Quandaries
and Queries 

Hi! My name is Hanna and I am having difficulty on my geometric proofs. I am in the 9th grade and am a student. Here is one problem I don’t understand. Given: ABCD is a quadrilateral; <A is congruent to <C; <B is congruent to <D Prove: ABCD is a parallelogram 

Hi Hanna, Different books might define a parallelogram differently but I am going to assume that your book defines a parallelogram as "a quadrilateral with both pairs of opposite sides parallel". I am also not sure what facts you know but let me try a proof. I have reproduced your diagram below with two additional constructed line segments. Extend the line segment DC to a point E and extend the line segment DA to a point F.
I can conclude that the sides AB abd CD are parallel if I can show that <CBA and <ECB are congruent. You know that the measure of the four interior angles of a quadrilateral add to 360 degrees, thus
But angles A and C are congruent and angles D and B are congruent. Hence
so
Also DCE is a straight line so measure(<BCD) + measure(<BCE) = 180 degrees. That is
Hence from equations 1 and 2
that is
Hence the sides AB and CD are parallel. Now use the extension of the line segment DA to F to show that the sides AD and CB are parallel. Penny 

