I don’t really understand how to solve proofs at all. Will someone help me?
Proofs are hard, because they are more of a puzzle than following a recipe, like algebra. Think of any multi-step puzzle game or even a computer puzzle game. First, how do you figure out how to win? You try to apply things that you know and see how they can help you progress. At any given moment, only one or two things will be useful. But, you have to string together each of these steps to get to the end. Sometimes, you will go the wrong way and you must go back to see where you went wrong. But, if you know enough “skills”, you can kind of work backwards from the end. What would you need to be able to get to the end? Well, what would you need to get to the place before the end, and before that? The more you practice, the more you will see all the different possible paths.
Writing a proof is done AFTER you have already figured out the steps to solving the puzzle. Basically the written proof is like a set of instructions you are leaving for a friend. If you were to tell the friend how to complete the puzzle, you would tell them each step, being careful not to leave any out, and you would give them reasons for doing that step. You might say, use your magic wand here, because the Ogres can’t be hurt by your sword (or something like that).
In this exercise, the steps have already been figured out for you, and you must supply the reasons. Usually, this is best done by looking at the prior step and seeing how one got to this step from that one. Sometimes it is obvious and direct. Like going from step 8 to step 9 you don’t need any other information, you are just using a definition. Like if one step said John had a coin worth 25¢ and the next step said John had a quarter, you simply are applying the definition of a quarter.
Sometimes you have to tie together several previous steps to get the next one. Like, step 5 is found by combining steps 3 and 4. Step 7 is found by combining steps 4, 5, and 6. The hard part is in remembering that you can’t skip steps. Like you might have been able to figure out steps 4 and 5. And, from those steps, you might intuitively know that step 8 is true, but you have to show the progressive steps you made to get there. Step 7 may seem obvious from the picture, but you first had to establish the addition of angles in step 6.
You are trying to establish an airtight case. That means that every step follows CONCLUSIVELY from the ones before, it doesn’t rely on “it’s obvious” or such things. Each step only relies on things you already know or have proven, and you can name the reason you are using.
How do you get better at proofs? Practice is the ONLY way. Like with puzzles, you start out with simpler ones until you have mastered them, meaning you know how to do them and do them quickly. Then you progress to slightly more complex ones. But, the complex ones will make absolutely no sense if you haven’t first mastered the simpler ones.
You will start seeing repeated strategies. like anytime you are trying to prove two things are congruent, usually that either means you first prove their measures are equal, or you use the transitive property after showing a chain of congruent things. Or, if you are trying to show an angle is right, usually that means that you are trying to show some series of steps that shows that the measure of the angle is 90 degrees.
My advice to you is to go back to page 110 and start practicing. If you aren’t getting proofs now, your life is about to get miserable, because the next few months are going to be almost entirely proofs and they are going to get more and more complex. Like when you memorized the multiplication tables, your ability to do proofs will rely on the relationships you have learned. You have to immediately be able to recognize them and know what they mean. You should know every definition, property, postulate and theorem practically by heart. Go back now, and write each of these down in their categories. Write it out how the book says it, then write out what it means in your own words. Make sure you REALLY understand each one, because every proof relies on that sound knowledge and quick recall. You will continue to get more and more of these relationships and if you haven’t memorized them and their meaning, you won’t be able to do the proofs.
It’s like if you play a role playing computer game. You might collect all of these tools and weapons and meet all of these friends and enemies. But, if you never memorize what each tool does, never learn who is a friend and who is an enemy, and never learn how each enemy is defeated, you will never win. Simply put, if you aren’t willing to keep lists and memorize, you will have no chance of solving proofs.