This applet demonstrates the principles behind side-angle-side congruency (SAS) and side-side-side congruency (SSS), and proves that knowing three angles (AAA) or two sides and an unincluded angle (SSA) are not sufficient to prove congruence.
Instructions for creating the applet:
[b]SAS[/b]
1. Hide the axes.
2. Using the input bar, enter “AC = 1,” “AD=1,” and “CAD=45.”
3. Click the radio next to these variables in the algebra pane to show their sliders.
4. Adjust the preferences for these sliders. Let 0<AC<5, 0<AD<1 (both with increments of 0.01) and 0<CAD<360 (increments of .1).
5. Create a point A.
6. Create a line through A passing through a second point, B.
7. Create a circle with center at A and radius AD.
8. Create a point D where this circle intersects the line AB.
9. Hide the line and circle (or make them dashed) and create a solid segment from A to D. Leave B visible.
10. Create an angle of size CAD counterclockwise from AD.
11. Create a line passing through AD’.
12. Create a circle with center at A and radius AC.
13. Hide or fade the circle and line AD’ and the point D’.
14. Create a solid segment from A to C.
15. Create a dashed segment from C to D (since we haven’t directly defined this segment).
[b]AAA[/b]
16. In the input bar, enter GEH=75, HGE=70.
17. Click the radio next to these variables in the algebra pane to show their sliders.
18. Adjust the preferences for these sliders so they can vary between 0 and 360 degrees.
19. Construct points E and F and a line through them.
20. Select the “Angle with Given Size” tool. Click on F, the E, then enter GEH in front of the degree sign. A new point, F’, will appear.
21. Construct a line through E and F’.
22. Construct a point G on the line EF’.
23. Hide the line EF’ and construct the segment EG.
24. Select the “Angle with Given Size” tool again. Select E, then G, then enter HGE in front of the degree symbol. A new point, E’, will appear.
25. Construct the line GE’ and create the point H where GE’ intersects FE.
26. Hide GE’ and E’ and construct the segment GH.
[b]SSS[/b]
27. In the input bar, enter IL =1.5, KL=1.6, and IK=2.3.
28. Click the radio next to these variables in the algebra pane to show their sliders.
29. Adjust the preferences for these sliders so they can vary between 0 and 5.
30. Construct the points I and J and the line IJ.
31. Construct a circle with center at I and radius IK.
32. Construct the point K where the circle intersects IJ.
33. Construct a circle with center at K and radius KL.
34. Construct a circle with center at I and radius IL.
35. Place the point L at the intersection of these circles.
36. Hide the circles and line IJ. Construct segments IK, KL, and IL.
[b]SSA[/b]
37. In the input bar, enter NQ =1.5, NO=1.6, and NOQ=30.
38. Click the radio next to these variables in the algebra pane to show their sliders.
39. Adjust the preferences for these sliders so NQ and NO can vary between 0 and 5 and NOQ can vary between 0 and 360.
40. Construct a circle with center at N and radius NO. Place the point O on the circle.
41. Create an angle by clicking “Angle with Given Size.” Select N, then O, and enter “NOQ” in the dialogue box in front of the degree sign. A point, N’, will appear.
42. Construct a line through O and N’.
43. Construct a circle with center at N and radius NQ.
44. Place Q1 and Q2 at the two places where ON’ intersects the circle with radius NQ.
45. Hide the lines, the circles, and N’. Construct the segments NO, NQ1, NQ2, OQ2, and OQ1.
46. Hide the algebra view.
