1. Unit 24
POLYGONS

2. TYPES OF POLYGONS
A polygon is a closed plane figure formed by three or more straight line segments
A convex polygon is a polygon in which no side, if extended, cuts inside the polygon
A concave polygon is a polygon in which two or more sides, if extended, cut inside the polygon

3. TYPES OF POLYGONS
An equilateral polygon is a polygon with all sides equal
An equiangular polygon is a polygon with all angles equal
A regular polygon is a polygon that is both equilateral and equiangular; all sides and all angles are equal
A quadrilateral is a polygon with four sides

4. TYPES OF QUADRILATERALS
A trapezoid is a quadrilateral that has only two sides parallel
A parallelogram is a quadrilateral whose opposite sides are parallel and equal
A rectangle is a special type of parallelogram whose angles each equal 90°

5. TYPES OF QUADRILATERALS
A square is a special type of rectangle whose sides are all equal
A rhombus is a special type of parallelogram whose sides are all equal, but whose angles are not all equal
A pentagon is a polygon with five sides
A hexagon is a polygon with six sides
An octagon is a polygon with eight sides

6. INTERIOR ANGLES OF A POLYGON
The sum of the interior angles of a polygon of n sides is equal to (n – 2) × 180°
Find the sum of the interior angles of the figure shown below:
The figure has six sides, so it is a hexagon
Since there are six sides, the sum of the
interior angles would be:
(6 – 2)  180 = 4  180 = 720 Ans

7. EXTERIOR ANGLES
If a side of a polygon is extended, the angle between the extended side and the adjacent side of the polygon is an exterior angle
The sum of the exterior angles of any polygon, formed as each side is extended in succession, is equal to 360

8. EXTERIOR ANGLES EXAMPLE
Find the number of sides of a regular polygon having an exterior angle of 24°
Since the sum of the exterior angles of any polygon is 360°, divide 360° by 24°
360°  24° = 15 sides Ans

9. MEDIAN OF A TRAPEZOID
A median of a trapezoid is a line that joins the midpoints of the nonparallel sides (legs)
The median of a trapezoid is parallel to the bases and equal to one half the sum of the bases A B E F C D
Line EF is the median

10. MEDIAN OF A TRAPEZOID EXAMPLE
Find the length of EF in the trapezoid shown below given that AB || CD, EF is the median, AB = 40 m, and CD = 26 m A B E F C D
Since EF is the median, it is equal to one
half the sum of the bases (AB plus CD)
EF = 1/2(40 m + 26 m) = 33 m Ans

11. PRACTICE PROBLEMS Define each of the following. Concave polygon
Regular polygon
Quadrilateral
Are all quadrilaterals parallelograms? Explain.
Are all squares rectangles? Explain.

12. PRACTICE PROBLEMS (Cont)
Identify each of the polygons below using at least one of the definitions given in this unit. 6. 7. 8. 9.

13. PRACTICE PROBLEMS (Cont)
Find the number of degrees in the sum of the interior angles of each of the following figures.
a. pentagon b. 7-sided polygon c. 9-sided polygon
Find the number of degrees in each exterior angle of the regular figures given below.
a. octagon b. 10-sided polygon c. 9-sided polygon

14. PRACTICE PROBLEMS (Cont)
Find the number of degrees in 1 in the polygon shown below.
72°
115°
70° 1

15. PRACTICE PROBLEMS (Cont)
Determine length CD in the figure at right given that CD || AB. C D A B E F
312 in
156 in
332 in
166 in
120 in
136 in

16. PROBLEM ANSWER KEY
A polygon in which two or more sides, if extended, cut inside the polygon
A polygon that is both equilateral and equiangular; all sides and all angles are equal
A polygon with four sides
No, a parallelogram is a quadrilateral whose opposite sides are parallel and equal. Not all quadrilaterals have parallel and equal opposite sides
Yes, squares are a special type of rectangle whose sides are all equal

17. PROBLEM ANSWER KEY (Cont)
Regular pentagon
Trapezoid
Hexagon
Parallelogram or quadrilateral
a. 540° b. 900° c °
a. 45° b. 36° c. 40°
103°
152 inches