![]() ![]() ![]() Recall that the altitude of a triangle is a line perpendicular to the base, passing through the opposite angle. Can you guess how?Ĭonstruct an altitude from side AK. We are about to turn those legs into hypotenuses of two right triangles. We know by definition that JA ≅ JK, because they are legs. To prove that two right triangles are congruent if their corresponding hypotenuses and one leg are congruent, we start with an isosceles triangle. Once proven, it can be used as much as you need. We have to enlist the aid of a different type of triangle. So we have to be very mathematically clever. Of course you can't, because the hypotenuse of a right triangle is always (always!) opposite the right angle. Hold on, you say, that so-called theorem only spoke about two legs, and didn't even mention an angle?Īha, 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. 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. The included angle has to be sandwiched between the sides. 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 converse of this, of course, is that if every corresponding part of two triangles are congruent, then the triangles are congruent. ![]() HL Theorem CPCTCĬPCTC reminds us that, if two triangles are congruent, then every corresponding part of one triangle is congruent to the other. 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. Notice the squares in the right angles.Įvery part of one triangle is congruent to every matching, or corresponding, part of the other triangle. Notice the hash marks for the three sides of each triangle. Notice the hash marks for the two acute interior angles. Here are two congruent, right triangles, △PAT and △JOG. It is shortened to CPCTC, which is easy to recall because you use three Cs to write it. CPCTCĬPCTC is an acronym for corresponding parts of congruent triangles are congruent. The longest side of a right triangle is called its hypotenuse. ![]()
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