How to find angles in a trapezoid

Updated April 17, 2017

In geometry, a trapezoid is a quadrilateral (four-sided figure) in which only one pair of opposite sides are parallel. Trapezoids are also known as trapeziums. The parallel sides of a trapezoid are called the bases. The nonparallel sides are called legs. A trapezoid, like a circle, has 360 degrees. Since a trapezoid has four sides, it has four angles. Trapezoids are named by their four angles, or vertices, such as "ABCD."

Determine if the trapezoid is an isosceles trapezoid. Isosceles trapezoids have a line of symmetry dividing each half. The legs of a trapezoid are equal in length, as are the diagonals. In an isosceles trapezoid, angles that share a base have the same measure. Supplementary angles, which are angles adjacent to opposite bases, have a sum of 180 degrees. These rules can be used to calculate an angle.

List the given measurements. You may be given the measurement of an angle or a base. Or, you may be given the measurement of a mid-segment, which is parallel to both bases and has a length equal to the average of the two bases. Use the given measurements to determine what measurements, if not the angle, can be calculated. These calculated measurements can then be used to calculate the angle.

Recall relevant theorems and formulas for solving measurements of bases, legs and diagonals. For instance, Theorem 53 states that base angles of an isosceles trapezoid are equal. Theorem 54 states that diagonals of an isosceles trapezoid are equal. The area of a trapezoid (whether or not isosceles) is half of the lengths of the parallel sides multiplied by the height, which is the perpendicular distance between the sides. The area of a trapezoid is also equal to the product of the mid-segment and the height.

Draw a right triangle, within the trapezoid, if necessary. The height of a trapezoid forms a right triangle that implicates an angle of the trapezoid. Use measurements, such as the area of the trapezoid, to calculate the height, leg or base that is shared by the triangle. Then solve for the angle using the rules of angle measurement that apply to triangles.

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About the Author

Audrey Farley began writing professionally in 2007. She has been featured in various issues of "The Mountain Echo" and "The Messenger." Farley has a Bachelor of Arts in English from the University of Richmond and a Master of Arts in English literature from Virginia Commonwealth University. She teaches English composition at a community college.