Two-Dimensional Figures

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Presentation transcript:

Two-Dimensional Figures Polygons & Angles of Polygons

Identify and name polygons. Find the sum of interior angles and a single interior angle of a regular polygon. Find the sum of exterior angles and a single exterior angle in a convex polygon. polygon concave convex n-gon Regular polygon

Number of Sides Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 12 dodecagon n n-gon

Tetracontakaihexagon What is an 46 sided figure called? Hint- go to the website http://www.math.com/tables/geometry/polygons.htm The answer is… Tetracontakaihexagon

Convex or Concave Polygons

Regular Polygon is… A convex polygon where: 1. All sides are congruent 2. All angles are congruent

Regular Hexagon Is it convex? All sides congruent? 4 120° 120° 4 4 Yes All sides congruent? 120° 120° 4 4 Yes 120° 120° 4 All angles congruent? Yes It’s a regular hexagon!

A. Name the polygon? quadrilateral B. Is it convex or concave? convex C. Is it regular or irregular? irregular

A. Name the polygon? nonagon B. Is it convex or concave? concave C. Is it regular or irregular? irregular

Interior Angle Sum (must be convex polygon) Interior Angle Sum Practical Method Triangle Quadrilateral Pentagon Hexagon 180° 180° 180° 180° 180° 180° 180° 180° 180° 180° Interior Angle Sum 180° 360° 540° 720°

Interior Angle Sum (must be convex polygon) S = mA + mB + mC + mD + mE + mF S = 180(n – 2) A B S = 180(6 – 2) F C S = 180(4) S = 720° E D

Interior Angle Sum Example (must be convex polygon) S = 180(n – 2) S = 180(5 – 2) S = 180(3) S = 540°

Single Interior Angle of a Regular Polygon (must be convex polygon) x = 180(n – 2) n x = 180(6 – 2) 6 A B 120° 120° x = 720 6 F C 120° 120° 120° 120° x = 120° E D

Single Interior Angle of a Regular Polygon (must be convex polygon) x = 180(n – 2) n x = 180(3 – 2) 3 x = 180 3 x = 60°

Exterior Angle Sum (must be convex polygon) 60° 120° 60° 90° 90° 60° Hexagon Quadrilateral Triangle 90° 60° 120° 90° 120° 60° 60° Always add up to 360°

Single Exterior Angle of a Regular Polygon 60° 120° 60° 90° 90° 60° Hexagon Quadrilateral Triangle 90° 60° 120° 90° 120° 60° 60° 120° 90° 60° 3 4 6

Given regular nonagon ABCDEFGHJ. A. Exterior Angle Sum? 360° B. Single Exterior Angle? 360/9= 40° (9 – 2)180 = 1260° (n - 2)180 = C. Interior Angle Sum? D. Single Interior Angle?

Multiply both sides by n ALGEBRAIC METHOD The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. 135 = 180(n – 2) n Multiply both sides by n 135n = 180(n – 2) 135n = 180n – 360 0 = 45n – 360 360 = 45n 8 = n Answer: The polygon has 8 sides.

= 8 sides 135˚ 45˚ OFF the CHAIN METHOD The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.   = 8 sides 135˚ 45˚

The measure of an interior angle of a regular polygon is 144 The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. A. 12 B. 9 C. 11 D. 10 A B C D

Find the measure of each interior angle. 60 3x x 180 180 60 60 How many sides? 5 Find interior angle sum = 180(5 – 2)= 180(n – 2) 540 540 = 3x + 3x + x + x + x 540 = 9x 60 = x

Find the value of x. A. x = 7.8 B. x = 22.2 C. x = 15 D. x = 10 A B C

All of the diagonals from one vertex are drawn for the regular polygons in the table to the right.   Fill in the table in order to find a pattern. How many triangles are formed by drawing all of the diagonals from one vertex of a 23-sided polygon? A 19 triangles B 20 triangles C 21 triangles D 23 triangles