Today, we will be investigating the properties of reflections. You are probably familiar with the reflection you see when you view yourself in a mirror. How far away from the mirror does your reflection appear to be? What size is your reflection in the mirror? A reflection in geometry is called that because it has many of the same properties as the reflections you see in a mirror. In this activity, we will be investigating the properties of geometric reflections across given lines.
Sketch 1
Step 1 : Construct line AB.
Step 2 : Construct points C, D, and E, with E on the opposite
side of the line as C and D.
Step 3 : Mark line AB as mirror in the Transform menu and reflect
points C, D, and E to create points C', D', and E'.
Step 4 : Construct segment CC', segment DD', and EE'.
Investigation 1
Move the points C, D, and E around. Do you see a relationship between the reflection axis line AB and the sement that connects a point and its reflection? Construct the points of intersection of these segments and the reflection axis. Measure the distances from the points of intersection to C and C', to D and D', and to E and E'. If Sketchpad didn't do reflections automatically, how could you find the reflection of a point across a line. Now go back the the original sketch and reflect points C', D', and E' across line AB. What happens? What are the reflection images of points C', D', and E'?
Sketch 2
Step 1 : Use the sketch you constructed in 
Sketch 1.
Step 2 : Label the points of intersection for segments CC', DD', and EE' with line AB as P, Q, and X, respectively.
Step 3 : Measure angles APC, PQD, and QXE.
Step 4 : Move points C, D, and E around and notice with each move the measurements of the angles mentioned in Step 3.
Step 5 : Move line AB around and, once again, notice the measurements of the angles.
Investigation 2
What is the relationship of the line of reflection to the segments created by the pre-image points and their corresponding image points? Even after you moved points C, D, and E around, what is the relationship of the line of reflection to the segments CC', DD', and EE'?
Assignment - Day 3
1. Answer the questions posed in the investigations above.
2. Describe the new ideas you learned about today.
3. Discuss any questions or points of misunderstanding you may have had in the course of this lesson.
4. On your own paper, create a scalene triangle, labelling the vertices A, B, and C. Draw any line on the same piece of paper, labelling it with points X and Y. Use the properties of reflections we have discussed today to accurately reflect triangel ABC over line XY. Be sure to show your work with lines and angle marks.
Resources: Some of this material is taken from Exploring Geometry with The Geometer's Sketchpad, Blackline Activity Masters for Use with The Geometer's Sketchpad. Key Curriculum Press. 1996.