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Chapter 6: Quadrilaterals

Chapter 6: Quadrilaterals. Fall 2008 Geometry. 6.1 Polygons. A polygon is a closed plane figure that is formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. Each endpoint of a side is a vertex of the polygon.

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Chapter 6: Quadrilaterals

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  1. Chapter 6: Quadrilaterals Fall 2008 Geometry

  2. 6.1 Polygons • A polygon is a closed plane figure that is formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. • Each endpoint of a side is a vertex of the polygon. • Name a polygon by listing the vertices in clockwise or counterclockwise order.

  3. Polygons • State whether the figure is a polygon.

  4. Identifying Polygons

  5. Polygons • A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon

  6. Polygons • A polygon is concave if it is not convex.

  7. Polygons • A polygon is equilateral if all of its sides are congruent. • A polygon is equiangular if all of its angles are congruent. • A polygon is regular if it is equilateral and equiangular.

  8. Polygons • Determine if the polygon is regular.

  9. Polygons • A diagonal of a polygon is a segment that joins two nonconsecutive vertices. B E L M R

  10. Polygons • The sum of the measures of the interior angles of a quadrilateral is 360 . • A + B + C + D = 360 A B D C

  11. Homework 6.1 • Pg. 325 # 12 – 34, 37 – 39, 41 – 46

  12. 6.2 Properties of Parallelograms • A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

  13. Theorems about Parallelograms • If a quadrilateral is a parallelogram, then its opposite sides are congruent. • AB = CD and AD = BC A B D C

  14. Theorems about Parallelograms • If a quadrilateral is a parallelogram, then its opposite angles are congruent. • A = C and D = B A B D C

  15. Theorems about Parallelograms • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. • D + C = 180 ; D + A = 180 • A + B = 180 ; B + C = 180 A B D C

  16. Theorems about Parallelograms • If a quadrilateral is a parallelogram, then its diagonals bisect each other. • AM = MC and DM = MB A B M D C

  17. Examples • FGHJ is a parallelogram. Find the unknown lengths. • JH = _____ • JK = _____ 5 F F G G K K 3 3 J J H H

  18. Examples • PQRS is a parallelogram. Find the angle measures. • m R = • m Q = P Q 70 S R

  19. Examples • PQRS is a parallelogram. Find the value of x. P Q 3x 120 S R

  20. 6.3 Proving Quadrilaterals are Parallelograms • For the 4 theorems about parallelograms, their converses are also true.

  21. Theorems • If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  22. Theorems • If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  23. Theorems • If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

  24. Theorems • If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

  25. One more… • If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

  26. Examples • Is there enough given information to determine that the quadrilateral is a parallelogram?

  27. Examples • Is there enough given information to determine that the quadrilateral is a parallelogram? 65 115 65

  28. Examples • Is there enough given information to determine that the quadrilateral is a parallelogram?

  29. How would you prove ABCD ? A B D C

  30. How would you prove ABCD ? A B D C

  31. How would you prove ABCD ? A B C D

  32. Homework 6.2 – 6.3 • Pg. 334 # 20 – 37 • Pg. 342 # 9 – 19, 32 – 33

  33. 6.4 Rhombuses, Rectangles, and Squares • A rhombus is a parallelogram with four congruent sides.

  34. Rhombuses, Rectangles, and Squares • A rectangle is a parallelogram with four right angles.

  35. Rhombuses, Rectangles, and Squares • A square is a parallelogram with four congruent sides and four right angles.

  36. Special Parallelograms • Parallelograms rectangles rhombuses squares

  37. Corollaries about Special Parallelograms • Rhombus Corollary • A quadrilateral is a rhombus iff it has 4 congruent sides. • Rectangle Corollary • A quadrilateral is a rectangle iff it has 4 right angles. • Square Corollary • A quadrilateral is a square iff it is a rhombus and a rectangle.

  38. Theorems • A parallelogram is a rhombus iff its diagonals are perpendicular.

  39. Theorems • A parallelogram is a rhombus iff each diagonal bisects a pair of opposite angles.

  40. Theorems • A parallelogram is a rectangle iff its diagonals are congruent.

  41. Examples • Always, Sometimes, or Never • A rectangle is a parallelogram. • A parallelogram is a rhombus. • A rectangle is a rhombus. • A square is a rectangle.

  42. Examples • (A) Parallelogram; (B) Rectangle; (C) Rhombus; (D) Square • All sides are congruent. • All angles are congruent. • Opposite angles are congruent. • The diagonals are congruent.

  43. Examples • MNPQ is a rectangle. What is the value of x? M N 2x Q P

  44. 6.4 Homework • Pg. 351 # 12-21, 25-43

  45. 6.5 Trapezoids and Kites • A trapezoid is a quadrilateral with exactly one pair of parallel sides. • The parallel sides are the bases. • A trapezoid has two pairs of base angles. • The nonparallel sides are called the legs. • If the legs are congruent then it is an isosceles trapezoid.

  46. Theorems • If a trapezoid is isosceles, then each pair of base angles is congruent. • A = B , C = D A B C D

  47. Theorems • If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A B C D

  48. Theorems • A trapezoid is isosceles iff its diagonals are congruent. • ABCD is isosceles iff AC = BD A B C D

  49. Midsegments of Trapezoids • The midsegment of a trapezoid is the segment that connects the midpoints of its legs. midsegment

  50. Midsegment Theorem • The midsegment of a trapezoid is parallel to each base and its length is ½ the sum of the lengths of the bases. • MN ll AD, MN ll BC, MN = ½ (AD + BC) B C N M D A

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