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October 29th, 2015, 02:25 PM   #1
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Triangles With Two Sides and One Angle In Common and Whether They Are Congruent

I learned that side-angle-side can prove triangles congruent and I have memorized the drawing of two triangles with angle-side-side in common that are not congruent. Besides providing a counterexample showing that angle-side-side doesn't work, is there an explanation of why two sides and one angle works only when the angle is in between the sides?

Last edited by skipjack; October 30th, 2015 at 01:42 AM.
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October 29th, 2015, 04:33 PM   #2
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Why can't be answered except by saying it can be proven.
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October 30th, 2015, 03:05 AM   #3
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Originally Posted by EvanJ View Post
. . . is there an explanation of why two sides and one angle works only when the angle is in between the sides?
It can't be proved, as it's not true that two sides and one angle works only when the angle is between the sides. For example, it works if the angle is 30°, the side opposite that angle has length 1 and the second given side has length 2 or 1 or 1/2 (or certain other values). It always works if the given angle is 90° (as Pythagoras gives the third side) or greater than 90°.
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October 30th, 2015, 03:11 PM   #4
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It can't be proved, as it's not true that two sides and one angle works only when the angle is between the sides. For example, it works if the angle is 30°, the side opposite that angle has length 1 and the second given side has length 2 or 1 or 1/2 (or certain other values). It always works if the given angle is 90° (as Pythagoras gives the third side) or greater than 90°.
Okay, but it is true that if the angle is between the sides the triangles have to be congruent and if the angle is not in between the sides the triangles may or may not be congruent. I learned about one side and one angle of a right triangle, but I never learned the greater than 90° part.
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October 30th, 2015, 04:21 PM   #5
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What you're referring to is probably one side and one angle in addition to the right angle, which is a particular case of one side and two angles. I was referring to two sides, one of which is the hypotenuse, and just the right angle. Another case that always works is when the two given sides equal each other, as that means that the triangles are isosceles, which allows the remaining angles to be calculated easily.
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