Idiot\'s Guides Basic Math and Pre-Algebra

182 Part 3: The Shape of the World

A parallelogram is a quadrilateral with two pairs of opposite sides parallel. Whenever you look at

one pair of parallel sides, the other sides can be thought of as transversals. You know a bit about

the angles that are formed when parallel lines are cut by a transversal, and you can see that you

have consecutive interior angles that are supplementary.

DEFINITION

A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral in

which both pairs of opposite sides are parallel and congruent.

In parallelogram ABCD, and in any parallelogram, consecutive angles are supplementary.

In ABCD, that means m’A + m’B =18 0r, m’B + m’C =18 0r, m’C + m’D =18 0r, and

m’D + m’A =18 0r. If you do a little algebra, saying that m’A + m’B = m’B + m’C, and

subtracting m’B from both sides, you can show that ’A and ’C are the same size. In any

parallelogram, opposite angles are congruent.

Draw a diagonal in any parallelogram and you form two triangles. Let’s draw diagonal AC in

parallelogram ABCD. That will form 'ACD and 'CAB. Because DCAB, alternate interior

angles ’DCA and ’BAC are congruent. Because AD BC, ’DAC and ’BCA are congruent.

That gives you two pairs of congruent angles, and AC is between those angles in both triangles

and equal to itself. If you rotate 'CAB so that ’BAC sits on top of ’DCA and ’BCA sits on top

of ’DAC, not only will the shared side match itself, but you’ll see AB matching CD and BC

matching AD. That tells you that the opposite sides are not only parallel, but also congruent.

B

A

C

D

D C

A B