Question

Trapezoids measuring and recording the angles in it, and writing a definition of it using its angle properties

Answer 1

A trapezoid is a quadrilateral in which at least one pair of opposite sides is parallel.

Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:

The degree measure of the four angles add up to 360 degrees. 

It has two adjacent angles that are supplementary, that is, they add up to 180 degrees.

The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.

The angles in the quadrilateral ABCD satisfy $\sin A\sin C=\sin B\sin D.$

The cosines of two adjacent angles sum to 0, as do the cosines of the other two angles.

The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.