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Given: $\triangle A B E$ and $\triangle C D E$ with $E$ the midpoint of $\overline{A C}$ and $\overline{B D}$ Prove: $\overline{A B} \| \overline{C D}$

$\overline{A B} \| \overline{C D}$

Geometry

Chapter 4

Triangle Congruence

Section 6

Triangle Congruence: CPCTC

Congruent Triangles

Relationships Within Triangles

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Hi. We're looking at question 15 which is a proof and it doesn't have a picture. So I'm gonna start with creating one, and it tells us that triangle A B e and triangle C D E R. There and ease the midpoint of a C and B D. So if is the midpoint of a see if I dress a c here, this would be my midpoint e roughly. And then I know that this is A and C, and it's also the midpoint of B D. But I know it makes try and go. So b d probably goes like this, and then I can connect those to see what my triangles look like, so that makes it easier to follow along. And then from here, we're gonna go to our two column proof, and we'll have our statements and reasons as usual, and some people like to do their statements first. And then the reasons other people like to do them together. I do both depending upon the mood I'm in. So we're going to do all the statements first, and then we'll fill in all the reasons and then But we'll talk about the reasons as we go. So we're gonna start with are given Triangle A B E Hopefully until it's a triangle, maybe e and triangle C d E. With Is the midpoint one of my ratings not the greatest here? And you can abbreviate these two like midpoint of a C and D d. Trying to write small still have room for the rest of the proof underneath. Okay, so that and that's in the given in your textbook. So if you can't read that, you can. I just literally took what's in your textbook. Wrote it down. I want to make sure the other parts are clearer for you to see. So for Step two, I would go with what are given tells us. And so, since it says that he is the midpoint of a C and B D, we know that E is congrats int t e c. And then for the same reason Because of the midpoint, we know that B E is going to be congratulated t e d. And since they're the same reason I could include them on the same statement, Unless your teacher tells you otherwise. In which case, then this would be two reasons and with two statements with the same reason for both that then for step three. We know we now have, and sometimes it helps to mark your diagram. We know that this is convenient to this and that this segment is congruent to this one. And looking at her picture, we can see that this angle must be congruent to this one because they're vertical angles. So I can say next that angle be a is congressional int to angle CED or D C whichever way you want to write it for my students, I don't make them line up corresponding part yet because we haven't actually proven the triangles. But if your teacher does, make sure you match up those corresponding parts. So here I have two sets asides, and here I have the angle that's between them. So I now how concurrent triangles by side angle side. So I know that trying will be a is congruent to triangle D. C. And then from there I'm trying to prove that the two lines a B and C D or parallel. So I need to get some congruent angles and I can see that angle A is concurrent angle see. You could also say that Angle B is congruent to angle D that would accomplish the same thing. So it doesn't matter which pair you pick, and you don't need both. You can write both if you want, but you don't need both. And that would be because the corresponding parts of congruent triangles or congruent, which then allows us to say that a B is parallel to see D, because those angles air alternate interior angles. And we know if there congruent, then our lines must be parallel from our previous unit that talked about that. Now we need to go back and fill in our reasons well, the 1st 1 is easy. One everybody's favorite given and for Step two were doing that because with 10 point, so depending upon your teacher, you can write definition a midpoint. Or you can say midpoint gives to immigrant segments or something like that. From what your teacher wants Here, you can either say definition of vertical angles or here's an example of typing out. Vertical angles are always congregant and then for number four were proving the triangles. Congrats again. That was by side angle side and then step fry was our corresponding Parts of congruent triangles are congruent from our angles because we know that angle a hacer beacon grunting and we'll see because they correspond. And then for six again, it depends on what your teacher wants if you want to do. Um just whatever version of this, I guess, is the best way to explain it That your teachers looking for So, um, two lines with congregants, alternate interior angles. Um, and, er trans versatile are parallel something along those lines, and then you're done.

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