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Chapter 6

Chapter 6. Quadrilaterals. Chapter Objectives. Define a polygon and its characteristics Identify a regular polygon Interior Angles of a Quadrilateral Theorem Properties of Parallelograms Using coordinate geometry to prove parallelograms Compare rhombuses, rectangles, and squares

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Chapter 6

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  1. Chapter 6 Quadrilaterals

  2. Chapter Objectives • Define a polygon and its characteristics • Identify a regular polygon • Interior Angles of a Quadrilateral Theorem • Properties of Parallelograms • Using coordinate geometry to prove parallelograms • Compare rhombuses, rectangles, and squares • Identify trapezoids and kites • Midsegment Theorem for Trapezoids • Calculate area of trapezoids, kites, rhombuses, rectangles, and squares

  3. Lesson 6.1 Polygons

  4. Lesson 6.1 Objectives • Identify a figure to be a polygon. • Recognize the different types of polygons based on the number of sides. • Identify the components of a polygon. • Use the sum of the interior angles of a quadrilateral.

  5. Definition of a Polygon • A polygon is plane figure (two-dimensional) that meets the following conditions. • It is formed by three or more segments called sides. • The sides must be straight lines. • Each side intersects exactly twoother sides, one at each endpoint. • The polygon is closed in all the way around with no gaps. • Each side must end when the next side begins. No tails. Polygons Not Polygons

  6. vertices sides Polygon Parts • Each segment that is used to close a polygon in is called a side. • Where each side ends is called a vertex. • A vertexis simply a corner of the polygon.

  7. Types of Polygons Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon

  8. A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that neverleave the interior of the polygon, then it is convex. A polygon is concave if a line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that does leave the interior of the polygon, then it is concave. Concave polygons have dentsin the sides, or you could say it caves in. Concave v Convex

  9. Example 1 Determine if the following are polygons or not.If it is a polygon, classify it as concave or convex. No! Yes Yes Concave Convex

  10. Regular Polygons • A polygon is equilateral if all of its sides are congruent. • A polygon is equiangular if all of its interior angles are congruent. • A polygon is regular if it is both equilateral and equiangular. The best way to draw these is to label each sides and angle with the proper congruent marks.

  11. Diagonals of a Polygon • A diagonalof a polygon is a segment that joins two nonconsecutive vertices. • A diagonal does not go to the point next to it. • That would make it a side! • Diagonals cut across the polygon to all points on the other side. • There is typically more than one diagonal.

  12. 4 3 1 2 Theorem 6.1:Interior Angles of a Quadrilateral Theorem • The sum of the measures of the interior angles of a quadrilateral is 360o. 360o m 1 +m 2 + m 3 + m 4 =

  13. Homework 6.1 • In Class • 1-11 • p325-328 • HW • 12-46, 54-59 • Due Tomorrow

  14. Lesson 6.2 Properties of Parallelograms

  15. Lesson 6.2 Objectives • Define a parallelogram • Identify properties of parallelograms • Use properties of parallelograms to determine unknown quantities of the parallelogram

  16. Definition of a Parallelogram • A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

  17. Theorem 6.2:Congruent Sides of a Parallelogram • If a quadrilateral is a parallelogram, then its opposite sides are congruent.

  18. Theorem 6.3:Opposite Angles of a Parallelogram • If a quadrilateral is a parallelogram, then its opposite angles are congruent.

  19.     Example 2 Find the missing variables in the parallelograms. d = 53 d + 15 = 68 m = 101 x = 11 y = 8 c – 5 = 20 c = 25

  20. Q R P S Theorem 6.4:Consecutive Angles of a Parallelogram • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. m Q + m R = 180o m P + m S = 180o m P + m Q = 180o m R + m S = 180o

  21. Theorem 6.5:Diagonals of a Parallelogram • If a quadrilateral is a parallelogram, then its diagonals bisect each other. • Remember that means to cut into two congruent segments.

  22. Example 3 Find the indicated measure in  HIJK • HI • 16 • Theorem 6.2 • GH • 8 • Theorem 6.6 • KH • 10 • Theorem 6.2 • HJ • 16 • Theorem 6.6 & Seg Add Post • m KIH • 28o • AIA Theorem • m JIH • 96o • Theorem 6.4 • m KJI • 84o • Theorem 6.3

  23. Homework 6.2 • HW • p333-336 • 20-37, 47-54, 60, 61 • Due Tomorrow • Quiz Wednesday • Lessons 6.1-6.3

  24. Lesson 6.3 Proving Quadrilaterals are Parallelograms

  25. Lesson 6.3 Objectives • Verify that a quadrilateral is a parallelogram. • Utilize coordinate geometry with parallelograms

  26. Theorem 6.6:Congruent Sides of a Parallelogram Converse • If both pairs of opposite sides are congruent, then it is a parallelogram.

  27. Theorem 6.7:Opposite Angles of a Parallelogram Converse • If both pairs of opposite angles are congruent, then it is a parallelogram.

  28. Q R P S m Q + m R = 180o m P + m S = 180o m P + m Q = 180o m R + m S = 180o Theorem 6.8:Consecutive Angles of a Parallelogram Converse • If an angle of a quadrilateral is supplementary to its consecutive angles, then it is a parallelogram.

  29. Theorem 6.9:Diagonals of a Parallelogram Converse • If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

  30. Theorem 6.10:Opposite Sides of a Parallelogram • If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

  31. Example 4 Which theorem would you use to show the following are parallelograms? Theorem 6.10 Theorem 6.6 Theorem 6.7 Theorem 6.8 or Theorem 6.7 Theorem 6.9 Theorem 6.6 or Theorem 6.10

  32. Homework 6.3 • In Class • 1-7 • p342-345 • HW • 9-29, 45-47 • skip 15-16 • Due Tomorrow • Quiz Friday • Lessons 6.1-6.3

  33. Lesson 6.4 Rhombuses, Rectangles, and Squares

  34. Lesson 6.4 Objectives • Identify characteristics of a rhombus. • Identify characteristics of a rectangle. • Identify characteristics of a square.

  35. Rhombus • A rhombusis a parallelogram with four congruent sides. • The rhombus corollary states that a quadrilateral is a rhombusif and only if it has four congruent sides.

  36. Theorem 6.11:Perpendicular Diagonals • A parallelogram is a rhombus if and only if its diagonals are perpendicular.

  37. Theorem 6.12:Opposite Angle Bisector • A parallelogram is a rhombusiff each diagonal bisects a pair of opposite angles.

  38. Rectangle • A rectangle is a parallelogram with four congruent angles. • The rectangle corollary states that a quadrilateral is a rectangleiff it has four right angles.

  39. Theorem 6.13:Four Congruent Diagonals • A parallelogram is a rectangleiff all four segments of the diagonals are congruent.

  40. Square • A square is a parallelogram with four congruent sides and four congruent angles.

  41. Square Corollary • A quadrilateral is a squareiff it s a rhombus and a rectangle. • So that means that all the properties of rhombuses and rectangles work for a square at the same time.

  42.   Example 5 Classify the parallelogram. Explain your reasoning. Must be supplementary Rhombus Square Rectangle Diagonals are perpendicular. Theorem 6.11 Square Corollary Diagonals are congruent. Theorem 6.13

  43. Homework 6.4 • In Class • 1, 3-11 • p351-354 • HW • 12-46 evens, 55-58, 66, 67 • Due Tomorrow

  44. Lesson 6.5 Trapezoids and Kites

  45. Lesson 6.5 Objectives • Identify properties of a trapezoid. • Recognize an isosceles trapezoid. • Utilize the midsegment of a trapezoid to calculate other quantities from the trapezoid. • Identify a kite.

  46. Trapezoid • A trapezoid is a quadrilateral with exactly onepair of parallel sides. • The parallel sides are called the bases. • The nonparallel sides are called legs. • The angles formed by the bases are called the base angles.

  47. Isosceles Trapezoid • If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

  48. Theorem 6.14:Bases Angles of a Trapezoid • If a trapezoid is isosceles, then each pair of base angles is congruent. • That means the top base angles are congruent. • The bottom base angles are congruent. • But they are not all congruent to each other!

  49. Theorem 6.15:Base Angles of a Trapezoid Converse • If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

  50. Theorem 6.16:Congruent Diagonals of a Trapezoid • A trapezoid is isosceles if and only if its diagonals are congruent. • Notice this is the entire diagonal itself. • Don’t worry about it being bisected cause it’s not!!

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