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Unit 6

Unit 6. Quadrilaterals. Lesson 6.1. Properties of Quadrilaterals. Lesson 6.1 Objectives. Identify a figure to be a quadrilateral. Use the sum of the interior angles of a quadrilateral. (G1.4.1). Definition of a Quadrilateral.

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Unit 6

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  1. Unit 6 Quadrilaterals

  2. Lesson 6.1 Properties of Quadrilaterals

  3. Lesson 6.1 Objectives • Identify a figure to be a quadrilateral. • Use the sum of the interior angles of a quadrilateral. (G1.4.1)

  4. Definition of a Quadrilateral • A quadrilateral is any four-sided figure with the following properties: • All sides must be line segments. • Each side must intersect only two other sides. • One at each of its endpoints, so that there are no: • Gaps that do not connect one side to another, or • Tails that extend beyond another side.

  5. Example 6.1 Determine if the figure is a quadrilateral. Yes No Too many intersecting segments No Yes No gaps Yes No Too many sides No tails No No No curves

  6. Interior Angles • Recall that the interior angles of any figure are located in the interior and are formed by the sides of the figure itself. Review: How manydegrees does a straightline measure? Review:What do youthink the sum of the interior angles of a quadrilateral might be? Review: What is the sum of the interior angles of anytriangle?

  7. 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 =

  8. Example 6.2 Find the missing angle.

  9. Example 6.3 Find the x.

  10. Lesson 6.1 Homework • Lesson 6.1 – Properties of Quadrilaterals • Due Tomorrow

  11. Lesson 6.2 Day 1: Parallelograms

  12. Lesson 6.2 Objectives • Define a parallelogram • Define special parallelograms • Identify properties of parallelograms (G1.4.3) • Use properties of parallelograms to determine unknown quantities of the parallelogram (G1.4.4)

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

  14. Theorem 6.2:Congruent Sides of a Parallelogram • If a quadrilateral is a parallelogram, then its opposite sides are congruent. • The converse is also true! • Theorem 6.6

  15. Theorem 6.3:Opposite Angles of a Parallelogram • If a quadrilateral is a parallelogram, then its opposite angles are congruent. • The converse is also true! • Theorem 6.7

  16.     Example 6.4 Find the missing variables in the parallelograms. c – 5 = 20 x = 11 m = 101 c = 25 y = 8 d + 15 = 68 d = 53

  17. Q R P S Theorem 6.4:Consecutive Angles of a Parallelogram • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. • The converse is also true! • Theorem 6.8 m Q + m R = 180o m P + m S = 180o m P + m Q = 180o m R + m S = 180o

  18. 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. • And again, the converse is also true! • Theorem 6.9

  19. Example 6.5 Find the indicated measure in  HIJK • HI • 16 • Theorem 6.2 • GH • 8 • Theorem 6.5 • KH • 10 • Theorem 6.2 • HJ • 16 • Theorem 6.5 & Seg Add Post • m KIH • 28o • AIA Theorem • m JIH • 96o • Theorem 6.4 • m KJI • 84o • Theorem 6.3

  20. Theorem 6.10:Congruent Sides of a Parallelogram • If a quadrilateral has onepair of opposite sides that are bothcongruent and parallel, then it is a parallelogram.

  21. Example 6.6 Is there enough information to prove the quadrilaterals to be a parallelogram.If so, explain. Yes! Yes! Yes! Both pairs of opposite sides are congruent.(Theorem 6.6) One pair of parallel andcongruent sides. (Theorem 6.10) Both pairs of opposite angles are congruent.(Theorem 6.7) Yes! Yes! Yes! OROne pair of parallel and congruent sides.(Theorem 6.10) Both pairs of opposite angles are congruent.(Theorem 6.7) All consecutive angles are supplementary.(Theorem 6.8) The diagonals bisect each other.(Theorem 6.9) Both pairs of opposite sides are congruent.(Theorem 6.6)

  22. Lesson 6.2a Homework • Lesson 6.2: Day 1 – Parallelograms • Due Tomorrow

  23. Lesson 6.2 Day 2: (Special) Parallelograms

  24. 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.

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

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

  27. 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.

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

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

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

  31.   Example 6.7 Classify the parallelogram. Explain your reasoning. Must be supplementary Rhombus Rectangle Square Diagonals are perpendicular. Theorem 6.11 Diagonals are congruent. Theorem 6.13 Square Corollary

  32. Lesson 6.2b Homework • Lesson 6.2: Day 2 – Parallelograms • Due Tomorrow

  33. Lesson 6.3 Trapezoids and Kites

  34. Lesson 6.3 Objectives • Identify properties of a trapezoid. (G1.4.1) • Recognize an isosceles trapezoid. (G1.4.1) • Utilize the midsegment of a trapezoid to calculate other quantities from the trapezoid. • Identify a kite. (G1.4.1)

  35. 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.

  36. Example 6.8 Find the indicated angle measure of the trapezoid. ConsecutiveInterior Angles are supplementary! CIA CIA Recall that a trapezoid has one set of parallel bases.

  37. Example 6.9 ConsecutiveInterior Angles are supplementary! Find x in the trapezoid. CIA CIA

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

  39. 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!

  40. 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.

  41. Theorem 6.16:Congruent Diagonals of a Trapezoid • A trapezoid is isoscelesif 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!!

  42.    Example 6.10 Find the measures of the other three angles. 53o Supplementary because of CIA 127o 127o 83o 97o 83o Supplementary because of CIA

  43. Midsegment • The midsegment of a trapezoid is the segment that connects the midpoints of the legs of a trapezoid.

  44. C D N M A B Theorem 6.17:Midsegment Theorem for Trapezoids • The midsegment of a trapezoid is • parallel to each baseand • its length is one half the sum of the lengths of the bases. • It is the averageof the base lengths!

  45. ? ? ? Example 6.11 Find the indicated length of the trapezoid. Multiply both sides by 2. Or essentially double the midsegment!

  46. Kite • A kite is a quadrilateral that has two pairs of consecutive sidesthat are congruent, but opposite sides are not congruent. • It looks like the kite you got for your birthday when you were 5! • There are no sides that are parallel.

  47. Theorem 6.18:Diagonals of a Kite • If a quadrilateral is a kite, then its diagonals are perpendicular.

  48. Theorem 6.19:Opposite Angles of a Kite • If a quadrilateral is a kite, then exactly onepair of opposite angles are congruent. • The angles that are congruent are between the two different congruent sides. • You could call those the shoulder angles. NOT

  49. Example 6.12 Find the missing angle measures. 125o But K M 64o 125o 88o 60 + K + 50 + M = 360 60 + M + 50 + M = 360 K = 88 110 + 2M = 360 2M = 250 88 + 120 + 88 + J = 360 M = 125 296 + J = 360 J = 64 K = 125

  50. Example 6.13 Find the lengths of all the sides of the kite. Round your answer to the nearest hundredth. a2 + b2 = c2 52 + 52 = c2 a2 + b2 = c2 7.07 7.07 25 + 25 = c2 52 + 122 = c2 50 = c2 25 + 144 = c2 c = 7.07 169 = c2 13 13 c = 13 Use Pythagorean Theorem! Cause the diagonals are perpendicular!! a2 + b2 = c2

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