Absolute geometry
Congruent triangles - SAS, ASA, SSS
Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 3.3, pp 139-150. The problems are all from section 3.3.
Proving SSS from SAS (fill in and submit as GR5 – part 2)
SSS Theorem
If, under some correspondence between their vertices, two triangles have the three sides of one congruent to the corresponding three sides of the other, then the triangles are congruent under that correspondence. [Kay, p 141]
Picture / Conclusions / Justifications/ Suppose you have two triangles which satisfy the hypothesis of SSS: three sides of one congruent to the corresponding three sides of the other. / Hypothesis (Given)
The goal is to show that having SSS (thing you want to prove) always leads to having SAS (the postulate); i.e. having the corresponding parts congruent that are marked below:
always leads to having the additional marked angles congruent:
which then allows us to conclude the triangles are congruent by the SAS Postulate (which we assume as the basis for this geometry). This establishes SSS as a theorem; if we start with theSSS hypothesis (“two triangles have the three sides of one congruent to the corresponding three sides of the other”), then the conclusion is the triangles must be congruent.
The trick in this one is to construct a copy onto , and show that the copy is congruent to . This is a slightly simplified version - you can use the betweenness relations apparent in the figures. Also assume that the figures are oriented so that the angles at the base are acute (i.e., if this were an obtuse triangle, I'd orient it with the obtuse angle at the top).
Picture / Conclusions / Justifications/
/ Hypothesis (Given)
/ Copy and onto segment, as shown with the vertices at and . / [Fill in 1]
/ / [Fill in 2]
/ This givesand therefore , and also [Fill in 3] / [Fill in 4]
/ (At this point you can ignore for a while.) Construct segment. / [Fill in 5]
/ What can you conclude about (what type of triangle is it and why)? [Fill in 6] / [Fill in 7]
/ Therefore, what do you know about and ?
[Fill in 8] / [Fill in 9]
/ Make similar statements about (you can condense the two steps into one here).
[Fill in 10] / [Fill in 8] and [Fill in 9], as above
/ Assuming betweeness as apparent…
[Fill in 11] / (you could throw in a couple extra steps and the Crossbar theorem, but just go straight to ...)
Angle addition postulate
/ Make a statement about angles and :
[Fill in 12] / [Fill in 13]
/ And conclude with the triangles
[Fill in 14] / [Fill in 15]
Which now gets us the SSS Theorem; if we assume SSS, we eventually get to SAS, so SSS is sufficient to show triangles congruent. Notice we also used the previously established ASA theorem along the way.