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Troy has written a proof showing that the diagonals of a parallelogram are equal in length. The diagonals of a parallelogram are not necessarily equal so Troy must have made an error. Identify and explain his error.

 

Statement   |  Justification

AB = ED       |     Opposite sides of parallelogram

BAE = AED   |      Alternate interior angles

ABD = BDE   |     Alternate interoir angles

ABC = EDC   |   ASA

BC = EC      |   Corresponding sides of congruent angles

AC = DC      |   Corresponding sides of congruent angles

AE = BD      |   AC + CE = DC + CB

3 years ago

Answered By Emily H

To figure out this problem, you will want to draw a diagram so that you can get a better idea of what the problem's asking.

Then, I would relable all the three letter angles as follows just for ease of following which is which:

 $EAB=\angle A$EAB=∠A ; $AED=\angle E$AED=∠E ; $BDE=\angle D$BDE=∠D ; $ABD=\angle B$ABD=∠B ; $ABC=\angle B_1$ABC=∠B1 ; $EDC=\angle D_2$EDC=∠D2 

So, then looking at the statements, we can see the first is true by definition, no matter the values of the angles.

The second and third statements, when rewritten become  $\angle A=\angle E$∠A=∠E and  $\angle B=\angle D$∠B=∠D . However, from the diagram we can see that these statements are not necessarily true, except when   $\angle A=\angle B=\angle D=\angle E=90$∠A=∠B=∠D=∠E=90, which would indicate a rectangle rather than a more general parallelogram.

As the remainder of this proof requires these two statements to hold, it swiftly falls apart except in the special case.

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