles Triangle Theorem, which makes the perpendicular bisector of the base the angle bisector of the vertex angle. We have two right angles at Point C, ∠JCA and ∠JCK. We have two right triangles, △JAC and △JCK, sharing side JC.
We know by the reflexive property that side JC ≅ JC (it is used in both triangles), and we know that the two hypotenuses, which began our proof as equal-length legs of an isosceles triangle, are congruent. So, we have one leg and a hypotenuse of △JAC congruent to the corresponding leg and hypotenuse of △JCK.
Now verify that AC ≅ CK and all the interior angles are congruent:
AC ≅ CK (the altitude of the base of an isosceles triangle bisects the base, since it is by definition the perpendicular bisector)
∠JCA ≅ ∠JCK (they are both right angles)
∠A ≅ ∠K (they were angles opposite to the legs in accordance with the Isosceles Triangle Theorem)
∠AJC ≅ ∠CJK (side JC was the angle bisector of original ∠AJK)
So, all three interior angles of each right triangle are congruent, and all sides are congruent. CPCTC! How about that, JACK?
We originally used the isosceles triangle to find the hypotenuse and a single leg congruent, and from that, we built proof that both triangles are congruent.
So, we have proven the HL Theorem, and can use it confidently now!
HL theorem practice proof
You have two suspicious-looking triangles, △MOP and △RAG.
You get out your mathematical detective's magnifying glass and notice that ∠O and ∠G are marked with the tell-tale little squares, □, indicating right angles.
Aha! These are two right triangles, because by definition a right triangle has one right angle.
You also notice, masterful detective that you are, the sides opposite the right angles are congruent:
Finally, you zero in on the little hash marks on sides OP and AG, which indicate they are congruent, too. So you have two right triangles, with congruent hypotenuses, and one congruent side.
You can whip out the ol' HL Theorem and state without fear of contradiction that these two right triangles are congruent.
After working your way through this lesson, you are now able to recall and state the Hypotenuse Leg (HL) Theorem of congruent right triangles, use the HL Theorem to prove congruence in right triangles, and recall what CPCTC means (corresponding parts of congruent triangles are congruent), using as needed.