G9_Triangle Congruency (demo)
Demonstrations of triangles with SAS, AAA, SSS, and SSA defined, respectively.
Side and angle measures are determined by sliders. The blue letters can be manipulated with the sliders constant to demonstrate how a well-determined triangle keeps its shape and size, but an insufficiently defined triangle does not.
1. In Quadrant II of the worksheet, a triangle has been defined with the radii of two circles, centered at A, and an angle, <CAD. The angle and the lengths of the sides can be manipulated by moving the three sliders.
a. How many triangles can be made with the two sides set to length 1 and 4, respectively, and the angle set to 126?
b. If you made more than one triangle with these settings, how are the triangles the same, and how are they different? Describe things like their shape, size, orientation, and so forth.
c. Move the triangle by dragging points B and A. How does the triangle change?
2. In Quadrant I of the worksheet, a triangle has been defined with two angles. These angles are determined by the two sliders.
a. Using what you know about triangles, explain why we can only choose the measures of two of the angles and not all three.
b. Try moving points E, G, and F. How does this triangle change? How are its changes different from the changes we got when we moved points B and A in triangle ACD?
c. Move the sliders controlling angle GEH and angle HGE. At what point does the triangle disappear?
d. What constraints are there on the values we can choose for GEH and HGE?
3. Behold the triangle in Quadrant III, triangle IKL. It has been defined using the radii of three circles. The length of the sides (and the radii of the circles) is set using the three sliders, labeled IL, KL, and IK.
a. Try manipulating the triangle by dragging points J and I. How does the triangle change in terms of size, shape, and orientation?
b. Try manipulating the three sliders. At what slider settings does the triangle disappear? Hint: Pay attention to where the circles cross.
c. Okay, this diagram really has two triangles, IKL1 and IKL2. How are these to triangles the same and how are they different in terms of size, shape, and orientation?
d. Prove that IKL1 and IKL2 are the same size.
4. Behold the triangle in Quadrant IV, triangle NOQ. This triangle has been defined using the radii of two circles (NQ and NO) and an angle, angle NOQ. The measure of the two sides and the angle can be manipulated using the three sliders.
a. Move points N and O. How does the triangle change in terms of shape, size, and orientation?
b. Try adjusting the settings on the sliders. At what point does the triangle disappear? (Hint: Pay attention to where the lines and circles cross).
c. This diagram also has two triangles, ONQ1 and ONQ2. Both of these triangles fit our definition: they both have a side of length NQ, a side of length NO, and an angle measuring NOQ. How are these triangles the same and how are they different in terms of shape, size, and orientation?
5. Create a new worksheet using Geogebra. Construct a triangle by defining two angles and the length of one side.
a. When you manipulate the vertices of this triangle, how does it change in terms of size, shape, and orientation?
b. If more than one triangle can be defined using the same settings, are the different triangles the same in shape and size?