OK guys, these are tough, particularly #17 and #18. Once you get them, however, they might become your favorite proofs of the year! I don't want to give away too much because I'll be robbing you of the joy that comes from figuring them out. (Yes, I did say 'joy'.)
OK, here goes... For each of these, you'll have to prove at least two sets of triangles congruent. Use the Givens and the diagrams to get the first sets. Then, carefully consider each part of those triangles until you find the parts that can be used ~ with CPCTC ~ to get the next set of triangles. Always keep your eye on which parts you're aiming for.
USE COLORED PENCILS TO OUTLINE TRIANGLES ONCE YOU HAVE THEM CONGRUENT.
14.) Subtraction property is quicker than the Congruent Complements Theorem. Start with the small triangles.
15.) Draw. I used SSS and ASA.
16.) This proof took me 11 steps and I did not combine Givens. From the Division Property, you'll need to get two conclusions. You can list them both right away and then use Restates if/when you need to. 'If angles, then sides' and its converse will be very useful.
17.) OK, now comes the real fun!! The first time through, this took me 19 steps but I was convinced I could be more efficient. Sure enough, I had neglected to use the obvious vertical angles. Fixing that saved me four steps! I used THREE sets of triangles on this one so that means I have six triangles listed in my proof. Two of them are RPU and SUQ (which are not congruent to each other, of course). I won't spoil the ending.
Some of my reasons include Subtraction property, Reflexive property, CPCTC, SAS and Vertical angles are congruent. (**Note: For the reflexive property, you can call the same angle by two different names. I did this for the first set of triangles.)
You might want to use backward braces for those statements which give you more than one important conclusion. For example, the congruence statement for triangle RPU gave me two parts that I wanted to use so I used a backward brace there.
18.) As soon as I used a colored pencil, I figured this one out. It's tough because there are so many triangles that you could try. This one took me 14 steps and two sets of triangles. The first set of triangles is easy because the Givens are all you need. Use a colored pencil to outline those.
Then, you move on to the next set of triangles. I chose a set that included angle DAC. You'll be able to use SAS for those. (Number the small angles by vertex D.)
Do not use any triangles that do not have a labeled point at each vertex.
Lastly, 'if sides, then angles' will come in handy with segments AD and BD. A little subtraction is all you'll need to close it up.
TO RECAP THE GRADING:
If you seriously attempt all five, you will be eligible for an A. If you try 3 or 4, you can earn a B. If you try 1 or 2, you can earn a C. I will be grading for content from there.
Have fun with these... every bit as enjoyable as a riddle, brain teaser or Sudoku puzzle. = + )
Happy Thanksgiving and go Cadets!!!!
Thursday, November 25, 2010
Monday, November 1, 2010
A Few New Tips - CPCTC
Hi everybody,
I hope that triangle proofs are becoming easy for you. I'm confident that you all can handle this next proof packet with little trouble. I'll include tips for those that are more than a few steps. You're on your own for the short ones - that is, unless you ask me about them at school.
4. Only 2/3 of the trisection in the Given is important. Number the right angles 1 and 2. Don't forget to go through the pedantic steps with perpendicularity. (Right, Kory?)
5. Start with the midpt. step.
6. Easy. If you choose to put numbers into angles, change the Givens to reflect the new angle names.
7. Remember your formulas?
8. This is not a system of equations, just three separate eqn's. To isolate a variable that is under a square root symbol, you have to do the opposite operation to both sides. Check your answers!
9. Easy. We've seen this diagram in the last section. Five-step proof.
10. Start with MP = RO. That alone leads to a needed step.
11. Shaded region problems? Imagine the whole diagram is shaded and find area. Then "cut out" the appropriate section.
12. Hmmm... midpt. is used twice. Even though this isn't a proof, I'd like to see the diagram drawn and marked. I'll leave it up to you to decide if this is a system of equations or just two equations to be solved separately. Remember this general rule: to solve for n variables, you need n equations. (There are exceptions.)
13. Remember our special definition of right angles? It was that if two angles form a right angle, then they are complementary.
14. Very similar to #13. This might be the last time you use the Linear Pair Postulate.
15. Turn around the second Given if you want to see the Subtraction Property work smoothly. Don't use extra steps.
16. I'll allow a double-transitive in one step. Make it look like a triple play.
17. With overlapping triangles, you should choose which ones to go for. You can complete this proof with the two smallest triangles or the overlapping ones. Which way is more efficient? The Prove statement usually leads you. Six-step proof.
18. Looks much harder than it is. Should I even mention the obvious triple play? I would suggest starting with those statements and reaching the first conclusion. Under the triple play, put the other Given that needs to be worked on a little. That way, your 3 important triangle statements should line up nicely.
19. Why is this here? Trust your common sense.
20. The first proof that goes beyond CPCTC - not very far beyond, though! Yes, you can draw them in.
21. OK, this is the last time you use LPP. With those included into one, I get nine steps.
18.
I hope that triangle proofs are becoming easy for you. I'm confident that you all can handle this next proof packet with little trouble. I'll include tips for those that are more than a few steps. You're on your own for the short ones - that is, unless you ask me about them at school.
4. Only 2/3 of the trisection in the Given is important. Number the right angles 1 and 2. Don't forget to go through the pedantic steps with perpendicularity. (Right, Kory?)
5. Start with the midpt. step.
6. Easy. If you choose to put numbers into angles, change the Givens to reflect the new angle names.
7. Remember your formulas?
8. This is not a system of equations, just three separate eqn's. To isolate a variable that is under a square root symbol, you have to do the opposite operation to both sides. Check your answers!
9. Easy. We've seen this diagram in the last section. Five-step proof.
10. Start with MP = RO. That alone leads to a needed step.
11. Shaded region problems? Imagine the whole diagram is shaded and find area. Then "cut out" the appropriate section.
12. Hmmm... midpt. is used twice. Even though this isn't a proof, I'd like to see the diagram drawn and marked. I'll leave it up to you to decide if this is a system of equations or just two equations to be solved separately. Remember this general rule: to solve for n variables, you need n equations. (There are exceptions.)
13. Remember our special definition of right angles? It was that if two angles form a right angle, then they are complementary.
14. Very similar to #13. This might be the last time you use the Linear Pair Postulate.
15. Turn around the second Given if you want to see the Subtraction Property work smoothly. Don't use extra steps.
16. I'll allow a double-transitive in one step. Make it look like a triple play.
17. With overlapping triangles, you should choose which ones to go for. You can complete this proof with the two smallest triangles or the overlapping ones. Which way is more efficient? The Prove statement usually leads you. Six-step proof.
18. Looks much harder than it is. Should I even mention the obvious triple play? I would suggest starting with those statements and reaching the first conclusion. Under the triple play, put the other Given that needs to be worked on a little. That way, your 3 important triangle statements should line up nicely.
19. Why is this here? Trust your common sense.
20. The first proof that goes beyond CPCTC - not very far beyond, though! Yes, you can draw them in.
21. OK, this is the last time you use LPP. With those included into one, I get nine steps.
18.
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