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Geometry

Students define an isosceles 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 an isosceles trapezoid as a trapezoid whose legs are congruent.
- Explain the properties of an isosceles trapezoid (such as, angles on the same base are congruent, the diagonals are congruent, the diagonals divide each other into congruent line segments).
- Explain sufficient conditions for a trapezoid to be isosceles (such as, angles on the same base are congruent and the diagonals divide each other into congruent line segments).
- Apply the properties of isosceles 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.
- Interior angles of a trapezoid which have the same base as a side are known as base angles.
- 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.

Some 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.

An isosceles trapezoid is a trapezoid whose legs are congruent. Isosceles trapezoids have some additional properties:

- Angles on the same base are congruent.
- The diagonals are congruent
- The diagonals divide each other into congruent line segments.

Trapezoids with the following properties are isosceles:

- Angles on the same base are congruent
- The diagonals divide each other into congruent line segments.

angles on the same base are congruent and the diagonals divide each other into congruent line segments.

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; define a trapezoid, its bases, and its legs; explain the properties of a trapezoid; understand angle-side-angle congruency, angle-angle similarity, and the triangle midsegment theorem; and identify corresponding angles and supplementary angles. |

Course | Geometry |

Type of Tutorial | Concept Development |

Key Vocabulary | base of an isosceles trapezoid, congruent angles, isosceles trapezoid |