## 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, *a* and *c* are vertical angles, and *b* and *d* are vertical angles. Vertical angles are **congruent,** that is, they have the same measure.

**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, *a* and *b*, *b* and *c*, *c* and *d*, and *d* and *a*−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, *a* and *c* are acute angles, and *b* and *d* are obtuse angles. Angles whose measures add to 180 degrees are **supplementary angles.** In the diagram above, *a* and *b* are supplementary angles.

An angle with measure 90 degrees, such as *RSU* below, is a **right angle.** Two angles whose measures add to 90 degrees, such as *m* and *n* below, are **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, *AOC* is a 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.

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.

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 *A* is congruent to *W,* *D* is congruent to *Z,* and so on. Also, side *AD* is congruent to side *WZ,* side *DC* is congruent to side *ZY,* and so on.

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.

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.

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'.*

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'.*