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. Show 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°). .
The longest side of a right triangle is called its hypotenuse CPCTCCPCTC 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:
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. SAS PostulateRecall 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. HL TheoremThe Hypotenuse Leg Theorem, or HL Theorem, tells us a suspiciously similar story:
The 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 TheoremWe 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:
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 ProofYou 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: MP ≅ RA 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.
That also means, thanks to CPCTC, the two as-yet-unidentified interior angles of one right triangle are congruent to the corresponding interior angles of the other triangle. Lesson SummaryAfter 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. Next Lesson:Perpendicular Bisector Theorem
In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation. In today’s geometry lesson, you’re going to learn how to use the Hypotenuse Leg Theorem. Up until now, we’ve have learned four out of five congruency postulates for triangles: Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Now it’s time to look at the final postulate for congruent triangles: Hypotenuse-Leg (HL). What is the HL Postulate?The HL Postulate states that if the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent. Hypotenuse Theorem Example Using the image above, if segment AB is congruent to segment FE and segment BC is congruent to segment ED, then triangle CAB is congruent to triangle DFE. Now, at first glance, it looks like we are going against our cardinal rule of not allowing side-side-angle…which spells the “bad word” (i.e., the reverse of SSA). But thanks to the Pythagorean Theorem, and our ability to find the measure of the third angle, we can conclude that for right triangles only, this type of congruence is acceptable. In other words, with right triangles we change our congruency statement to reflect that one of our congruent sides is indeed the hypotenuse of the triangle. Phew! No bad words here! And with the last piece of the congruency puzzle finally unearthed we are going to combine our knowledge of triangle congruence with our understanding of both Isosceles Triangles and Equilateral Triangles. Why? Because right triangles are not the only types of triangles that have special properties – Isosceles and Equilateral Triangles are both pretty special. And right triangles, isosceles triangles, and equilateral triangles can work together to prove congruence and help us solve for missing sides and angles of triangles. Now if we remember from when we learned to classify triangles, a triangle is isosceles if two sides of a triangle are congruent. And just as we have two equal legs, an isosceles triangle has two equal legs (sides), as Math is Fun nicely points out. Well, if a triangle has exactly two congruent sides, then the base angles are congruent. The base angles are the two angles formed between the legs of the triangle and the non-congruent side. And more importantly, these base angles are congruent. Isosceles Triangle TheoremThe Isosceles Triangle Theorem, sometimes called the Base Angle Theorem, states that if two sides of a triangle are congruent, then the angles opposite them are also congruent. Equilateral Triangle TheoremMoreover, the Equilateral Triangle Theorem states if a triangle is equilateral (i.e., all sides are equal) then it is also equiangular (i.e., all angles are equal). And if a triangle is equiangular, then it is also equilateral. Markedly, the measure of each angle in an equilateral triangle is 60 degrees. Armed with our new information and knowledge, we are are going to:
The HL Theorem – Lesson & Examples (Video)37 min
Get access to all the courses and over 450 HD videos with your subscription Monthly and Yearly Plans Available Get My Subscription Now Still wondering if CalcWorkshop is right for you? |
What does HL mean in triangles?
Postagens relacionadas
direito autoral © 2024 mesadeestudo Inc.
