answer for two-column proof

Question

Given: Two concentric circles with center O and line ABCD intersecting both. Lines BC and AD are chords of circle O.

 

To Prove: Line AB is congruent to line CD.

Answer 1

     SSA does NOT prove that triangles are congruent!!!! 

 

Draw radius OEFG, E being a point on AD, F being a point on the inner circle and G being a point on the outer circle, so that OEFG is perpendicular to AD. there is a theorem stating that a radius drawn perpendicular to a chord bisects the chord.                                   So AD and BC are bisected.

AE =DE and BE=CE

AE-BE=DE-CE

AB = DC

Answer 2

since the circles are concentric, and the points A, D lie on outer circle and points B, C lie on the inner circle, then line OB and OC are both equal to the radius of the inner circle and likewise OA and OD are equal to the radius of the outer circle.

 

so the line AB =OA-OB and line CD = OD-OC, but OA = OD and OB = OC

line AB is equal or congruent to line CD 

Answer 3

We prove AB = CD by showing congruent triangles ABO = DCO using SAS = SAS