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Properties of Quadrilaterals 3.2

Properties of Quadrilaterals 3.2. Homework. Any four sided polygon is a quadrilateral. Angles sum to be 360  We’ll study special quadrilaterals in this section: Trapezoid Isosceles Trapezoid Parallelogram Rhombus Rectangle Square Kite. Opposite sides of a parallelogram are parallel

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Properties of Quadrilaterals 3.2

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  1. Properties of Quadrilaterals 3.2

  2. Homework Any four sided polygon is a quadrilateral. Angles sum to be 360 We’ll study special quadrilaterals in this section: Trapezoid Isosceles Trapezoid Parallelogram Rhombus Rectangle Square Kite

  3. Opposite sides of a parallelogram are parallel Opposite sides are congruent Opposite angles of parallelograms are congruent. Diagonals of a parallelogram bisect each other Consecutive angles of a parallelogram are supplementary Alternate interior angles are congruent Homework Properties of Parallelograms

  4. || || Homework

  5. Homework

  6. Homework

  7. Alternate interior Homework

  8. Homework Find a and b so that the quadrilateral is a parallelogram State the property. a. mMJK b. mJML c. mJKL d. mKJL e. a f. b 100 80 80 30 21 7

  9. Homework Find d so that the quadrilateral is a parallelogram. State the property. • mPLM • b. mLMN • c. d = 108 72 11

  10. Homework Find x and y so that the quadrilateral is a parallelogram State the property. a. x b. y y = 21 x = 12

  11. Homework Find x and y so that the quadrilateral is a parallelogram. State the property. a. x b. y x = 7 y = 4

  12. Homework Find the value of x that makes the figure a parallelogram. State the property. a. x x = 46

  13. Homework Find the valuesso that the figure is a parallelogram State the property. a. x b. y c. a d. b y = 15 a = 7 x = 25 b = 7 g. w h. z e. x f. y z = 4½ w = 4 y = 65 x = 8

  14. 6.13 Homework Find x, y, w, and z so that the quadrilateral is a parallelogram. State the property . a. mMNP b. mNRP c. mRNP d. mRMN e. mMQN f. mMQR g. x h. y i. w j. z 71 33 38 109 97 83 6.45 3.525 8

  15. Assignment Geometry: Properties of a Parallelogram

  16. A rhombus is a parallelogram (this means it has ALL of the characteristics of a parallelogram) In addition: A rhombus has four congruent sides The diagonals of a rhombus are perpendicular The diagonals bisect opposite angles Homework Properties of a Rhombus (Rhombi)

  17. Homework Rhombus

  18. a. f. c. d. e. b. JM m KJL m KNL m KJM NM JN Homework Find the indicated measure in rhombus JKLM KM = 8 and JL = 6. State the property. 4 90° 3 5 53° 37 106°

  19. A rectangle is a parallelogram (this means it has ALL the characteristics of a parallelogram) IN ADDITION: Four right angles The diagonals of a rectangle are congruentandthey bisect each other Homework Properties of Rectangles

  20. Homework Rectangles

  21. Homework In rectangle JKLM shown below, JL and MK are diagonals. If JL = 2x + 5 and MK = 4x – 11, what is x? x = 8 If mMNL = 140 answer the following? d. mMJK e. mNLK f. mNLM • mJNK • b. mMNJ • c. mLNK g. mLJK h. mLJM 90° 20° 140° 70° 70° 40° 20° 40°

  22. (2z) + 11) Homework In rectangle ABCD shown below, find the value of x, y, and z.State the property. a. x b. y c. z z = 12.5 y = 9 x = 5

  23. Homework WXYZ is a rectangle. Find each measure if m1 = 35. State the property. a.m1 b. m2 c. m3 d. m4 e. m5 f. m6 g. m7 h. m8 i. m9 j. m10 k. m11 l. m12 55° 55° 35° 35° 55° 35° 35° 55° 70° 110° 110° 70°

  24. Homework Quadrilateral JKMN is a rectangle. Find each measure. State the property. 36 a. If NQ = 5x +3 & QM =4x +6, find NK. b. If NQ =2x +3 & QK 5x -9, find JQ. c. If NM =2x +14 & JK =x2 -1, find JK. d. If mNJM =2x +3 & mKJM =x +6, find x. e. If mNKM =x2 +4 & mKNM =x +30, find mJKN. f. If mJKN =16x & mNKM = 14x, find x. 11 8 or 24 27 37 3

  25. in. c =  1737 Homework Television screens are rectangles and are measured by their diagonals. Find the length of the diagonal. a² + b² = c² 21² + 36² = c² 1737 = c² c  41.6773

  26. A square is a parallelogram, a rectangle, and a rhombus (It has ALL those characteristics!!!) Has four congruent sides Has four right angles The diagonals of a square: bisect each other are congruent are perpendicular. bisect opposite angles Homework Properties of Squares

  27. Homework

  28. A B 10 in. c =  200 C D Homework Parallelogram ABCD is a square.Find x and y. a² + b² = c² 10² + 10² = c² 200 = c² c  14.14 • x • b. y x = 45 y  14.14

  29. Assignment Geometry: Rectangles, Rhombus & Squares

  30. Homework Inheritanceof Properties

  31. Kites Trapezoids Homework Isosceles Trapezoid

  32. Homework

  33. 2 pair of consecutive congruent sides Opposite sides are NOT congruent Angles are congruent as marked (also mK  mT) Diagonals are perpendicular Notice only ONE diagonal is bisected Homework Properties of a Kite:A quadrilateral with NO parallel sides.

  34. Homework Kites

  35. x + 4 14 y + 16 2x + 12 Homework Find the value of x and y. Find the lengths of the sides. • x • b. y 10 16 c. IT d. KE 14 32

  36. Homework Find the value of x and y in the kite below. a² + b² = c² 24² + (SO)² = 27² 576 + (SO)² = 729 (SO)² = 153 SO =  153 SO  12.4 12.4 • x b. y 4x + 3 = 15 2x + 5y = 12.4 6 + 5y = 12.4 4x = 12 x = 3 5y = 6.4 y = 1.28

  37. 2 6 5 4 3 1 52 7 27 Homework In kite ABCD, find the measures of the numbered angles. a.m1 b. m2 c. m3 d. m4 e. m5 f. m6 g. m7 90° 52° 38° 27° 63° 38° 63°

  38. Homework Trapezoid

  39. Homework Isosceles Trapezoid

  40. Homework Properties of a Trapezoid • A trapezoid has one and only one pair of parallel sides. • The median of a trapezoid is parallel to the bases, and the length of the median equals one-half the sum of the lengths of the bases. Base Median Base

  41. 6 65 18 Homework For isosceles trapezoid XYZW, Find the length of the median, mX and mZ. • Median • b. mZ • c. mX 12 115° 65°

  42. Homework In trapezoid QRST, A and B are midpoints of the legs. Find AB, mQ, and mS. a. AB b. mQ c. mS 135° 60° 16

  43. Homework PQRS is an isosceles trapezoid; find x. 2a2 – 54 = a2 + 27 a2 = 81 a = 9 or a = –9

  44. 17x 22.5x + 9 30x + 12 Homework XY is the midsegment of trapezoid ABCD; find x. x = 3

  45. Homework 1. Opposite sides parallel. 2. Opposite sides congruent. 3. Opposite angles are congruent. 4. Consecutive angles are supplementary. 5. Diagonals bisect each other. 1. Has 4 right angles. 2. Diagonals are congruent. 3. All properties of parallelogram. 1. Has 4 Congruent sides 2. Diagonals bisect opposite angles. 3. Diagonals are perpendicular. 4. All properties of parallelograms. 1. 4 congruent sides and 4 congruent (right) angles 2. All properties of parallelogram, rectangle, and rhombus QUADRILATERALS 1. One pair of parallel sides 2. Leg angles supplementary 3. Midsegment = ½ (b1 + b2) 1. 2 pairs of consecutive sides congruent 2. 1 pair of opposite angles congruent 3. Diagonals perpendicular 4. Small diagonal bisected 5. Non-congruent angles are bisected • 1. 2 pairs of congruent base angles • 2. Diagonals are congruent • 3. One pair of parallel sides • 4. Leg angles supplementary • 5. Midsegment = ½ (b1 + b2)

  46. Homework Quadrilateral Characteristics Summary Convex Quadrilaterals 4 sided polygon 4 interior angles sum to 360 4 exterior angles sum to 360 Parallelograms Trapezoids Bases Parallel Legs are not Parallel Leg angles are supplementary Median is parallel to basesMedian = ½ (base + base) Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Rectangles Rhombi IsoscelesTrapezoids All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Angles all 90° Diagonals congruent Legs are congruent Base angle pairs congruent Diagonals are congruent Squares Diagonals divide into 4 congruent triangles

  47. Homework In parallelogram PNWL, NW = 12, PM = 9, and mWLP = 144°. Find each measure. 1.PW2.mPNW 18 144°

  48. Homework QRST is a parallelogram. Find each measure. a.TQb.mT 71° 28

  49. Assignment Geometry: Trapezoids & Kites

  50. Assignment Geometry: 3.2A and 3.2B Section 9 - 41

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