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Angles of Polygons

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**Polygons can be CONCAVE or CONVEX**

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**Concave and Convex Polygons**

If a polygon has an indentation (or cave), the polygon is called a concave polygon. Any polygon that does not have an indentation is called a convex polygon. Any two points in the interior of a convex polygon can be connected by a line segment that does not cut or cross a side of the polygon. Concave polygon Convex polygon We will only be discussing CONCAVE polygons

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**Triangle Octagon Quadrilateral Nonagon Pentagon Decagon Dodecagon**

Hexagon n-gon Heptagon Hendecagon

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**Important Terms A VERTEX is the point of intersection of two sides F A**

CONSECUTIVE VERTICES are two endpoints of any side. F A B C D E A segment whose endpoints are two nonconsecutive vertices is called a DIAGONAL. Sides that share a vertex are called CONSECUTIVE SIDES.

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Tear off two vertices….

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**Line up the 3 angles (all vertices touching)**

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A straight line = 180°

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**ALWAYS!!! Angle sum of a Triangle 180° <1 + <2 + <3 = 180° 2**

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**Consider a Quadrilateral**

What is the angle sum? <1 + <2 + <3 + <4 = ?

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**Quadrilateral Draw a diagonal…what do you get? Two triangles 2 3 5 1 4**

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**Quadrilateral 180° 180° Each triangle = 180°**

2 3 5 180° 180° 1 4 Therefore the two triangles together = 360° 6

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**Angle sum of a Quadrilateral**

180° + 180° = 360°

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Consider a Pentagon What is the angle sum?

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Pentagon Draw the diagonals from 1 vertex How many triangles?

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**Angle sum of a Pentagon 180° 180° 180°**

Draw the diagonals from 1 vertex 180° 180° 180°

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**Continue this process through Decagon**

Draw the diagonals from 1 vertex

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**Continue this process through Decagon**

Draw the diagonals from 1 vertex

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**What about a 52-gon? What is the angle sum? Can you find the pattern?**

1 180° 2 360° 3 540° 4 720° 5 900° 6 1080°

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Find the nth term 7 1260° 8 1440° n - 2 (n – 2)(180)

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**pentagon 5(20) - 5 m1 = = 95 Find m1. (4x + 15) 2 (5x - 5)**

3 110 (5x - 5) (4x + 15) (8x - 10) m1 = 5(20) - 5 = 95 540 17x + 200= 540 17x = 340 5x x x = x = 20

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**More important terms Interior Angles**

Exterior Angles the SUM of an interior angle and it’s corresponding exterior angle = 180o

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**Sums of Exterior Angles**

1 2 3 4 5 6 180 180 180•3 = 540 180 Sum of Interior & Exterior Angles = 540 Sum of Interior Angles = 180 Sum of Exterior Angles = 540- 180= 360

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**Sums of Exterior Angles**

180 180 180 180•4 = 720 180 Sum of Interior & Exterior Angles = 720 Sum of Interior Angles = 360 Sum of Exterior Angles = 720- 360= 360

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**Sums of Exterior Angles**

Sum of Interior & Exterior Angles = 180•5 = 900 Sum of Interior Angles = 180•3 = 540 Sum of Exterior Angles = 360 900- 540=

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**What conclusion can you come up with regarding the exterior angle sum of a CONVEX n-polygon??**

Sum of Interior & Exterior Angles = 180n Sum of Interior Angles = 180(n-2) = 180n - 360 Sum of Exterior Angles = 180n – (180n – 360)

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**The exterior angle sum of a CONVEX polygon =**

360°

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**Important Terms EQUILATERAL - All sides are congruent**

EQUIANGULAR - All angles are congruent REGULAR - All sides and angles are congruent

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**Interior Angle Measure of a REGULAR polygons**

60° 90° Equilateral Triangle Angle measure = 60° Square Angle measure = 90° These are measurement that we generally know at this time, But what about the other regular polygons? How do we calculate the interior angle measure?

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Pentagon 72° 108° 72° 108° 108° 72° 72° 72° 108° 108°

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**Interior Angle Measure of a REGULAR polygons**

108° 120° Calculate by: Angle Sum Number of sides 135°

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