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Geometry Unit 7: Polygons

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Geometry Unit 7: Polygons

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  1. Geometry Unit 7:Polygons Mr. Andrew Volk Spring 2014

  2. Unit 7 Preview in Numbers • Unit Lessons: 6 • Instructional Days: 6 • Review Days: 1 • Test Days: 1 • Total Homework Problems: ? • Total Objectives: 17

  3. Unit 7 Preview of Topics • Two-Dimensional Figures • Parallelograms • Tests of Parallelograms • Rectangles, Rhombus, and Square • Trapezoid and Kites • Area of Parallelograms and Triangles • Area of Trapezoids, Rhombi, and Kites

  4. Two-Dimensional Figures Lesson 7.1

  5. Lesson 7.1 Objectives The student will be able to… • Identify and name polygons. • Find perimeter, circumference and area of two dimensional figures. • Find and use the sum of measures of the interior angles. • Find and use the sum of the measures of the exterior angles.

  6. Virginia SOL Standard G.9 and G10 • G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • G.10 The student will solve real-world problems involving angles of polygons.

  7. Naming Polygons by sides • 3 Triangle • 4 Quadrilateral • 5 Pentagon • 6 Hexagon • 7 Heptagon • 8 Octagon • 9 Nonagon • 10 Decagon • 11 Hendecagon • 12 Dodecagon

  8. Finding the sum of the interior angles of a polygon • 3 Triangle: 180 • 4 Quadrilateral: 360 • 5 Pentagon: 180*3=540 • 6 Hexagon: 180*4=720 • 7 Heptagon • 8 Octagon • 9 Nonagon • 10 Decagon????? (10-2)*180 = 8*180=1440 • 11 Hendecagon • 12 Dodecagon

  9. In a regular polygon, all the angles and sides are the same size. To find the measure of a single angle: Divide the Sum by the Number of Sides. For example: In a regular pentagon, the sum of the interior angles is 540 and the number of sides is 5. °

  10. In your notebooks:Find the measure of a single angle in each regular polygon. 180 720 360 To find the measure of a single angle: Divide the Sum of the Interior Angles by the Number of Sides/Angles.

  11. To Find the Missing Angle, with an irregular polygon: Step 1: Find the sum of the interior angles for the polygon (count number of sides, and use formula) Step 2: Add up all the given angles. Step 3: Set two sums equal to each other. Step 4: Solve for x

  12. To Find the Missing Angle: 2 135 1 x Step 1: Find the sum of the interior angles for the polygon (ignore given angles, count number of sides, and use formula) 180(n – 2) = 180(7 – 2) = 180(5) = 900° 3 90 100 7 145 4 95 170 6 5

  13. To Find the Missing Angle: 135 x Step 2: Add up all the given angles. 100 + 95 + 170 + 145 + 90 + 135 + x = 735 + x 90 100 145 95 170

  14. To Find the Missing Angle: 135 x Step 3: Set two sums equal to each other. Step 4: Solve for x (you will need to subtract) 735 + x = 900 -735 -735 x = 165° 90 100 145 95 170

  15. Find the sum of the exterior angles of a polygon • ALWAYS SUMS TO 360 DEGREES!

  16. Exterior Angle Example 70+60+65+40+y = 360 235+y = 360 y = 125

  17. Finding Perimeter and Area • Perimeter is the sum of all the side lengths • Area can be found using the appropriate formula.

  18. Guided Practice Practice this now by completing the guided practice problems: Worksheet 7.1

  19. Independent Practice To practice this and solidify your learning please complete Homework Set 7.1

  20. Parallelograms Lesson 7.2

  21. Lesson 7.2 Objectives The student will be able to… • Recognize and apply the properties of the sides and angles of parallelograms. • Recognize and apply the properties of the diagonals of parallelograms.

  22. Virginia SOL Standard G.9 and G10 • G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • G.10 The student will solve real-world problems involving angles of polygons.

  23. Consecutive Angles • Two angles whose vertices are the endpoints of the same side are called consecutive angles.

  24. Opposite Angles • Two angles that are not consecutive are called opposite angles.

  25. Consecutive Sides • Two sides of a quadrilateral that meet are called consecutive sides.

  26. Opposite Sides • Two sides that are not consecutive are called opposite sides.

  27. Properties of Parallelograms: Sides and Angles • A parallelogram is a type of quadrilateral whose pairs of opposite sides are parallel.

  28. Properties of Parallelograms: Sides and Angles • If a quadrilateral is a parallelogram, then… • (1) its opposite sides are congruent, • (2) its opposite angles are congruent, • (3) its consecutive angles are supplementary. • (4) if one angle is a right angle then all angles are right angles.

  29. Example 1 • Find x and y.

  30. Example 2

  31. Properties of Parallelograms: Diagonals • When we refer to the diagonals of a parallelogram, we are talking about lines that can be drawn from vertices that are not connected by line segments.

  32. Properties of Parallelograms: Diagonals • If a quadrilateral is a parallelogram, then… • (1) its diagonals bisect each other, and • (2) each diagonal splits the parallelogram into two congruent triangles.

  33. Example 3

  34. Example 4 • Find the value of each variable

  35. Guided Practice Practice this now by completing the guided practice problems: Worksheet 7.2

  36. Independent Practice To practice this and solidify your learning please complete Homework Set 7.2

  37. Tests of Parallelograms Lesson 7.3

  38. Lesson 7.3 Objectives The student will be able to… • Recognize the conditions that ensure a quadrilateral is a parallelogram. • Prove that a set of points form a parallelogram in the coordinate plane.

  39. Virginia SOL Standard G.9 and G10 • G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • G.10 The student will solve real-world problems involving angles of polygons.

  40. Tests for Parallelograms • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • A pair of opposite sides is both parallel and congruent.

  41. Example 4a: Find x so that the quadrilateral is a parallelogram. 3(x-2) 4x-1 Opposite sides of a parallelogram are congruent.

  42. Example 4a: Substitution Distributive Property Subtract 3x from each side. Add 1 to each side. Answer: When x is 7, ABCD is a parallelogram.

  43. D E G F Example 4b: Find y so that the quadrilateral is a parallelogram. Opposite angles of a parallelogram are congruent.

  44. Example 4b: Substitution Subtract 6y from each side. Subtract 28 from each side. Divide each side by –1. Answer: DEFG is a parallelogram when y is 14.

  45. a. b. Answer: Answer: Your Turn: Find m and n so that each quadrilateral is a parallelogram.

  46. Parallelograms on the Coordinate Plane • We can use the Distance Formula and the Slope Formula to determine if a quadrilateral is a parallelogram on the coordinate plane. • Just pick one of the tests… and apply either or both of the formulas.

  47. Example 5a: COORDINATE GEOMETRYDetermine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and D(1, –1) is a parallelogram. Use the Slope Formula.

  48. Answer: Since opposite sides have the same slope, Therefore, ABCD is a parallelogram by definition. Example 5a: If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.

  49. Example 5b: COORDINATE GEOMETRYDetermine whether the figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and S(1, –3)is a parallelogram. Use the Distance and Slope Formulas.

  50. Example 5b: First use the Distance Formula to determine whether the opposite sides are congruent.