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Parallelograms

A quadrilateral is a polygon that has four sides.

A unique property of polygons you learned in the previous lesson is that the interior angles of a convex quadrilateral have a sum of 360°.

S = 180 °(4 - 2) or S = 360°

87°+139°+93°+41°=360°

A 4 sided figure with angles 87, 139, 93, and 41

90°+90°+90°+90°=360°

A 4 sided figure with all 90 degree angles

44°+136°+136°+44°=360°

A 4 sided figure with angles 89, 111, 99, and 61.

89°+111°+99°+61°=360°

A 4 sided figure with angles 44, 136, 136, and 44

Parallelograms

The first unique quadrilateral that you will learn is parallelogram. The key word in "parallelogram" is parallel.

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

Remember that arrow marks are symbols used to identify parallel lines and segments. Matching arrows identify which lines are parallel.

A quadrilateral with opposite sides parallel to each other

Special Parallelograms

These figures are also parallelograms. Do you know the names of these?

The first figure has 4 right angles and opposite sides parallel. The second figure has 4 equal sides with opposite angles congruent. The third figure has 4 right angles and 4 congruent sides.

Answer: Rectangle, Rhombus, Square

For this lesson, we will use the shape of the parallelogram on the previous slide because the above parallelograms will be discussed in further detail later in the unit.

Properties - Opposite Sides

The following are properties based on theorems about parallelograms. Write these properties down for future use.

In a parallelogram, opposite sides are congruent.

line CF ≅ line DE

line FE ≅ line CD

Parallelogram with vertices C D E F

Properties - Opposite Angles

In a parallelogram, opposite angles are congruent.

∠c ≅ ∠e

∠f ≅ ∠d

Properties - Consecutive Angles

In a parallelogram, consecutive angles (angles that share a side) are supplementary.

m∠f + m∠e = 180°

m∠e + m∠d = 180°

m∠d + m∠c = 180°

m∠c + m∠ f = 180°

Properties - Diagonals

In a parallelogram, diagonals bisect (divide into two equal sections) each other.

Parallelogram with vertexes C D E F. The diagonals D F and C E intersect at point L.

Solving Parallelogram Problems

When solving problems with parallelograms, pay close attention to the given information, such as k = ∠120 in the next example. When the measurement of a line or angle is given, that means you will need that information to solve the parallelogram. To solve the following problem and find ∠L, you needed to know that consecutive angles in a parallelogram are supplementary. This means that any two angles that are connected by a single line segment, like ∠L and ∠k, added together equal 180°. Opposite angles, like ∠L and ∠n, are equal.

Example #1

Open Apply Properties of Parallelograms to Find Unknown Angle Measures in a new tab

Brightstorm: Parallelogram Properties is a video that explains the properties you learned in this lesson along with how to solve problems similar to the previous problems we covered. Problem 3 is an excellent example to help you solve task problems that involve two variables.