SOLUTION: Hello. I was wondering if I could get some assistance on this proof for my geometry class. I have been trying to find out the best way to do the problem, but this is new ground for

Algebra ->  Geometry-proofs -> SOLUTION: Hello. I was wondering if I could get some assistance on this proof for my geometry class. I have been trying to find out the best way to do the problem, but this is new ground for      Log On


   

Question 1010765: Hello. I was wondering if I could get some assistance on this proof for my geometry class. I have been trying to find out the best way to do the problem, but this is new ground for me.
Given: The diagonals of parallelogram PARL are congruent.
Prove: PARL is a rectangle.
Your help is much appreciated.
Thanks so much in advance.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's draw out the picture. Let's assume PARL is a rectangle. This drawing (below) does NOT prove it is a rectangle. The table after the drawing is what will prove the claim. The drawing is simply a tool to help visualize the proof.




Proof Table

NumberStatementReason
1PARL is a parallelogramGiven
2The diagonals of PARL are congruentGiven
3PR = ALDiagonals are congruent
4PA = RLOpposite sides of a parallelogram are congruent
5AR = LPOpposite sides of a parallelogram are congruent
6Triangle APL = Triangle RLPSSS Property of Congruence
7Angle APL = Angle RLPCPCTC
8Triangle PAR = Triangle LRASSS Property of Congruence
9Angle PAR = Angle LRACPCTC
10(Angle APL)+(Angle RLP) = 180Same Side Interior angles are supplementary
11(Angle APL)+(Angle APL) = 180Substitution (lines 10 and 7)
122*(Angle APL) = 180Combine like terms
13Angle APL = 90 degreesDivision Property of Equality
1490 degrees = Angle RLPSubstitution (lines 7 and 13)
15Angle RLP = 90 degreesSymmetric Property of Equality
16Angle ARL = Angle APLOpposite angles of a parallelogram are congruent
17Angle ARL = 90 degreesSubstitution (lines 16 and 13)
18Angle PAR = Angle RLPOpposite angles of a parallelogram are congruent
19Angle PAR = 90 degreesSubstitution (lines 18 and 15)
20PARL has 4 right anglesSee lines 13,15,17,19
21PARL is a rectangleDefinition of Rectangle (see note)


Note: A rectangle is simply a quadrilateral with 4 right angles (aka 90 degree angles).