Congruent Triangles, Discovery Day 2
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- Garey Conley
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1 Congruent Triangles, Discovery Day 2 Name: October, Triangles! We ve talked about congruence of segments and angles. Notice that an angle only has one measure, and a segment only has one length. So how do we define congruence when there s more then one measurable aspect of a figure? 1. Carefully examine the figures below and decide which pairs are congruent. Remember that congruence is still a relationship between two objects. (a) (e) (i) 60 (b) (f) 2 (j) (c) (g) (d) (h) 2 (k) 2. How did you decide if two objects were congruent?
2 There are four ways we can manipulate an object without changing the actual structure of the object (changing the structure might include adding or removing sides and angles): 1. Rotation - rotating the object means to spin it around. For instance the end of the minute hand on a clock rotates as time goes by. 2. Translation - translating an object means to slide it. For instance when you drag a window on your computer screen, the window is being translated. 3. Reflection - reflecting an object means to flip it over a line. For instance, the elementary school project when you painted a picture on one side of a paper and then folded it and it made a butterfly shape? That was a reflection! 4. Dilation - dilating an object means scaling it. It can get smaller or bigger. If you ve ever been to the eye doctor they use drops to dilate your pupils; you re pupils get very large!
3 Definition of Congruence for Figures: Two figures are congruent if all their corresponding parts are congruent. OR If a figure can be rotated, translated, or reflected to be the same as another figure, then the figures are congruent. When we are proving triangles congruent, it is useful to use the first definition. 2 Proving Triangles Congruent : When we try to prove triangles congruent we need to show that all the corresponding parts are congruent. For example: Now if we actually had to prove all six things (three angles and three sides) that could take a loooooong time! Yesterday you discovered how many pieces of information you need to determine a triangle. 3. How many corresponding parts have to congruent for two triangles to be congruent? 4. With three sides, and three angles, how many combinations of things can you make? List each combination.
4 Game Time! Yesterday, you worked to discover what information determined a triangle, but sometimes that same information could create different looking triangles. Today we want to discover what information determines a unique triangle. This means that those three pieces create a specific triangle, with specific measures and no triangle with different measures can be created. 5. How does creating a unique triangle relate to congruent triangles? Consider the following: Pick up three sides, and make a triangle, is it a unique triangle? In other words, can you put those three sides together to create a triangle that is fundamentally different from the first one? 6. If you picked up three sides, and your partner picked up the same three sides, would you make the same triangle? Try it out! Conclusion: Now it is time for you to decide which combinations will prove two triangles congruent! BUT first, the rules of the game! Segments and angles can only be connected at their endpoints and vertices indicated by the black dots. Angles are created by a union of infinite rays so remember the angle doesn t end when the paper ends! β δ You have to combine the information in the same order! If you pick a combination like SAS, the angle is included, while in ASS the angle is not included.
5 Choose a combination to explore. Make sure you and your partner are using the same tools. Challenge your partner to make a triangle that looks different than yours! Combination Tool Combinations tried Will this prove If not, why? triangles congruent? SSS ASS, SSA SAS AAS, SAA ASA AAA
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