Congruence: More than meets the eye. Sometimes we think that figures are congruent because they look like exact copies, but you can’t always trust your eyes. In geometry it’s important to be precise so we say that a figure is congruent to another figure if we can get one by a combination of reflections (flips), translations (slides), and rotations (turns) performed on the other.
Lorna thought she had created 2 congruent rectangles within a square.
However, since she knew shapes could look like exact copies when they were not, she decided she should investigate further. She folded her paper so she could check to see if the rectangle on top could be reflected onto the one below. If the rectangles were truly congruent, one rectangle should cover the other one exactly. When she folded she found that wasn’t the case. Her rectangles looked congruent but they weren’t!
Next, Lorna decided she would try to create 2 congruent triangles within the square. She drew one of the diagonals of the square and exclaimed, “I have congruent isosceles right triangles!”
Again, she knew she better check to make sure because figures that look congruent aren’t always congruent. She folded the square along the diagonal and the triangles sat exactly on top of each other – they were congruent because one was the reflection of the other!
1. Now it’s your turn to compare some shapes. Start with a piece of paper that is a non-square rectangle, and be sure it is a different size than your neighbor’s rectangle. Fold your paper so that point A is directly on top of point C as shown in the picture below.
a) As in our picture above, you should see some triangles. Outline the triangles on your paper using a marker. How many did you find?
b) Compare triangles ∆EDC and ∆FGC on your paper. How are they alike?
c) Your neighbor folded a different rectangle. Ask your neighbor what they noticed. What did your neighbor discover about how ∆EDC and ∆FGC are alike?
d) Julio thinks the two triangles are congruent, but cannot show it. Help Julio show congruence. Remember how Lorna thought about congruence – one triangle is congruent to another if we can get one by reflecting, translating, or rotating the other. Can you show Julio how to get ∆EDC by reflecting, translating, or rotating ∆FGC?
e) Look at ∆CEF in our picture. What is special about this triangle? Compare this triangle from your paper with your neighbor’s. What do you notice?
2. Research and advise Morgana. Morgana is starting a small business, making and selling geometry materials. She wants your advice on a new product, isosceles folding paper.
She tells you: "I want to sell paper that creates 2 isosceles triangles that are congruent to each other. That means when you fold point A to point C, ∆EDC and ∆FGC should be isosceles triangles that are congruent to each other. Not all rectangular paper can create these isosceles triangles so I need your help. Help me find 3 different sizes of rectangular paper that will create pairs of congruent isosceles triangles."
Your job is to find 3 different-sized rectangles that fit Morgana's conditions. Attach the 3 rectangles to this sheet along with a brief note (1 or 2 paragraphs) to her describing:
- how you know each paper size produces triangles that are isosceles and congruent to each other
- how you found the 3 paper sizes
- two important things you discovered as you conducted your research
3. Explore some more. Start with another non-square rectangular piece of paper.
a) Instead of folding one corner onto the opposite corner (A onto C or B onto D), like you did in number 1, fold the paper so that A’ lies somewhere on side . There are three triangles showing from your fold. They are all right triangles. What else do you notice about these triangles?
b) Slide point A’ along side and fold the paper so that A’ is at a new location. Examine the triangles that are created. Slide point A’ along side and fold the paper once again. Examine the triangles that are created. Use the chart below to record what you noticed and what you wondered about.
c) If you used a different sized piece of paper, do you think you would notice the same things you noticed in part b? Why or why not?