Investigation: Angle angle similarity

Objective

Use informal arguments to establish facts about the angle-angle criterion for similarity of triangles

Vocabulary

Write the definition of each of the words listed below.  Be sure to use these terms when you discuss the activity below with your classmates.

• transformation
• translation
• rotation 
• dilation
• reflection
• congruent
• similar

Procedure

1. Use the bottom left slider to adjust the size of the white triangle in the applet below, such that it is not the same size as the pink triangle.  You may also choose to drag one of the big green vertices in either triangle to re-position the two triangles.  Notice that there are two pairs of angles that are marked congruent in the two triangles.
2. Next drag the bottom right slider far enough across that only one transformation of the white triangle occurs.  Describe the properties of this transformation with a partner (ie.  Are the side lengths of the triangle preserved under this transformation?, Are the angle measures in the triangle preserved under this transformation?, Has the orientation of the triangle changed under this transformation?).  Would it be possible to use the slider to transform the white triangle such that one of its vertices corresponds with a vertex of the pink triangle - using only one transformation - regardless of where you initially had these two triangles positioned?
3. Then, continue to drag the bottom right slider far enough across that a second transformation occurs.  Describe the properties of this transformation with a partner.  Would it be possible to use the slider to transform the white triangle such that the angles marked with two lines will lie on top of one another, regardless of the position that the white triangle started off in?
4. What if the orientation of the white triangle were different than the orientation of the pink triangle?  Is there another type of transformation you could do first, that would allow steps 2 and 3 above to result in the congruent angles lying on top of one another?
5. Next, drag the bottom right slider completely across to the right side.  Describe the properties of the transformation that occurred.  Would that transformation have been possible regardless of the original size and position of the white triangle?

Conclusion

Does the series of transformations that were displayed by moving the slider in the applet above prove that given a pair of triangles with two pairs of congruent angles, the triangles must be similar?  Why or why not?  Discuss your reasoning with a partner.