1. Write a two-column proof for the following conjecture.

Question

Given:ABCD is a parallelogram

Prove:Angle A and angle B are supplementary. Angle C and angle D are supplementary.

 

Answer 1

 Statement                                               Reason

 

 

Diagonals AC and BD                              line between 2 points axiom

 

Parallelogram ABCD                                 given

 

 

Let angle BAC = x                                   construction axiom

              DAC = z

             ABD = y

             DBC = k

 

AD and BC are parallel;                            definition of parallelogram: opposite sides are parallel

AB and CD are parallel;

 

Angles BAC = CDA = x                       parallel lines cut by a stray line, alternate interior angles are congruent

          DAC = BCA = z

           ABD = CDB = y

            ADB = CBD = k

 

A = x+z = C                                          angle addition property

B = y +k = D

 

 

 

360 = A+B+C+D                                   quadrilateral contains 360 degrees

 

360 = A+B+A+B                                     substitution

 

360 = 2A + 2B                                      combines like terms

 

360 = 2(A+B)                                      factors out 2

 

180 = A+B                                               division property of equality; divides both sides by 2

 

A and B are supplementary                      definition of supplementary angles; they add up 180

 

                   [first statement proven]

 

360 = A+B+C+D                                      quadrilateral contains 360 degrees

 

360 = C + D+ C+ D                                   substitution

 

360 = 2C + 2D                                          combines like terms

 

360 = 2(C+D)                                         factors out 2

 

180  = C+D                                                division property of equality; divides both sides by 2

 

C and D are supplementary                        definition of supplementary angles; they add up to 180

 

 [end of proof]