I have been working by myself out of a math book. I love the book, but recently I have run across a very annoying sentence: "Though the HL Postulate can be proved, we shall treat it as a Postulate." Can anyone explain to me why this is the case?
Also, it appears to me that the HL Postulate is just a variation of the SAS Postulate where the corresponding congruent angle is a right angle. Why do you need an HL Postulate when you could use the SAS Postulate?
Also, it appears to me that the HL Postulate is just a variation of the SAS Postulate where the corresponding congruent angle is a right angle. Why do you need an HL Postulate when you could use the SAS Postulate?
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Very good question. Reading geometry with an eye for getting rid of useless postulates is exactly the right sort of thinking to get you through the course. A few points:
(1) HL isn't quite a version of SAS, since the right angle in a right triangle doesn't lie between the hypotenuse and a leg. Rather, HL is a version of ***, which isn't in general true.
(2) If you'll admit an algebraic proof of HL, it's rather easy to prove. The pythagorean theorem lets us find the length of the missing leg from the given hypotenuse and leg. So if the hypotenuse and leg of two right triangles are congruent, so are the remaining legs. So HL is true by SSS congruence.
(3) There are geometric proofs of HL floating out there on the internet, but they're long and nasty. Here's an example: http://www.youtube.com/watch?feature=pla… (disclaimer; I didn't actually wade through the proof. What can I say, I'm lazy. But I presume it works.)
I hope this helps!
(1) HL isn't quite a version of SAS, since the right angle in a right triangle doesn't lie between the hypotenuse and a leg. Rather, HL is a version of ***, which isn't in general true.
(2) If you'll admit an algebraic proof of HL, it's rather easy to prove. The pythagorean theorem lets us find the length of the missing leg from the given hypotenuse and leg. So if the hypotenuse and leg of two right triangles are congruent, so are the remaining legs. So HL is true by SSS congruence.
(3) There are geometric proofs of HL floating out there on the internet, but they're long and nasty. Here's an example: http://www.youtube.com/watch?feature=pla… (disclaimer; I didn't actually wade through the proof. What can I say, I'm lazy. But I presume it works.)
I hope this helps!