Parallel Lines        Congruent Tri.        Congruent R. Tri.        Isosc. and Equil.        Quadrilaterals    Parallelograms        Ratios        Similar Polygons        Special Triangles       Circles        Area        Coordinate Geo.        Triangle Ineq.        Solids        Computer Fun On this page, we hope to clear up problems that you might have with parallelograms.  A parallelogram is a special kind of quadrilateral.  There are many special rules and theorems that apply to parallelograms only.  By scrolling down or clicking on the link below, you will be on your way to understanding parallelograms! How to tell if a quadrilateral is a parallelogram Quiz on Parallelograms A parallelogram is so named because it has two pairs of opposite sides that are parallel. There are four theorems that apply to parallelograms only.  They are outlined below. 1.  A diagonal of any parallelogram forms two congruent triangles.  Example: ``` Problem: Prove triangle ABC is congruent to triangle CDA. Solution: Since the figure is a parallelogram, segment AB is parallel to segment DC and the two segments are also congruent. Angle 2 is congruent to angle 4 and angle 1 is congruent to angle 3. This is true because alternate interior angles are congruent when parallel lines are cut by a transversal. Segment AC is congruent to segment CA by the Reflexive Property of Congruence, which says any figure is congruent to itself. Triangle ABC is congruent triangle CDA by Angle-Side-Angle``` 2.  Both pairs of opposite sides of a parallelogram are congruent. 3.  Both pairs of opposite angles of a parallelogram are congruent. 4.  The diagonals of any parallelogram bisect each other.  Example: ``` Problem: Prove segment AE is congruent to segment CE and segment DE is congruent to segment BE. Solution: By the definition of a parallelogram, segment AD and segment BC are parallel and congruent. Angle 1 is congruent to angle 3 and angle 2 is congruent to angle 4. This is true because alternate interior angles are congruent when parallel lines are cut by a transversal. Triangle AED and triangle CEB are congruent by Angle-Side-Angle. The segments we were asked to prove as congruent are congruent by CPCTC.``` In this section, we hope to clear up problems associated with figuring out if a given quadrilateral is a parallelogram.  Most of the theorems that help us figure out if a shape is a parallelogram are the converses of the theorems stated above. The three theorems that tell us how to find a parallelogram are outlined below. 1.  If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram 2.  If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 3.  If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram. Take the Quiz on parallelograms.  (Very useful to review or to see if you've really got this topic down.)  Do it!

Math for Morons Like Us - Geometry: Parallelograms
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