Angle-Side-Side Does Not Work

Date: 11/12/2001 at 11:18:12
From: Dana Hess
Subject: Angle-Side-Side does not prove 2 triangles congruent?

Can you give me a construction to show that Angle-Side-Side does not 
prove two triangles congruent? I would like to show my students why it 
does not work, and for some reason I keep coming up with two congruent 
triangles when I am done. It would be great if they could even do 
this on the Geometer's Sketchpad to see that SSS, ASA, SAS, and AAS 
work, but AAA and SSA do not.  

Thank you!
Dana

Date: 11/12/2001 at 11:37:33
From: Doctor Mitteldorf
Subject: Re: Angle-Side-Side does not prove 2 triangles congruent?

Dana -

For most combinations, there is only one triangle that can be drawn 
from an angle-side-side specification. But here's an example of the 
ambiguous case:

Specify angle = 30, adjacent side = sqrt(3), opposite side = 1.  
Draw two triangles: 
 
 - one with the angles 30-30-120 and the sides 1,1,sqrt(3)
 - the other with angles 30-90-60 and sides 1,2,sqrt(3)

  

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/