Quadrilaterals and Other Polygons
Lessons in this chapter
About Quadrilaterals and Other Polygons
Quadrilaterals and Other Polygons extends angle and side relationships from triangles to four-sided figures and beyond.
Angles of Polygons finds the angle sum of any polygon by splitting it into triangles, before Properties of Parallelograms establishes the defining features every parallelogram shares: opposite sides parallel and equal, opposite angles equal, diagonals bisecting each other.
Tests for Parallelograms reverses that logic, using a subset of those properties to prove a given quadrilateral IS a parallelogram, while Properties of Special Parallelograms narrows in on rectangles, rhombuses, and squares.
Trapezoids and Kites closes the chapter with the two common quadrilaterals that aren't parallelograms, each with its own distinct set of properties.
All five lessons here open as Geometry worksheets in the page, including Properties of Parallelograms, Tests for Parallelograms and Trapezoids and Kites. Each question is scored as you answer it, which keeps the two halves of this chapter straight: what you may assume once a shape is named a parallelogram, and what you have to show before you may name it one. The worksheets are free to try, with no login needed for your first 3.
Frequently asked questions
Angles of Polygons gives the interior and exterior angle sums for a polygon with any number of sides. Properties of Parallelograms lists what follows once a shape is known to be a parallelogram, and Tests for Parallelograms runs the argument the other way, giving the conditions that prove a quadrilateral is one. Properties of Special Parallelograms separates rhombuses, rectangles and squares, and Trapezoids and Kites covers the two quadrilaterals that are not parallelograms. The chapter sits in the high school Geometry conceptual category (G).
Split the polygon into triangles by drawing diagonals from one vertex, then multiply the number of triangles by 180°: a polygon with n sides always splits into (n-2) triangles.
In every parallelogram, opposite sides are parallel and equal in length, opposite angles are equal, consecutive angles are supplementary, and the diagonals bisect each other.
Show that just ONE of several sufficient conditions holds, like both pairs of opposite sides being equal, or the diagonals bisecting each other: any one of several specific tests is enough to prove it, without needing to check every property.
A rectangle adds four right angles and equal diagonals; a rhombus adds four equal sides and perpendicular diagonals; a square (both a rectangle and a rhombus) has all of these properties at once.
A trapezoid has exactly one pair of parallel sides; a kite has two pairs of adjacent (not opposite) equal sides and no parallel sides at all: they're both non-parallelogram quadrilaterals but with very different structures.