Quadrilaterals and Other Polygons Lesson 7 Quadrilaterals and Other Polygons

Quadrilaterals A quadrilateral is a four-sided figure like the one shown. The sides are line segments connected together at their endpoints, called the vertices (plural of vertex) of the quadrilateral. We call this quadrilateral ABCD or BADC or CDAB etc. (We start with a vertex and then proceed around the quadrilateral, either clockwise or counterclockwise.) The line segments shown are called diagonals. D C A B

Rectangles A rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are congruent and parallel. Each diagonal of a rectangle divides the rectangle into two congruent triangles. The diagonals bisect each other and are congruen to each other. The width of a rectangle is the common length of one pair of opposite sides. The length of a rectangle is the common length of another pair of opposite sides. Instead of width and length, we sometimes use base and height. D C A B

Example The length of a rectangle is 12 and the width is 5. What is the length of one of its diagonals? It’s a good idea to draw a picture. The diagonal is the hypotenuse of a right triangle whose legs measure 5 and 12. So, we use the Pythagorean Theorem: c 12 5

Squares A square is a rectangle in which all four sides are congruent. Each diagonal divides the square into two 45-45-90 triangles. Both diagonals together divide the square into four 45-45-90 triangles. If a side of the square measures then each diagonal measures The diagonals bisect each other and are congruent to each other.

Example In the figure, ABCD and AFDE are squares. If then what is First note that is a 45-45-90 triangle. So, So, But So, too. A B C D E F G P

Parallelograms A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Each pair of opposite sides in a parallelogram are not only parallel, but congruent as well. Each diagonal divides the parallelogram into two congruent triangles. The diagonals of a parallelogram bisect each other. Opposite angles of a parallelogram are congruent. Consecutive angles of a parallelogram are supplementary.

Rhombi A rhombus (plural rhombi) is a parallelogram with four congruent sides. Since a rhombus is a parallelogram, it has all the properties a parallelogram has. In addition, the diagonals of a rhombus are perpendicular to each other and they divide the rhombus into four congruent right triangles.

Trapezoids A trapezoid is a quadrilateral such that one pair of opposite sides is parallel. The parallel sides are called the bases of the trapezoid. The nonparallel sides are called the legs of the trapezoid. If the legs of a trapezoid are congruent, the the trapezoid is called isosceles. In an isosceles trapezoid, the diagonals bisect each other.

Kites A kite is a quadrilateral that has two pairs of congruent sides. The congruent sides are consecutive, not opposite. A kite has one pair of opposite angles that are congruent. One of the diagonals bisects the other. The diagonals are perpendicular to each other.

Polygons A polygon is a closed figure, like the one shown, made up of joining line segments together at their endpoints. Triangles and quadrilaterals are examples of polygons. A polygon with 5 sides is called a pentagon. A polygon with 6 sides is called a hexagon. A polygon with 7 sides is called a heptagon. A polygon with 8 sides is called an octagon. A polygon with 9 sides is called a nonagon. A polygon with 10 sides is called a decagon. A regular polygon is a polygon in which all of its sides are congruent and all of its interior angles are congruent.

Interior Angles of a Polygon If a polygon has n sides, then the sum of the measures of its n angles is Each interior angle of a regular polygon with n sides measures

Exterior Angles of a Polygon An exterior angle of a polygon is formed by a side of the polygon and an extension of an adjacent side. in the figure is an exterior angle of a regular hexagon. An exterior angle is the supplement of an interior angle. In any polygon if we draw one exterior angle at each vertex then the sum of the measures of these exterior angles is 1

Diagonals of a Polygon A diagonal of a polygon is a line segment connecting two non-consecutive vertices. For example, a triangle has no diagonals and a quadrilateral has two diagonals. In general, if a polygon has n sides, then it has diagonals.