Math Background

Properties of Polygons: Overview

A polygon is a simple closed plane figure made up of three or more line segments. Examples of polygons can be found everywhere. Doors, chalkboards, and walls are usually rectangular in shape. Triangles can be found in buildings, bridges, and other structures. Squares can be found in tiles on a floor or a ceiling. Hexagons and other polygons can be found in signs, in windows, and in nature. Understanding the properties of polygons can help students make sense of the world around them.

It is important that students understand the geometric terminology associated with polygons. A point is a location in space. A line is a straight, continuous, unending set of points. Since a point has no dimensions and a line has only one dimension, it is impossible to see either of them. However, we can see representations of these ideas. For example, the corner of a room where two walls and the ceiling come together or the end of a sharp pencil could represent a point. A line segment is a part of a line with two endpoints. The part of a room where two walls meet and the edge of a ruler are representations of line segments. A plane is a collection of points that forms a flat, continuous, and unending surface (imagine a wall or the surface of a book cover continuing forever in all directions). Space is the collection of all points.

A ray is a half-line−a straight, continuous, and unending set of points with one endpoint. If two rays have a common endpoint they form an angle, and the common endpoint is called the vertex of the angle. When two lines intersect, the angles opposite one another are called vertical angles. In the diagram below, anglea and anglec are vertical angles, and angleb and angled are vertical angles. Vertical angles are congruent, that is, they have the same measure.

intersecting lines

Adjacent angles are angles that share a common vertex and a common ray and whose interiors do not intersect. The angles next to each other in the diagram above-namely, anglea and angleb, angleb and anglec, anglec and angled, and angled and anglea−are adjacent angles.

Acute angles have measures less than 90 degrees. Obtuse angles have measures greater than 90 degrees but less than 180 degrees. In the diagram above, anglea and anglec are acute angles, and angleb and angled are obtuse angles. Angles whose measures add to 180 degrees are supplementary angles. In the diagram above, anglea and angleb are supplementary angles.

An angle with measure 90 degrees, such as angleRSU below, is a right angle. Two angles whose measures add to 90 degrees, such as anglem and anglen below, are complementary angles.

complementary angles

A central angle is an angle whose vertex is the center of the circle and whose sides intersect the circle. In the diagram below, angleAOC is a central angle.

central angle

A triangle is a polygon with three sides. The triangle does not include the region enclosed by its sides; that region is called the interior of the triangle. One way to classify triangles is by the lengths of their sides. A scalene triangle, such as triangle RST below, has three sides with different lengths. A triangle with at least two congruent sides is an isosceles triangle. In the diagram below, triangles ABC and XYZ are both isosceles triangles. Triangle ABC is also called an equilateral triangle, since all three of its sides are congruent. In fact, every equilateral triangle is an isosceles triangle.

scalene triangle, isosceles triangle, equilateral triangle

Triangles can also be classified according to their angle measures. A right triangle has one right angle. An acute triangle has three acute angles. An obtuse triangle has one obtuse angle. The sum of the angle measures for any triangle is 180 degrees. If we combine these two ways to classify triangles, we come up with seven different types of triangles: acute scalene, right scalene, obtuse scalene, acute isosceles, right isosceles, obtuse isosceles, and acute equilateral. There are no right equilateral or obtuse equilateral triangles.

A quadrilateral is a polygon with four sides. There are many types of quadrilaterals, and many of them share some common properties. A quadrilateral with exactly one pair of parallel sides is called a trapezoid. In the diagram below, ABCD is a trapezoid. If a quadrilateral has two pairs of parallel sides, then it is called a parallelogram. In a parallelogram the opposite sides and opposite angles are congruent. All of the figures below except ABCD are parallelograms. A rectangle is a parallelogram with four right angles. In the diagram below, figures IJKL and QRST are both rectangles. A parallelogram with four congruent sides is called a rhombus. Figures MNOP and QRST below are rhombuses. A square is a rectangle that is also a rhombus. In other words, a square is a parallelogram with four right angles and four congruent sides. Figure QRST is a square.

trapezoid, parallelogram, rectangle, rhombus, square

Two polygons are congruent if they are the same size and shape. If two figures are congruent, then their corresponding parts are also congruent. For example, quadrilateral ABCD is congruent to quadrilateral WXYZ, so angleA is congruent to angleW, angleD is congruent to angleZ, and so on. Also, side AD is congruent to side WZ, side DC is congruent to side ZY, and so on.

congruent polygons

Polygons that have the same shape, but that are not necessarily the same size, are said to be similar. For example, all equilateral triangles are similar to one another and all squares are similar to one another. If two figures are similar, the corresponding angles are congruent, and the corresponding sides are proportional. All congruent figures are similar, but not all similar figures are congruent. In the figure below, quadrilateral ABCD is similar to quadrilateral MNOP, since the corresponding angles are congruent and the corresponding sides are proportional, with a ratio of 3 to 2.

similar polygons

A geometric transformation changes the position of a figure. A translation slides a figure along a straight line in one direction. Translations are sometimes referred to as slides. In the diagram below, PQRS was translated to HIJK. The figure resulting from a translation is congruent to the original figure.

translated figure

A rotation rotates, or turns, a figure about a given point. Rotations are sometimes referred to as turns. A figure and its image after a rotation are congruent to one another. In the diagram below, PQRS has been rotated 90 degrees counterclockwise about point P to figure PQ'R'S'.

rotated figures

A reflection is a transformation that reflects a figure across a line. The figures resulting from a reflection, or flip, are congruent to the original figures. In the figure below, ABCD has been reflected about line MN to figure A'B'C'D'.

reflected figures

Houghton Mifflin Math Grade 6