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October 29th, 2015, 01:25 PM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 637 Thanks: 85  Triangles With Two Sides and One Angle In Common and Whether They Are Congruent
I learned that sideangleside can prove triangles congruent and I have memorized the drawing of two triangles with anglesideside in common that are not congruent. Besides providing a counterexample showing that anglesideside 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 12:42 AM. 
October 29th, 2015, 03:33 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,710 Thanks: 675 
Why can't be answered except by saying it can be proven.

October 30th, 2015, 02:05 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,388 Thanks: 2015  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°.

October 30th, 2015, 02:11 PM  #4  
Senior Member Joined: Oct 2013 From: New York, USA Posts: 637 Thanks: 85  Quote:
 
October 30th, 2015, 03:21 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,388 Thanks: 2015 
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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angle, common, congruent, sides, triangles 
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