Need help with a quadrilateral proof please

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In summary, the given statements and reasons lead to the conclusion that JUNE is a parallelogram, using the properties of congruent and similar triangles, and the definition of a parallelogram.
  • #1
shyguy10918
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Homework Statement



E 1\---------------------------------1N
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
K 1---------\-M---------------------1L
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
1 \ 1
J 1---------------------------------\1U
JUNE is a quadrilateral
K is the midpoint of line JE
L is the midpoint of line UN
line KL and line UE bisect each other at point M
Prove:JUNE i a parallelogram


Homework Equations





The Attempt at a Solution



statement 1 reason
--------------------------------------------------
1)K is the midpoint of line 1 1) given
JE
2)line KE is congruent to 1 2) midpoint is the center of a line segment
line KJ
3)L is the midpoint of line 1 3) given
UN
4)line NL is congruent to line 1 4) midpoint is the center of a line segment
LU
5)line KL and line UE bisect 1 5) given
each other at M
6)line EM is congruent to line 1 6) line bisector splits the line segment in half
MU
7)line KM is congruent to line 1 7) line bisector splits the line segment in half
ML
8)Triangle EKM is congruent to1 8) theorem of Side,Side,Side
triangle ULM
9)line EK is congruent to line 1 9)corresponding parts of congruent triangles and congruent
UL
10)line KJ is congruent to line 1 10)substitution
NL
11)line EJ is congruent to line 1 11)substitution
NU
This is where i am stuck,how can I prove that either line EN is congruent to line JU or that line EJ is parallel to line NU?
 
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  • #2
sorry the picture didn't come out the way it was suppose to. it is suppose to have a diagnol EU and line NU.
 
  • #3
I'm going to have [tex]\overline{XY}[/tex] denote the length of some line segment XY.

let: [tex]a = \overline{KM} = \overline{ML}[/tex].

By similar triangles (KEM and JEU), [tex]\overline{JU} = 2a[/tex].

By similar triangles (MUL and EUN), [tex]\overline{EN} = 2a[/tex].

It follows that [tex]\overline{JU} = \overline{EN}[/tex].



let: [tex]b = \overline{UL} = \overline{NL}[/tex].

By addition, [tex]\overline{NU} = 2b[/tex].

By congruent triangles (UML and EMK), [tex]\overline{EK} = b[/tex].

It follows that [tex]\overline{KJ} = \overline{EK} = b[/tex].

By addition, [tex]\overline{EJ} = 2b[/tex].

It follows that [tex]\overline{NU} = \overline{EJ}[/tex].



[tex]\overline{JU} = \overline{EN}[/tex] and [tex]\overline{NU} = \overline{EJ}[/tex]; therefore, JUNE is a parallelogram.



Also, don't worry about the picture too much, the text below it defines your problem perfectly.
 
Last edited:
  • #4
thank you alot. i didn't even notice those bigger traingles.
 
  • #5
Hi shyguy10918! :smile:

That's a proof using only lengths (and it's fine).

You might like to try an alternative proof using angles, and the definition of a parallelogram as having parallel sides. :smile:
 

1) What is a quadrilateral proof?

A quadrilateral proof is a mathematical argument that shows the validity of a statement or theorem about a quadrilateral. It involves using geometric properties and logical reasoning to support the statement.

2) What are the steps for solving a quadrilateral proof?

The steps for solving a quadrilateral proof may vary depending on the specific problem, but generally they include identifying and labeling the given information, applying relevant geometric theorems and properties, and using logical reasoning to reach a conclusion.

3) How do I know if my quadrilateral proof is correct?

You can check the validity of your quadrilateral proof by making sure that you have used accurate labeling, applied relevant theorems and properties correctly, and used logical reasoning to justify each step. You can also check your work by comparing it to known proofs or asking a teacher or peer for feedback.

4) What are some common mistakes to avoid when solving a quadrilateral proof?

Common mistakes in quadrilateral proofs include forgetting to label or correctly identify the given information, using incorrect or irrelevant theorems and properties, and making incorrect assumptions or conclusions. It is also important to double check your calculations and make sure they are accurate.

5) How can I improve my skills in solving quadrilateral proofs?

The best way to improve your skills in solving quadrilateral proofs is to practice regularly and seek help when needed. You can also review geometric theorems and properties, as well as study examples of successful proofs. Additionally, breaking down the problem into smaller steps and using logical reasoning can help make the process easier.

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