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Geometry Properties and Attributes of Polygons

Geometry Properties and Attributes of Polygons. Warm up. Solve by factoring: 1) x 2 + 3x – 10 = 0 2) x 2 - x – 12 = 0 3) x 2 - 12x = - 35. Properties and Attributes of Polygons. Today you will learn about the parts of polygon and the ways to classify polygons.

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Geometry Properties and Attributes of Polygons

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  1. Geometry Properties and Attributes of Polygons CONFIDENTIAL

  2. Warm up Solve by factoring: 1) x2 + 3x – 10 = 0 2) x2 - x – 12 = 0 3) x2 - 12x = - 35 CONFIDENTIAL

  3. 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 B side vertex C E D diagonal CONFIDENTIAL

  4. 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. CONFIDENTIAL

  5. 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

  6. 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 CONFIDENTIAL

  7. 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

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

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

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

  11. 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

  12. 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

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

  14. 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

  15. Finding Interior Angle Measures and Sums in Polygons 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

  16. Finding Interior Angle Measures and Sums in Polygons C) Find the measure of each interior angle of a quadrilateral PQRS. R Q 3c° c° (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. 3c° P c° S Polygon ∕ Sum thm. Substitute. m∕P =m∕R = 45° m∕Q = m∕S = 360° CONFIDENTIAL

  17. 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. CONFIDENTIAL

  18. 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°. 41° 81° 55° 111° 147° 132° 147° + 81° + 132° = 360° 110° 43° 43° + 111° + 41° + 55° + 110° = 360° CONFIDENTIAL

  19. 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

  20. 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

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

  22. 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 4r° 7r° K M 8r° 5r° L CONFIDENTIAL

  23. 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? ∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior measures is 360°. A B C D m ∕CBD = 360° =36° 10 A regular decagon has 10 congruent ext. angles. So, divide the sum by 10. CONFIDENTIAL

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

  25. Now some problems for you to practice ! CONFIDENTIAL

  26. Assessment Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides: 1) 2) CONFIDENTIAL

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

  28. 5) Find the measure of each interior angle of pentagon ABCDE. C 5x° B 4x° D 3x° 5x° E 3x° A 6) Find the measure of each interior angle of a regular dodecagon. CONFIDENTIAL

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

  30. 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 S Q T P CONFIDENTIAL

  31. 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 B side vertex C E D diagonal CONFIDENTIAL

  32. 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. CONFIDENTIAL

  33. 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

  34. 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

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

  36. Finding Interior Angle Measures and Sums in Polygons 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

  37. 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°. 41° 81° 55° 111° 147° 132° 147° + 81° + 132° = 360° 110° 43° 43° + 111° + 41° + 55° + 110° = 360° CONFIDENTIAL

  38. 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

  39. 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? ∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior measures is 360°. A B C D m ∕CBD = 360° =36° 10 A regular decagon has 10 congruent ext. angles. So, divide the sum by 10. CONFIDENTIAL

  40. You did a great job today! CONFIDENTIAL

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