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Geometry

Students define a trapezoid, explore its properties and their proofs, and use these properties to solve problems.

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After completing this tutorial, you will be able to complete the following:

- Define a trapezoid as a quadrilateral with exactly one pair of parallel sides.
- Explain the properties of a trapezoid (such as adjacent angles of a trapezoid are supplementary, the median of a trapezoid is parallel to the bases and its length is the average of the base lengths, the diagonals cut each other in the same ratio, and so on).
- Apply the properties of trapezoids.

Trapezoids are quadrilaterals with one pair of parallel sides. There has been some debate about the number of parallel sides allowed in a trapezoid. If a trapezoid is defined as having at least one pair of parallel sides, then all parallelograms would also be considered to be trapezoids. However, it is more common for trapezoids to be defined as having exactly one pair of parallel sides, and this is the definition applied here. As a result, the hierarchy of quadrilaterals is as follows:

Trapezoids derive their name from the ancient Greek "trapezion," meaning "little table," and are sometimes called "trapeziums" in countries outside North America.

When describing a trapezoid, we use the following definitions:

- A trapezoid is a quadrilateral with exactly one pair of parallel sides.
- The parallel sides of a trapezoid are known as its bases.
- The nonparallel sides of a trapezoid are called legs.
- A median (also known as a midsegment) is a line segment connecting the midpoints of two legs of a trapezoid.

The legs of a trapezoid are transversals of the bases and many of the properties of a trapezoid are derived from this fact along with facts about triangles.

Properties of a trapezoid:

- Two adjacent angles of a trapezoid are supplementary.
- The median is parallel to the bases.
- The median's length is half the sum of the bases.
- The diagonals divide each other in the same ratio.

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should be able to define and explain the properties of a quadrilateral; identify the diagonal of a quadrilateral; understand the angle-side-angle congruency, angle-angle similarity, and triangle midsegment theorems; and identify corresponding and supplementary angles. |

Course | Geometry |

Type of Tutorial | Concept Development |

Key Vocabulary | base of a trapezoid, trapezoid, interior angles of a trapezoid |