need help with parallelogram proof

[sc00t34]

Hello, we are learning about similar triangles and this was a problem. So I know that opposite sides of a parallelogram are congruent as are opposite angles, so I can establish similarity with triangles WYS and STW, but I don't understand how that proves SX x YW = SV x WT because the proportions don't match up when I compare similar triangles for their corresponding parts.

Any help is greatly appreciated... am I on the right track?

This is what I have so far.

Statement Reason

1. WYST is Parallelogram. 1. Given
2. angle Y and angle T are congruent. 2. Def of parallelogram, opposite angles congruent.
3. WT and YS congruent, WY and TS congruent. 3. Def of parallelogram, opposite sides congruent.
4. WYS and STW are similar. 4. SAS
?????

 

[R.M.]

While WYS and STW are triangles, they are not explicitly drawn in your picture. Focus on triangles SVT and SXY. You already know that T and Y are congruent, and that each of those triangles has a right angle. Therefore those triangles are similar by AA. See if you can create a proportion from those and then make a substitution to get the relationship that you are trying to prove.

 

[sc00t34]

While WYS and STW are triangles, they are not explicitly drawn in your picture. Focus on triangles SVT and SXY. You already know that T and Y are congruent, and that each of those triangles has a right angle. Therefore those triangles are similar by AA. See if you can create a proportion from those and then make a substitution to get the relationship that you are trying to prove.

Hi and thanks so much for the reply. I understand what you mention about the triangles being similar by AA, but I guess I’m still confused about how to get to SX YW = SV WT. SX and SV are corresponding parts of similar triangles, but aren’t YW and WT completely outside those triangles? Can we assume that they are also proportional ? And if so why? Thank you

 

[R.M.]

If triangle SVT and triangle SXY are similar, we can make the following proportion: SV/SX = ST/SY, which leads to (SV)(SY) = (SX)(ST). You also know that YSTW is a parallelogram. You are one substitution away from finishing.

 

[LCKurtz]

If you would like a geometric argument, consider the image below. You may still need to verify a couple of congruences.