Properties and Attributes of Polygons

Name: Properties and Attributes of Polygons
Uploaded: 2017-07-07T01:01:18+00:00
Duration: PTM21S14
Channel: Alvin Fitzgerald
Description: Properties and Attributes of Polygons

Properties and Attributes of Polygons
Geometry Properties and Attributes of Polygons CONFIDENTIAL

Warm up Solve by factoring: 1) x2 + 3x – 10 = 0 2) x2 - x – 12 = 0
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Today you will learn about the parts of polygon and the ways to classify polygons. Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal. A E B C D side vertex diagonal CONFIDENTIAL

Number of Slides Name of Polygon
You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE in the previous slide is a pentagon. Number of Slides Name of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n - gon CONFIDENTIAL

Identifying Polygon Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides: Polygon Octagon Polygon Pentagon Not a Polygon CONFIDENTIAL

Now you try! Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides: 1a 1b 1c 1a) not a polygon ; 1b) polygon; decagon ; 1c) not a polygon CONFIDENTIAL

concave quadrilateral
All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. convex quadrilateral concave quadrilateral CONFIDENTIAL

Classifying Polygons Tell whether each polygon is regular or irregular. Tell whether it is concave or convex: A irregular convex Next page -> CONFIDENTIAL

Tell whether each polygon is regular or irregular
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex: C B regular convex irregular concave CONFIDENTIAL

Now you try! Tell whether each polygon is regular or irregular. Tell whether it is concave or convex: 2a 2b 2a) regular convex ; 2b) irregular concave CONFIDENTIAL

To find the sum of the interior angles measure of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum measures of the polygon. Quadrilateral Triangle Pentagon Hexagon CONFIDENTIAL

Sum of Interior Angle Measures
Polygon Number of Slides Number of Triangles Sum of Interior Angle Measures Triangle 3 1 (1) 180° = 180° Quadrilateral 4 2 (1) 180° = 360° Pentagon 5 (1) 180° = 540° Hexagon 6 (1) 180° = 720° n - gon n n - 2 (n - 2) 180° In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n - 2) 180°. CONFIDENTIAL

Polygon Angle Sum Theorem
The sum of the interior angle measures of a convex polygon with sides n is (n - 2) 180°. CONFIDENTIAL

Finding Interior Angle Measures and Sums in Polygons
A) Find the sum of the interior angle measures of a convex octagon. (n - 2) 180° = (8 - 2) 180° = 1080° Polygon ∕ Sum thm. An octagon has 8 sides. So, substitute 8 for n. Simplify. CONFIDENTIAL

B) Find the measure of each interior angle of a regular nonagon. Step1: Find the sum of the interior angle measures. (n - 2) 180° = (9 - 2) 180° = 1260° Polygon ∕ Sum thm. Substitute 9 for n. Simplify. Step2: Find the measure of one interior angle. 1260° = 140° 9 The int. ∕s are congruent, so divide by 9. CONFIDENTIAL

Q R S c° 3c° C) Find the measure of each interior angle of a quadrilateral PQRS. (4 - 2) 180° = 360° m∕P + m∕Q + m∕R + m∕S = 360° c + 3c + c + 3c = 360° 8c = 360° => c = 45° Polygon ∕ Sum thm. Polygon ∕ Sum thm. Substitute. m∕P =m∕R = 45° m∕Q = m∕S = 360° CONFIDENTIAL

Now you try! 3a) Find the sum of the interior angle measures of a convex 15 - gon. 3b) Find the measure of each interior angle of a regular decagon. 3a) 2340° ; 3b) 144° CONFIDENTIAL

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measure is 360°. 81° 132° 147° 41° 55° 110° 43° 111° 147° + 81° + 132° = 360° 43° + 111° + 41° + 55° + 110° = 360° CONFIDENTIAL

Polygon Exterior Angle Sum Theorem
The sum of the exterior angle measures , one angle at each vertex, of a convex polygon with sides n is 360°. CONFIDENTIAL

Finding Exterior Angle Measures in Polygons
A) Find the measure of each exterior angle of a regular hexagon. A hexagon has 6 sides and 6 vertices. Sum of the exterior angle = 360° Measure of one exterior angle = 360° = = 60° 6 Polygon ext ∕ Sum thm. A regular hexagon has 6 ext ∕s. So, divide the sum by 6. The measure of each exterior angle of a regular hexagon = 60° CONFIDENTIAL

Finding Exterior Angle Measures in Polygons
B) Find the value of a in polygon RSTUV. 2a° 3a° 6a° 7a° R S T U V 7a° + 2a° + 3a° + 6a° + 2a° = 360° 20a°= 360° a = 60° Polygon ext ∕ Sum thm. Combine like terms. So, divide the sum by 20. CONFIDENTIAL

4b) Find the value of r in polygon JKLM.
Now you try! 4a) Find the sum of the measures of exterior angle of a regular dodecagon. 4b) Find the value of r in polygon JKLM. J K L M 4r° 7r° 5r° 8r° 4a) 30°; 4b) 15° CONFIDENTIAL

Photography Application
The appearance of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is the measure of ∕CBD? B D A C ∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior measures is 360°. m ∕CBD = 360° =36° 10 A regular decagon has 10 congruent ext. angles. So, divide the sum by 10. CONFIDENTIAL

Now you try! 5) Suppose the shutter of the camera were formed by 8 blades. What would the measure of each exterior angle be? 5) 45° CONFIDENTIAL

Now some problems for you to practice !
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Assessment Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides: 1) 2) 1) Not a polygon 2) Polygon; dodecagon CONFIDENTIAL

Tell whether each polygon is regular or irregular
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex: 3) 4) 3) regular ; convex ; 4) irregular ; concave CONFIDENTIAL

5) Find the measure of each interior angle of pentagon ABCDE.
3x° 4x° 5x° 6) Find the measure of each interior angle of a regular dodecagon. 5) angles A & D = 81°; angle B = 108° ; angles C & E = 135° 6) 150° CONFIDENTIAL

7) Find the value of y in polygon JKLM.
8) Find the measure of each exterior angle of a regular pentagon. J K L M 4y° 2y° 6y° 7) 22.5° 8) 72° CONFIDENTIAL

9) Name the polygon by the number of its sides.
10) In the polygon, /P, /R and /T are right angles and /Q is congruent to /S. What are m/Q and m/S? R P Q S T 9) Pentagon; 10) 135° CONFIDENTIAL

Let’s review Properties and Attributes of Polygons Today you will learn about the parts of polygon and the ways to classify polygons. Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal. A E B C D side vertex diagonal CONFIDENTIAL