I have been working through this book (Plane Euclidean Geometry) and am at around chapter four. However, I was very curious about the first exercise, which asks you to prove:
i) Every triangle is congruent to itself
ii) If triangle ABC is congruent to triangle A'B'C', then A'B'C' is congruent to ABC
iii) If ABC is is congruent to A'B'C', and A'B'C' is congruent to A''B''C'', then ABC is congruent to A''B''C''.
How do you prove these? Is it enough to say for i) AB=AB, AC=AC,
angle BAC=angle BAC, triangles are congruent by SAS, or am I missing something?
These results must be proved using only the following axioms
1) It is possible to draw exactly one line through any two points
2) All straight angles (180) are equal to each other
3) If a straight line cuts two straight lines m,n so that the interior angles on one side add up to less than one straight angle, the lines m,n meet on that side
4)If X,Y lie on the same segment with AX=AY, then X=Y
Also, if A,B lie on the same side of line PQ with angle BAX=angle BAY, then A,X,Y lie on one straight line
5) Two triangles ABC and A'B'C' are congruent if AB=A'B', AC=A'C' and angle BAC=angle B'A'C'
Plane Euclidean Geometry Watch
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Last edited by j.alexanderh; 01-12-2010 at 22:53.
- 01-12-2010 22:42