Parallelograms are four-sided shapes that have two pairs of parallel sides. Rectangles, squares and rhombuses are all classified as parallelograms. The classic parallelogram looks like a slanted rectangle, but any four-sided figure that has parallel and congruent pairs of sides can be classified as a parallelogram. Parallelograms have six key properties that distinguish them from other shapes.
Opposite Sides Are Congruent
Opposite sides of all parallelograms -- including rectangles and squares -- must be congruent. Given parallelogram ABCD, if side AB is on the top of the parallelogram and is 9 centimeters, side CD on the bottom of the parallelogram must also be 9 centimeters. This also holds true for the other set of sides; if side AC is 12 centimeters, side BD, which is opposite of AC, must also be 12 centimeters.
Opposite Angles Are Congruent
Opposite angles of all parallelograms -- including squares and rectangles -- must be congruent. In parallelogram ABCD, if angles B and C are located in opposite corners -- and angle B is 60 degrees -- angle C must also be 60 degrees. If angle A is 120 degrees -- angle D, which is opposite angle A -- must also be 120 degrees.
Consecutive Angles Are Supplementary
Supplementary angles are a pair of two angles whose measures add up to 180 degrees. Given parallelogram ABCD above, angles B and C are opposite and are 60 degrees. Therefore, angle A -- which is consecutive to angles B and C -- must be 120 degrees (120 + 60 = 180). Angle D -- which is also consecutive to angles B and C -- is also 120 degrees. Additionally, this property supports the rule that opposite angles must be congruent, as angles A and D are found to be congruent.
Right Angles in Parallelograms
Although students are taught that four-sided figures with right angles -- 90 degrees -- are either squares or rectangles, they are also parallelograms, but with four congruent angles instead of two pairs of two congruent angles. In a parallelogram, if one of the angles is a right angle, all four angles must be right angles. If a four-sided figure has one right angle and at least one angle of a different measure, it is not a parallelogram; it is a trapezoid.
Diagonals in Parallelograms
Parallelogram diagonals are drawn from one opposite side of the parallelogram to the other. In parallelogram ABCD, this means that one diagonal is drawn from vertex A to vertex D and another is drawn from vertex B to vertex C. When drawing the diagonals, students will find that they bisect each other, or meet at their midpoints. This occurs because the opposite angles of a parallelogram are congruent. The diagonals themselves will not be congruent to each other unless the parallelogram is also a square or a rhombus.
In parallelogram ABCD, if a diagonal is drawn from vertex A to vertex D, two congruent triangles, ACD and ABD, are created. This also holds true when drawing a diagonal from vertex B to vertex C. Two more congruent triangles, ABC and BCD, are created. When both diagonals are drawn, four triangles are created, each with a midpoint E. However, these four triangles are only congruent if the parallelogram is a square.
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