Question 694844: please help me with this proof
Given: triangle PWR is isosceles with the base segment PR
segment PU is congruent to segment RA
prove: triangle AWU is Isosceles
thanks i need it by thursday please. if you can
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! I'm assuming that your diagram looks kind of like an hourglass and looks like the two triangles are meeting at the middle near vertex W. It is difficult for me to know exactly unless I have more information or a picture is shown. I have drawn out a diagram with everything you've given me and this is solvable. But if my assumptions are not correct, then it would be better to disregard my entire answer than to accept it. I've tried drawing it on here, but I am limited. Hopefully our diagrams are similar. Compare my work to your diagram. If they don't match, then disregard my answer and message me. I'll do my best to give a clearer answer next time.
Obviously the first statement and reason are from the given. A good tip is if you don't know where to start, always look at what we are given. After all, we can't go anywhere unless we are given information to start with. The given tells us that PU and RA are congruent. So, writing it in two-column proof format:
Statement Reason
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1. △ PWR is isosceles; PR is base; PU ≅ RA. |1. Given.
Let us continue by analyzing our triangle from what we are given. The triangle PWR is isosceles with base PR. Since the base is PR, the other two sides, or the legs PW and RW, are congruent by isosceles definition. This may be a hint to find a way to prove that the same two sides of the other triangle are also congruent. We could definitely use that, but let us examine everything from the given before we try to take a route to prove the other triangle is isosceles.
Looking at the unique piece PU ≅ RA from the given diagram, we can see that PU and RA are made up of two segments each: PU is made up of PW and WU; RA is made up of RW and WA. We can expound on this idea and say that by segment addition, PU = PW + WU and RA = RW + WA. This looks like a good path to venture on because it involves the four sides between both triangles that we need to deal with (two from the given and the two we need to prove are congruent). We need to prove that WU and WA are congruent in order for triangle AWU to be isosceles by definition. Let's see what we can do from the segment addition.
PU = PW + WU and RA = RW + WA may look like a jumbled mess right now. But we can use some algebra in order to bring everything into sight. Since we've PU and RA are congruent, their parts also must be congruent. That is, since PU = RA and PU = PW + WU and RA = RW + WA, then PW + WU = RW + WA. We can call this substitution (PU with RA or RA with PU). Furthermore, since we know that the given triangle is isosceles (thus RW and PW are congruent), we could use substitution again in order to make things simpler.
PW + WU = RW + WA can be taken to PW + WU = PW + WA or RW + WU = RW + WA (the first substitutes PW and the latter RW). This actually helps us because now we can see that we can cancel out something. (If you use a variable for the repeating terms, you can see that that variable can be cancelled out through subtraction on both sides. The same goes for using side or angle names rather than simple variables.) So, with either PW or RW gone (depending on which term you substituted), we are left with WU = WA. By definition of congruence (when dealing with measurements), WU = WA can be written as WU ≅ WA. Now we use a special theorem reserved for isosceles triangles.
Here is the definition from mathwarehouse.com:
"If two sides of a triangle are congruent, then the angles opposite those sides are congruent (Base Angle Theorem)."
The converse of this theorem is also true. So, since we've proven that triangle AWU has two congruent sides WU and WA, the angles opposite of them must be congruent (the base angles). By definition of isosceles triangles, triangle AWU is isosceles. What have proven it! Here's the rest of the work in two-column proof:
Statement Reason
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1. △ PWR is isosceles; PR is base; PU ≅ RA. 1. Given.
2. PW ≅ RW 2. Def. of Iso. Tri.
3. PW = RW and PU = RA 3. Def. of Cong. 4. PU = PW + WU and RA = RW + WA 4. Segment Addition Post.
5. PW + WU = RW + WA 5. Substit. (PU = RA)
6. PW + WU = PW + WA 6. Substit. (PW = RW)
7. WU = WA 7. Subtract. Prop. of Equ.
8. WU ≅ WA 8. Def. of Cong.
9. ∠WUA ≅ ∠WAU 9. Base Angle Theorem
10. △AWU is isosceles 10. Def. of Iso. Tri.
There are probably other ways to go about proving this triangle to be isosceles, but the core of it is in this work. Statements 2 and 8 are more of technical jargon that may or may not need to be present in your work. This depends on your teacher. This is just showing the process of going back and forth from naming measurements and naming pieces.
It is always good to look at everything we are given and analyze with all the knowledge we have of proofs, theorems and postulates. The more we can tell about diagrams by looking at it the better. We may not be able to get to the finish line of proving what we need by taking our first route. We may run into proof difficulty and need to start back at the given. Notwithstanding, do not give up if you don't get it the first or second try of trying to prove it.
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