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°

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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

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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

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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

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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

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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

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PRACTICE PROBLEMS

Define each of the following.1. Concave polygon

2. Regular polygon

3. Quadrilateral

4. Are all quadrilaterals parallelograms? Explain.

5. Are all squares rectangles? Explain.

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PRACTICE PROBLEMS (Cont)

Identify each of the polygons below using at least one of the definitions given in this unit.

7.

9.

6.

8.

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PRACTICE PROBLEMS (Cont)

10. 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

11. 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

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PRACTICE PROBLEMS (Cont)

12. Find the number of degrees in 1 in the polygon shown below.

72°

115°

70°

1

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PRACTICE PROBLEMS (Cont)

13. Determine length CD in the figure at right given that CD || AB.

C

D

A

B

E

F

312 in156 in

332 in166 in

120 in136 in

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PROBLEM ANSWER KEY

1. A polygon in which two or more sides, if extended, cut inside the polygon

2. A polygon that is both equilateral and equiangular; all sides and all angles are equal

3. A polygon with four sides4. No, a parallelogram is a quadrilateral whose

opposite sides are parallel and equal. Not all quadrilaterals have parallel and equal opposite sides

5. Yes, squares are a special type of rectangle whose sides are all equal

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PROBLEM ANSWER KEY (Cont)

6. Regular pentagon7. Trapezoid8. Hexagon9. Parallelogram or quadrilateral10. a. 540° b. 900° c. 1260°11. a. 45° b. 36° c. 40°12. 103°13. 152 inches