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Geometry Chapter 6

Geometry Chapter 6 . Quadrilaterals Lesson 1: Angles of Polygons. Warm Up 1. A ? is a three-sided polygon. 2. A ? is a four-sided polygon. Evaluate each expression for n = 6. 3. ( n – 4) 12 4. ( n – 3) 90 Solve for a . 5. 12 a + 4 a + 9 a = 100. triangle.

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Geometry Chapter 6

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  1. Geometry Chapter 6 Quadrilaterals Lesson 1: Angles of Polygons

  2. Warm Up 1.A ? is a three-sided polygon. 2. A ? is a four-sided polygon. Evaluate each expression for n = 6. 3. (n – 4) 12 4. (n – 3) 90 Solve for a. 5. 12a + 4a + 9a = 100 triangle quadrilateral 24 270 4

  3. Your Math Goal Today… Classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons.

  4. Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex

  5. 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.

  6. You can name a polygon by the number of its sides. The table shows the names of some common polygons.

  7. Remember! A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.

  8. Example 1A: Identifying Polygons If it is a polygon, name it by the number of sides. polygon, hexagon

  9. Example 1B: Identifying Polygons If it is a polygon, name it by the number of sides. polygon, heptagon

  10. Example 1C: Identifying Polygons If it is a polygon, name it by the number of sides. not a polygon

  11. In Your Notes If it is a polygon, name it by the number of its sides. not a polygon

  12. In Your Notes If it is a polygon, name it by the number of its sides. polygon, nonagon

  13. In Your Notes If it is a polygon, name it by the number of its sides. not a polygon

  14. All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygonis one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.

  15. 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. A regular polygon is always convex.

  16. Example 2A: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex

  17. Example 2B: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

  18. Example 2C: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex

  19. In Your Notes Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex

  20. In Your Notes Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

  21. To find the sum of the interior angle measures 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 of the angle measures of the polygon.

  22. Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.

  23. 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°.

  24. Example 3A: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of a convex heptagon. (n – 2)180° Polygon  Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify.

  25. Example 3B: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon  Sum Thm. Substitute 16 for n and simplify. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.

  26. Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. Polygon  Sum Thm. (5 – 2)180° = 540° Polygon  Sum Thm. mA + mB + mC + mD + mE = 540° 35c + 18c+ 32c+ 32c+ 18c= 540 Substitute. 135c= 540 Combine like terms. c= 4 Divide both sides by 135.

  27. Example 3C Continued mA = 35(4°)= 140° mB = mE = 18(4°)= 72° mC = mD = 32(4°)= 128°

  28. In Your Notes Find the sum of the interior angle measures of a convex 15-gon. (n – 2)180° Polygon  Sum Thm. (15 – 2)180° A 15-gon has 15 sides, so substitute 15 for n. 2340° Simplify.

  29. In Your Notes Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon  Sum Thm. Substitute 10 for n and simplify. (10 – 2)180° = 1440° Step 2 Find the measure of one interior angle. The int. s are , so divide by 10.

  30. In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.

  31. Remember! An exterior angle is formed by one side of a polygon and the extension of a consecutive side.

  32. measure of one ext.  = Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20 sides and 20 vertices. sum of ext. s = 360°. Polygon  Sum Thm. A regular 20-gon has 20  ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°.

  33. Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext.  Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b°= 360° 120b= 360 Combine like terms. b= 3 Divide both sides by 120.

  34. measure of one ext. In Your Notes Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. Polygon  Sum Thm. A regular dodecagon has 12  ext. s, so divide the sum by 12. The measure of each exterior angle of a regular dodecagon is 30°.

  35. In Your Notes Find the value of r in polygon JKLM. 4r° + 7r° + 5r° + 8r°= 360° Polygon Ext.  Sum Thm. 24r= 360 Combine like terms. r= 15 Divide both sides by 24.

  36. Example 5: Art Application Ann is making paper stars for party decorations. What is the measure of 1? 1 is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°. A regular pentagon has 5  ext. , so divide the sum by 5.

  37. In Your Notes What if…? Suppose the shutter were formed by 8 blades instead of 10 blades. What would the measure of each exterior angle be? CBD is an exterior angle of a regular octagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°. A regular octagon has 8  ext. , so divide the sum by 8.

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