Geometry Quadrilaterals

Transcription

1 1

2 Geometry Quadrilaterals

3 Table of Contents Polygons Properties of Parallelograms Proving Quadrilaterals are Parallelograms Rhombi, Rectangles and Squares Trapezoids Click on a topic to go to that section. Kites Constructing Quadrilaterals Families of Quadrilaterals Proofs Coordinate Proofs PARCC Released Questions & Standards 3

4 Polygons Return to the Table of Contents 4

5 Polygons A polygon is a closed figure made of line segments connected end to end at vertices. Since it is made of line segments, none its sides are curved. The sides of simple polygons intersect each other at the common endpoints and do not extend beyond the endpoints. The sides of complex polygons intersect each other and extend to intersect with other sides. However, we will only be considering simple polygons in this course and will refer to them as "polygons." Click to Reveal Would a circle be considered a polygon? Why or why not? 5

6 Polygons Simple Polygons Complex Polygons Click to Reveal This course will only consider simple polygons, which we will refer to as "polygons." 6

7 Types of Polygons Polygons are named by their number of sides, angles or vertices; which are all equal in number. Number of Sides, Angles or Vertices Type of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon gon 12 dodecagon n n-gon 7

8 Convex or Concave Polygons Convex Polygons interior All interior angles are less than 180º. No side, if extended, will cross into the interior. interior 8

9 Convex or Concave Polygons Concave Polygons At least one interior angle is at least 180º. At least two sides, if extended, will cross into the interior. interior A way to remember the difference is that concave shapes "cave in". interior 9

10 1 The figure below is a concave polygon. True False 10

11 2 Indentify the polygon. A B C D E F Pentagon Octagon Quadrilateral Hexagon Decagon Triangle 11

12 3 Is the polygon convex or concave? A B Convex Concave 12

13 4 Is the polygon convex or concave? A B Convex Concave 13

14 Equilateral Polygons A polygon is equilateral if all its sides are congruent. 14

15 Equiangular Polygons A polygon is equiangular if all its angles are congruent. 15

16 Regular Polygons A polygon is regular if it is equilateral and equiangular. 16

17 5 Describe the polygon. (Choose all that apply) A Pentagon B Octagon C Quadrilateral D Hexagon E Triangle 4 F Convex G Concave H Equilateral 60 I Equiangular J Regular 4 60 o 4 60 o 17

18 6 Describe the polygon. (Choose all that apply) A B C D E F G H I J Pentagon Octagon Quadrilateral Hexagon Triangle Convex Concave Equilateral Equiangular Regular 18

19 7 Describe the polygon. (Choose all that apply) A B C D E F G H I J Pentagon Octagon Quadrilateral Hexagon Triangle Convex Concave Equilateral Equiangular Regular 19

20 Angle Measures of Polygons In earlier units, we spent a lot of time studying one specific polygon: the triangle. We won't have to spend that same amount of time on each new polygon we study. Instead, we will be able to use what we learned about triangles to save a lot of time and effort. 20

21 Angle Measures of Polygons Every polygon is comprised of a number of triangles. The sum of the interior angles of each triangle is 180. A triangle is (obviously) comprised of one triangle, and therefore the sum of its interior angles is 180. What is the sum of the interior angles of this polygon? 21

22 Angle Measures of Polygons Every polygon can be thought of as consisting of some number of triangles. A quadrilateral is comprised of two triangles What is the sum of the interior angles of a quadrilateral? 22

23 Angle Measures of Polygons Every polygon can be thought of as consisting of some number of triangles. A pentagon is comprised of three triangles. 180º 180º What is the sum of the interior angles of a pentagon? 180º 23

24 Angle Measures of Polygons Every polygon can be thought of as consisting of some number of triangles. An octagon is comprised of six triangles. 180º 180º 180º What is the sum of the interior angles of an octagon? 180º 180º 180º 24

25 Angle Measures of Polygons Fill in the final cell of this table with an expression for the number of triangular regions of a polygon based on the number of its sides. Number of sides Number of triangular regions 3 1 Math Practice n 25

26 Angle Measures of Polygons The sum of the interior angles of a triangle is 180 degrees. This will also be true of each triangular region of a polygon. Using that information, complete this table. Math Practice Number of sides Number of triangular regions Sum of Interior Angles º º º 6 4 n 26

27 8 What is the sum of the interior angles of this polygon? 27

28 9 How many triangular regions can be drawn in this polygon? 28

29 10 What is the sum of the interior angles of this polygon? 29

30 11 How many triangular regions can be drawn in this polygon? 30

31 12 What is the sum of the interior angles of this polygon? 31

32 13 Which of the following road signs has interior angles that add up to 360? A Yield Sign B No Passing Sign C National Forest Sign D Stop Sign E No Crossing Sign 32

33 Polygon Interior Angles Theorem Q1 The sum of the measures of the interior angles of a convex polygon with n sides is 180(n-2). Polygon Complete the table. Number of Sides Sum of the measures of the interior angles. hexagon 6 180(6-2) = 720 heptagon 7 octagon 8 nonagon 9 decagon gon 11 dodecagon 12 33

34 14 What is the measurement of each angle in the diagram below? L M (3x+4) 146 P (2x+3) (3x) x N O 34

35 Interior Angles of Equiangular Polygons Regular polygons are equiangular: their angles are equal. So, it must be true that each angle must be equal to the sum of the interior angles divided by the number of angles. m = m = Sum of the Angles Number of Angles (n - 2)180º n 35

36 Interior Angles of Regular Polygons The measure of each interior angle of an regular polygon is given by: m = (n-2)180º n regular polygon number of sides sum of interior angles measure of each angle triangle quadrilateral pentagon hexagon octagon decagon (n-2)(180 ) 36

37 15 If the stop sign is a regular octagon, what is the measure of each interior angle? 37

38 16 What is the sum of the measures of the interior angles of a convex 20-gon? A 2880 B 3060 C 3240 D

39 17 What is the measure of each interior angle of a regular 20-gon? A 162 B 3240 C 180 D 60 39

40 18 What is the measure of each interior angle of a regular 16-gon? A 2520 B 2880 C 3240 D

41 19 What is the value of x? (10x+8) (8x) (5x+15) (9x-6) (11x+16) 41

42 Polygon Exterior Angle Theorem The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is In this case, m 1 + m 2 + m 3 + m 4 + m 5 =

43 Proof of Exterior Angles Theorem Given: Any polygon has n exterior angles, where n is the number of its sides, interior angles, or vertices Prove: The sum of those exterior angles is 360 We will prove this for any polygon, not for a specific polygon. To do that, it will help if we introduce one piece of notation: the Greek letter Sigma, Ʃ. Ʃ represents the phrase "the sum of." So, the phrase, Math Practice "the sum of the interior angles" becomes "Ʃ interior s." 43

44 Proof of Exterior Angles Theorem Given: Any polygon has n interior angles, where n is the number of its sides, interior angles, or vertices Prove: Ʃ external s = 360 This proof is based on the fact that each exterior angle is supplementary to an interior angle. And, that the sum of the measures of the interior angle of a polygon with n sides is (n-2) x 180, Or, with our Sigma notation: Ʃ internal s = (n-2) x

45 Polygon Exterior Angle Theorem External angles are formed by extending one side of a polygon. 5 Take a look at 1 in the diagram. The external and internal angles at a vertex form a linear pair. 1external 1internal 4 What is the sum of their measures?

46 Polygon Exterior Angle Theorem Any polygon with n sides has n vertices and n internal s. That also means if we include one external at each vertex, that there are n linear pairs. Since 1 linear pair has a measure of 180, then n of them must have a measure of n x 180. The sum of the measures of a polygon's linear pairs of internal and external s must be n x

47 Polygon Exterior Angle Theorem Each linear pair includes an external and an internal. Adding up all of those means that: The sum of all the external and internal s of a polygon yields a measure of n x 180. Symbolically: Ʃ external s + Ʃ internal s = n x

48 Polygon Exterior Angle Theorem Ʃ external s + Ʃ internal s = n x 180 But recall that the internal angle theorem tell us that: Ʃ internal s = (n-2) x 180 Substituting that in yields Ʃ external s + [(n-2) x 180 ] = n x 180 Multiplying out the second term yields: 48

49 Polygon Exterior Angle Theorem Ʃ external s + [n x x 180 ] = n x 180 Subtracting n x x 180 from both sides yields Ʃ external s = 2 x 180 Ʃ external s = 360 Notice that n is not in this formula, so this result holds for all polygons with any number of sides. This is an important, and surprising, result. 49

50 Proof of Exterior Angles Theorem Given: Any polygon has n interior s, one at each vertex, where n is the number of sides, vertices or angles Prove: The sum of those n exterior s = 360 Statement Reason 1 Any polygon has n interior s Given Each interior and exterior are a linear pair Each interior is supplementary to an exterior The sum of the measures of each linear pair is 180 Given in diagram Linear pair angles are supplementary Definition of supplementary 5 There are n linear pairs Substitution 6 The sum of the linear pairs = n x 180 = sum of exterior s + interior s Multiplication Property 50

51 Proof of Exterior Angles Theorem Given: Any polygon has n interior s, one at each vertex, where n is the number of sides, vertices or angles Prove: The sum of those n exterior s = 360 Statement Reason 7 The sum of interior s is (n - 2) x 180 Interior Angle Theorem 8 9 n x 180 = sum of exterior s + (n - 2) x 180 The sum of the exterior s = n x [(n - 2) x 180 ] Substitution Property Subtraction Property 10 The sum of the exterior s = 180 [n - (n - 2)] 11 The sum of the exterior s = 180 (n - n + 2) 12 The sum of the exterior s = 180 (2) = 360 Subtraction Property Distributive Property Simplifying 51

52 Polygon Exterior Angle Theorem Corollary The measure of each exterior angle of a regular polygon with n sides is 360º/n The sum of the external angles of any polygon is 360º. The number of external angles of a polygon equals n. All angles are equal in a regular polygon, so their external angles must be as well. So, each external angle of a regular polygon equals 360º/n. 52

53 Polygon Exterior Angle Theorem Corollary The measure of each exterior angle of a regular polygon with n sides is 360º/n This is a regular hexagon. n = 6 a a = a = 60 53

54 20 What is the sum of the measures of the exterior angles of a heptagon? A 180 B 360 C 540 D

55 21 If a heptagon is regular, what is the measure of each exterior angle? A 72 B 60 C D 45 55

56 22 What is the sum of the measures of the exterior angles of a pentagon? 56

57 23 What is the measure of each exterior angle of a regular pentagon? 57

58 24 The measure of each interior angle of a regular convex polygon is 172. How many sides does the polygon have? 58

59 25 The measure of each interior angle of a regular convex polygon is 174. Find the number of sides of the polygon. A 64 B 62 C 58 D 60 59

60 26 The measure of each interior angle of a regular convex polygon is 162. Find the number of sides of the polygon. 60

61 Properties of Parallelograms Return to the Table of Contents 61

62 Click on the link below and complete the two labs before the Parallelogram lesson. Lab - Investigating Parallelograms Lab - Quadrilaterals Party Teacher Notes 62

63 Parallelograms By definition, the opposite sides of a parallelogram are parallel. D E G F In parallelogram DEFG, DG? DE? 63

64 Theorem The opposite sides a parallelogram are congruent. D A C B Math Practice If ABCD is a parallelogram, then AB DC and DA CB 64

65 Proof of Theorem The opposite sides a parallelogram are congruent. Since we are proving that this theorem is true, we are not using it directly in the proof. Instead, we need to use other properties/theorems that we know. A B D C Construct diagonal AC as shown and consider it a transversal to sides AD and BC, which we know are. What does that tell about the pairs of angles: ACD & CAB and ACB & CAD Why? 65

66 Proof of Theorem The opposite sides a parallelogram are congruent. A B D C Given that ACD CAB and ACB CAD Name a pair of congruent Δs For what reason are they congruent? 66

67 Proof of Theorem The opposite sides a parallelogram are congruent. A B D C Given that ΔACD ΔCAB Which pairs of sides are must be congruent? Why? 67

68 27 The opposite sides of a parallelogram are A Congruent B Parallel C Both a and b 68

69 Theorem The opposite angles of a parallelogram are congruent. D A C B Math Practice If ABCD is a parallelogram, then m A = m C and m B = m D 69

70 Proof of Theorem The opposite angles of a parallelogram are congruent. A B D C As we did earlier, construct diagonal AC. We earlier proved ΔACD ΔCAB What does this tell you about B and D? Why? 70

71 28 ΔDAB ΔBCD. Prove your answer. True A B False D C 71

72 29 Based on the previous question, m A =? Prove your answer. A m B B m C A B C m D D None of the above D C 72

73 Theorem The consecutive angles of a parallelogram are supplementary. D y A x x C y B Math Practice If ABCD is a parallelogram, then x o + y o = 180 o 73

74 Proof of Theorem The consecutive angles of a parallelogram are supplementary. A B x y D y x C Consider AD and BC to be transversals of AB and DC, which are. What are these pairs of angles called: A & D and B & C? What is the relationship of each pair of angles like that? 74

75 Proof of Theorem The consecutive angles of a parallelogram are supplementary. A B x y D y x C Repeat the same process, but this time consider AB and DC to be transversals of AD and BC, which are. What are these pairs of angles called: A & B and D & C? What is the relationship of each pair of angles like that? 75

76 30 ABCD is a parallelogram. Find the value of x. A 65 2y w B 12 x-5 D 9 (5z) C 76

77 31 ABCD is a parallelogram. Find the value of y. A 2y B 65 w 12 x-5 D 9 (5z) C 77

78 32 ABCD is a parallelogram. Find the value of z. A 65 2y w B 12 x-5 D 9 (5z) C 78

79 33 ABCD is a parallelogram. Find the value of w. A 65 2y w B 12 x-5 D 9 (5z) C 79

80 34 DEFG is a parallelogram. Find the value of w. D 21 E 70 G 15 (2w) (z + 12) 3x - 3 F y 2 80

81 35 DEFG is a parallelogram. Find the value of x. D 21 E 70 G 15 (2w) (z + 12) 3x - 3 F y 2 81

82 36 DEFG is a parallelogram. Find the value of y. D 21 E 70 G 15 (2w) (z + 12) 3x - 3 F y 2 82

83 37 DEFG is a parallelogram. Find the value of z. D 21 E 70 G 15 (2w) (z + 12) 3x - 3 F y 2 83

84 Theorem The diagonals of a parallelogram bisect each other. A B D E C Math Practice If ABCD is a parallelogram, then AE EC and BE ED 84

85 Proof of Theorem The diagonals of a parallelogram bisect each other. Since we are proving that this theorem is true, we are not using it directly in the proof. Instead, we need to use other properties/theorems that we know. Steps to follow: D 1) Identify pairs of triangles in the above diagram that you need to prove congruent to prove this theorem. A E C B 2) Then, using what you know about the relationships between pairs of angles and pairs of sides in a parallelogram and the properties of the angles formed by parallel lines & transversals, prove that the triangles are congruent. 3) Then prove the theorem by stating that corresponding parts of congruent triangles are congruent (CPCTC). 85

86 38 LMNP is a parallelogram. The diagonals bisect each other. Find QN P L 6 4 Q N M 86

87 39 LMNP is a parallelogram. Find MP L 4 M P 6 Q N 87

88 40 BEAR is a parallelogram. Find the value of x. B x 4y E 8 S 10 R A 88

89 41 BEAR is a parallelogram. Find the value of y. B x 4y E R 8 S 10 A 89

90 42 BEAR is a parallelogram. Find ER. B x 4y E R 8 S 10 A 90

91 43 In a parallelogram, the opposite sides are parallel. A B C sometimes always never 91

92 44 The opposite angles of a parallelogram are. A B C D bisect congruent parallel supplementary 92

93 45 The consecutive angles of a parallelogram are. A B C D bisect congruent parallel supplementary 93

94 46 The diagonals of a parallelogram each other. A B C D bisect congruent parallel supplementary 94

95 47 The opposite sides of a parallelogram are. A B C D bisect congruent parallel supplementary 95

96 48 MATH is a parallelogram. AH = 12 and MR = 7. Find RT. A 6 M A B 7 7 C 8 D 9 R 12 H T 96

97 49 MATH is a parallelogram. Find AR. A 6 B 7 C 8 M 7 A D 9 12 R H T 97

98 50 MATH is a parallelogram. Find m H. M 14 (3y + 8) A H 2x T 98

99 51 MATH is a parallelogram. Find the value of x. M x 2-24 (3y + 8) A H 2x 98 T 99

100 52 MATH is a parallelogram. Find the value of y. M 14 (3y + 8) A H 2x T 100

101 Proving Quadrilaterals are Parallelograms Math Practice Return to the Table of Contents 101

102 Theorem If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. In quadrilateral ABCD, AB DC and AD BC, A B so ABCD is a parallelogram. D C 102

103 Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. In quadrilateral ABCD, A D and B C, A B so ABCD is a parallelogram. D C 103

104 53 PQRS is a Parallelogram. Explain your reasoning. True False S 4 R 6 P 6 4 Q 104

105 54 Is PQRS a parallelogram? Explain your reasoning. Yes No P Q S R 105

106 55 Is this quadrilateral definitely a parallelogram? Yes No

107 56 Is this quadrilateral definitely a parallelogram? Yes No

108 57 Is this quadrilateral definitely a parallelogram? Yes No 108

109 Theorem If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. A B D 75 C In quadrilateral ABCD, m A + m B = 180 and m B + m C = 180, so ABCD is a parallelogram. 109

110 Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. A B E D C In quadrilateral ABCD, AE CE and DE BE, so ABCD is a quadrilateral. 110

111 Theorem If one pair of sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. A B D C In quadrilateral ABCD, AD CB and AD BC, so ABCD is a parallelogram. 111

112 58 Is this quadrilateral definitely a parallelogram? Yes No 112

113 59 Is this quadrilateral definitely a parallelogram? Yes No

114 60 Is this quadrilateral definitely a parallelogram? Yes No

115 61 Is this quadrilateral definitely a parallelogram? Yes No 115

116 62 Three interior angles of a quadrilateral measure 67, 67 and 113. Is this enough information to tell whether the quadrilateral is a parallelogram? Explain. Yes No 116

117 Fill in the blank parallel perpendicular supplementary bisect congruent In a parallelogram... the opposite sides are and, the opposite angles are, the consecutive angles are, and the diagonals each other. 117

118 Fill in the blank parallel perpendicular supplementary bisect congruent To prove a quadrilateral is a parallelogram... both pairs of opposite sides of a quadrilateral must be, both pairs of opposite angles of a quadrilateral must be, an angle of the quadrilateral must be to its consecutive angles, the diagonals of the quadrilateral each other, or one pair of opposite sides of a quadrilateral are and. 118

119 6(7-3) 63 Which theorem proves the quadrilateral is a parallelogram? A B C D E F The opposite angles are congruent. The opposite sides are congruent. An angle in the quadrilateral is supplementary to its consecutive angles. The diagonals bisect each other. One pair of opposite sides are congruent and parallel. Not enough information. 3(2) 3 119

120 64 Which theorem proves this quadrilateral is a parallelogram? A B C D E F The opposite angles are congruent. The opposite sides are congruent. An angle in the quadrilateral is supplementary to its consecutive angles. The diagonals bisect each other. One pair of opposite sides are congruent and parallel. Not enough information. 120

121 65 Which theorem proves the quadrilateral is a parallelogram? 6 A B The opposite angles are congruent. The opposite sides are congruent. 3(6-4) 6 C D E F An angle in the quadrilateral is supplementary to its consecutive angles. The diagonals bisect each other. One pair of opposite sides are congruent and parallel. Not enough information. 121

122 Rhombi, Rectangles and Squares Math Practice Return to the Table of Contents 122

123 Three Special Parallelograms All the same properties of a parallelogram apply to the rhombus, rectangle, and square. Rhombus Rectangle Square 123

124 Rhombus A rhombus is a quadrilateral with four congruent sides. A B If AB BC CD DA, then ABCD is a rhombus. D C 124

125 66 What is the value of y that will make the quadrilateral a rhombus? A 7.25 B 12 C 20 D y 125

126 67 What is the value of y that will make this parallelogram a rhombus? A 7.25 B 12 C 20 D 25 2y+29 6y 126

127 Theorem The diagonals of a rhombus are perpendicular. A B If ABCD is a rhombus, then AC BD. D C 127

128 Theorem The diagonals of a rhombus are perpendicular. A B O Since ABCD is a parallelogram, its diagonals bisect each other. AO CO and BO DO. D C 128

129 Theorem The diagonals of a rhombus are perpendicular. A O B By definition, the sides of a rhombus are of equal measure, which means that AB = BC = CD = DA. Then by SSS, ΔAOB ΔBOC ΔCOD ΔDOA D C And due to corresponding parts of Δs being (or CPCTC) AOB COB COD AOD If the adjacent angles formed by intersecting lines are congruent, then the lines are perpendicular. 129

130 Theorem If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. A B If ABCD is a rhombus, then D C DAC BAC BCA DCA and ADB CDB ABD CBD 130

131 Theorem If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. A B We proved earlier by SSS, that ΔAOB ΔBOC ΔCOD ΔAOD D O C Corresponding parts of Δs are ; so ABO CBO ADO CDO AND DAO BAO BCO DCO So each diagonal bisects a pair of opposite angles 131

132 68 EFGH is a rhombus. Find the value of x. E 72 2x - 6 z F 5y G 10 H 132

133 69 EFGH is a rhombus. Find the value of y. E 72 2x - 6 z F 5y G 10 H 133

134 70 EFGH is a rhombus. Find the value of z. E 2x - 6 F 5y 72 z G 10 H 134

135 71 This quadrilateral is a rhombus. Find the value of x. 8 z 86 3x + 2 1y

136 72 This quadrilateral is a rhombus. Find the value of y. 8 z 86 3x + 2 1y

137 73 This quadrilateral is a rhombus. Find the value of z. 8 z 86 3x + 2 1y

138 74 This is a rhombus. Find the value of x. x 138

139 75 This is a rhombus. Find the value of x. 1 3 x

140 76 This is a rhombus. Find the value of x. 126 x 140

141 77 HJKL is a rhombus. Find the length of HJ. H J 6 M 16 L K 141

142 Rectangles A rectangle is a quadrilateral with only right angles. A B D C If ABCD is a rectangle, then A, B, C and D are right angles. If A, B, C and D are right angles, then ABCD is a rectangle. 142

143 78 What value of y will make the quadrilateral a rectangle? (6y)

144 Theorem The diagonals of a rectangle are congruent. A B M D C If ABCD is a rectangle, then AC BD. Also, since a rectangle is a parallelogram, AM BM CM DM 144

145 Theorem The diagonals of a rectangle are congruent. A D B C Since ABCD is a rectangle (and therefore a parallelogram) AB DC Because of the reflexive property, BC BC. Since ABCD is a rectangle m B = m C = 90º Then ΔABC ΔDCB by SAS And, AC DB since they are corresponding parts of Δs 145

146 79 RECT is a rectangle. Find the value of x. R 63 (9y) E 2x T C 146

147 80 RECT is a rectangle. Find the value of y. R 63 (9y) E 2x T C 147

148 81 RSTU is a rectangle. Find the value of z. R S U (8z) T 148

149 82 RSTU is a rectangle. Find the value of z. R S 4z U T 149

150 Square A square is a quadrilateral with equal sides and equal angles. It is, therefore both a rhombus and a rectangle. A square has all the properties of a rectangle and rhombus. 150

151 83 This quadrilateral is a square. Find the value of x. 6 z - 4 (5x) (3y) 151

152 84 This quadrilateral is a square. Find the value of y. 6 z - 4 (5x) (3y) 152

153 85 This quadrilateral is a square. Find the value of z. z (5x) (3y) 153

154 86 This quadrilateral is a square. Find the value of x. y 2 (12z) 10-3y (x 2 + 9) 154

155 87 This quadrilateral is a square. Find the value of y. y y (12z) (x 2 + 9) 155

156 88 This quadrilateral is a square. Find the value of z. y 2 (12z) 10-3y (x 2 + 9) 156

157 89 This quadrilateral is a square. Find the value of y. (18y) 157

158 90 This quadrilateral is a rhombus. Find the value of x. 4x 2x

159 91 The quadrilateral is parallelogram. Find the value of x. 112 (4x) 159

160 92 The quadrilateral is a rectangle. Find the value of x. 10x 3x

161 Fill in the Table with the Correct Definition rhombus rectangle square Opposite sides are Diagonals are Diagonals bisect opposite 's Has 4 right 's Has 4 sides Diagonals are 161

162 Lab - Quadrilaterals in the Coordinate Plane Math Practice 162

163 Trapezoids Return to the Table of Contents 163

164 Trapezoid A trapezoid is a quadrilateral with one pair of parallel sides. base leg base angles leg base The parallel sides are called bases. The nonparallel sides are called legs. A trapezoid also has two pairs of base angles. 164

165 Isosceles Trapezoid An isosceles trapezoid is a trapezoid with congruent legs. 165

166 Theorem If a trapezoid is isosceles, then each pair of base angles are congruent. A B D C If ABCD is an isosceles trapezoid, then A B and C D. 166

167 Theorem If a trapezoid is isosceles, then each pair of base angles are congruent. Given: ABCD is an isosceles trapezoid Prove: A B and C D A B D E F C 167

168 Theorem If a trapezoid is isosceles, then each pair of base angles are congruent. Given: ABCD is an isosceles trapezoid Prove: A B and C D. A B Math Practice D E F C continued

169 A B Theorem D E F Statement 1 ABCD is an isosceles trapezoid 2 AB DC, AD BC C Given: ABCD is an isosceles trapezoid Prove: A B and C D. Reason 3 4 Draw segments from A and B perpendicular to AB and DC. These segments are AE and BF. EAB, AEF, BFE, FBA, BFC and AED are right angles Geometric Construction 5 All right angles are 6 Definition of congruent and right angles continued

170 Theorem 7 ABFE is a rectangle 8 AE BF 9 ΔAED ΔBFC 10 C D DAE CBF 11 m DAE = m CBF 12 m DAE + 90 = m CBF m DAE + m EAB = m CBF + m FBA m DAE + m EAB = m DAB m CBF + m FBA = m CBA 15 m DAB = m CBA 16 DAB CBA, or A B 170

171 Theorem If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. A B D C If in trapezoid ABCD, A B. Then, ABCD is an isosceles trapezoid. 171

172 Theorem If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. Case 1 - Longer Base s are Given: In trapezoid ABCD, C D Prove: ABCD is an isosceles trapezoid A B Math Practice D E F C continued

173 Statement 1 In trapezoid ABCD, C D Theorem Reason 2 AB DC 3 4 Draw segments from A and B perpendicular to AB and DC. These segments are AE and BF. EAB, AEF, BFE, FBA, BFC and AED are right angles Geometric Construction 5 All right angles are 6 ABFE is a rectangle 7 AE BF 8 ΔAED ΔBFC 9 AD BC 10 ABCD is an isosceles trapezoid 173

174 Theorem If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. A B Case 2 - s of shorter base are In this case, extend the shorter base and draw a perpendicular line from each vertex of the longer base to the extended base. D C 174

175 Theorem F If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. A B E D C Case 2 - s of shorter base are In this case, extend the shorter base and draw a perpendicular line from each vertex of the longer base to the extended base, as shown. Can you now see how to extend the approach we used for the first case to this second case? 175

176 Theorem If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. F A B E Case 2 - s of shorter base are Given: In trapezoid ABCD, A B Prove: ABCD is an isosceles trapezoid D C Steps to follow: 1) Include the constructions of our new perpendicular segments CE & DF in your proof. 2) Identify pairs of triangles in the above diagram that you need to prove congruent to prove this theorem. 3) Then, using what you know about the relationships between pairs of angles and sides in isosceles trapezoids and rectangles, prove that the triangles are congruent. 4) Then prove the theorem by stating that corresponding parts of congruent triangles are congruent (CPCTC). 176

177 93 The quadrilateral is an isosceles trapezoid. Find the value of x. 6 A 3 B 5 C 7 D x

178 94 The quadrilateral is an isosceles trapezoid. Find the value of x. A 3 B 5 C 7 D 9 64 (9x + 1) 178

179 Theorem A trapezoid is isosceles if and only if its diagonals are congruent. A B D C In trapezoid ABCD, If AC BD. Then, ABCD is isosceles. If ABCD is isosceles Then, AC BD. 179

180 Theorem A trapezoid is isosceles if and only if its diagonals are congruent. A B D Statement 1 Trapezoid ABCD is isosceles 2 AB DC, AD BC, A B, C D 4 CD CD C Reason 5 SAS 6 AC BD Given: Trapezoid ABCD is isosceles Prove: AC BD Teacher's Note 180

181 95 PQRS is a trapezoid. Find the m S. P Q S (6w + 2) (3w) R 181

182 96 PQRS is a trapezoid. Find the m R. P Q S (6w + 2) (3w) R 182

183 97 PQRS is an isosceles trapezoid. Find the m Q. P 123 (9w - 3) Q S (4w + 1) R 183

184 98 PQRS is an isosceles trapezoid. Find the m R. P 123 (9w - 3) Q S (4w + 1) R 184

185 99 PQRS is an isosceles trapezoid. Find the m S. P 123 (9w - 3) Q S (4w + 1) R 185

186 100 The trapezoid is isosceles. Find the value of x x + 3 2x

187 101 The trapezoid is isosceles. Find the value of x. 137 x 187

188 102 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals? Yes No H I K J 188

189 Midsegment of a Trapezoid The midsegment of a trapezoid is a segment that joins the midpoints of the legs Teacher Notes Click on the link below and complete the lab. Lab - Midsegments of a Trapezoid 189

190 Theorem The midsegment of a trapezoid is parallel to both the bases. A B E F AB EF DC D C 190

191 D E The midsegment of a trapezoid is parallel to both the bases. A Proof of Theorem B F C It's given that E is the midpoint of AD; F is the midpoint of BC; and AB DC. Since AB DC, they have the same slope and therefore the distance between them, along any perpendicular line, will be the same. Any line that connects the midpoints of the legs will also be the same distance from each base if measured along a perpendicular line. Which means that the lines that connect the midpoints will have the same slope as the bases. If the slopes are the same, the lines are parallel. That means that AB EF DC, where EF is the midsegment of the trapezoid. 191

192 A B Theorem The length of the midsegment is half the sum of the bases. E F EF = 0.5(AB + DC) D C 192

193 A Proof of Theorem The length of the midsegment is half the sum of the bases. B D E 2x x G H I J y 2y F C Given: E is the midpoint of AD F is the midpoint of BC AB DC EF Prove: EF = 0.5(AB + DC) Math Practice continued

194 1 Statement E is the midpoint of AD F is the midpoint of BC AB DC EF 2 AEG ADH, BFI BCJ 3 EAG DAH, FBI CBJ 4 AEG ~ ADH, BFI ~ BCJ 5 Proof of Theorem AD AH DH BC BJ CJ AE = AG = EG, BF = BI = IF 6 AD = 2AE, BC = 2BF 7 8 AD = 2 BC = 2 AE, BF The scale factor in both of our similar triangles is 2. Reason The midpoints E and F divide AD and BC into 2 congruent segments, so AD and BC are twice the size of AE and BF respectfully. continued

195 Proof of Theorem Statement If EG = x, then DH = 2x. 9 If IF = y, then CH = 2y. 10 EF = AB + x + y 11 2EF = 2AB + 2x + 2y 12 DC = AB + 2x + 2y 13 2x + 2y = DC - AB 14 2EF = 2AB + DC - AB 15 2EF = AB + DC Reason 16 EF = 0.5(AB + DC) 195

196 103 PQRS is a trapezoid. Find LM. Q 15 R L P 7 S M 196

197 104 PQRS is a trapezoid. Find PS. Q 20 R L 14.5 M P S 197

198 105 PQRS is an trapezoid. ML is the midsegment. Find the value of x. S 10 P y 7 14 M L 5 x z R Q 198

199 106 PQRS is an trapezoid. ML is the midsegment. Find the value of y. y M 5 R S x P 7 L z Q 199

200 107 PQRS is an trapezoid. ML is the midsegment. Find the value of z. y M 5 R S x P 7 L z Q 200

201 108 EF is the midsegment of trapezoid HIJK. Find the value of x. H 6 I E x F K 15 J 201

202 109 EF is the midsegment of trapezoid HIJK. Find the value of x. J 19 K F 10 E x I H 202

203 110 Which of the following is true of every trapezoid? Choose all that apply. A B C D Exactly 2 sides are congruent. Exactly one pair of sides are parallel. The diagonals are perpendicular. There are 2 pairs of base angles. 203

204 Kites Return to the Table of Contents 204

205 Kites A kite is a quadrilateral with two pairs of adjacent congruent sides whose opposite sides are not congruent. Math Practice Lab - Properties of Kites 205

206 If a quadrilateral is a kite, then it has one pair of congruent opposite angles. B Theorem A C In kite ABCD, B D and A C D 206

207 Proof of Theorem If a quadrilateral is a kite, then it has one pair of congruent opposite angles. A B D C Draw line AC and use the Reflexive Property to say that AC AC. ΔABC ΔADC by SSS. B D since corresponding parts of Δs are. m A m C since AD CD. In kite ABCD, m B = m D and m A m C 207

208 111 LMNP is a kite. Find the value of x. M (x 2-1) L N P 208

209 112 READ is a kite. RE. A B EA AD E C DR R A D 209

210 113 READ is a kite. A. A E B D C R R E A D 210

211 114 Find the value of z in the kite. z 5z

212 115 Find the value of x in the kite. 68 (8x + 4)

213 116 Find the value of x. 36 (3x 2-2)

214 Theorem The diagonals of a kite are perpendicular. A D B In kite ABCD AC BD C 214

215 Proof of Theorem The diagonals of a kite are perpendicular. Given that AD AB and DC BC A AC AC and AE AE by reflexive property D E B ΔADC ΔABC by SSS DAE BAE by CPCTC ΔADE ΔABE by SAS AED AEB by CPCTC C In kite ABCD AC BD AE is perpendicular to BD since intersecting lines that create adjacent angles are perpendicular 215

216 117 Find the value of x in the kite. x 216

217 118 Find the value of y in the kite. (12y) 217

218 Constructing Quadrilaterals Math Practice Return to the Table of Contents 218

219 Construction of a Parallelogram 1. Draw two intersecting lines. 219

220 Construction of a Parallelogram 2. Place a compass at the point of intersection and draw an arc where it intersects one line in two places

221 Construction of a Parallelogram 3. Keeping the compass at the point of intersection, either enlarge, or shorten the radius of the compass and draw an arc where it intersects the other line in 2 places

222 Construction of a Parallelogram 4. Connect the four points where the arcs intersect the lines. 222

223 Construction of a Parallelogram Which property of parallelograms does this construction technique depend upon? 223

224 Try this! Construct a parallelogram using the lines below. 1) Construction of a Parallelogram 224

225 Try this! Construct a parallelogram using the lines below. 2) Construction of a Parallelogram 225

226 Videos Demonstrating the Constructions of a Parallelogram using Dynamic Geometric Software Click here to see parallelogram video: compass & straightedge Click here to see parallelogram video: Menu options 226

227 Construction of a Rectangle 1. Draw two intersecting lines. 227

228 Construction of a Rectangle 2. Place a compass at the point of intersection and draw an arc where it intersects each line, keeping the radius of the compass the same all the way around

229 Construction of a Rectangle 3. Connect the four points where the arcs intersect the lines. 229

230 Construction of a Rectangle Which properties of rectangles does this construction technique depend upon? 230

231 Construction of a Rectangle Try this! Construct a rectangle using the lines below. 3) 231

232 Construction of a Rectangle Try this! Construct a rectangle using the lines below. 4) 232

233 Videos Demonstrating the Constructions of a Rectangle using Dynamic Geometric Software Click here to see rectangle video: compass & straightedge Click here to see rectangle video: Menu options 233

234 Construction of a Rhombus 1. Construct a segment of any length. 234

235 Construction of a Rhombus 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points a. Place the point of the compass on A and open it so the pencil is more than halfway, it does not matter how far, to point B 235

236 Construction of a Rhombus 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points b) Keeping the compass locked, place the point of the compass on B and draw an arc so that it intersects the one you just drew 236

237 Construction of a Rhombus 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points C c. Draw a line through the arc intersections. The intersection point created with the two perpendicular lines is the center of your rhombus. 237

238 Construction of a Rhombus 3. Place the tip of your compass on your center point (midpoint) C. Extend your compass to any length. Draw 2 arcs where the compass intersects the perpendicular bisector C Note: If you wanted to, you could use the intersection points of the 2 arcs from step #2 as your extra points. 238

239 Construction of a Rhombus 4. Connect the intersection points of the 2 arcs and the perpendicular bisector from step #3 with the endpoints of your original segment. C 239

240 Construction of a Rhombus Which properties of rhombii (pl. for rhombus) does this construction technique depend upon? C 240

241 Videos Demonstrating the Constructions of a Rhombus using Dynamic Geometric Software Click here to see rhombus video: compass & straightedge Click here to see rhombus video: Menu options 241

242 Construction of a Rhombus Try this! Construct a rhombus using the segment below. 5) 242

243 Construction of a Rhombus Try this! Construct a rhombus using the segment below. 6) 243

244 Construction of a Square 1. Construct a segment of any length. 244

245 Construction of a Square 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points a. Place the point of the compass on A and open it so the pencil is more than halfway, it does not matter how far, to point B. Draw an arc. 245

246 Construction of a Square 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points b) Keeping the compass locked, place the point of the compass on B and draw an arc so that it intersects the one you just drew 246

247 Construction of a Square 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points C c. Draw a line through the arc intersections. The intersection point created with the two perpendicular lines is the center of your circle (and future square). 247

248 Construction of a Square 3. Place your compass tip on point C and the pencil point on either A or B and construct your circle C 248

249 Construction of a Square 4. Connect the intersection points of your perpendicular bisector and the circle with the initial vertices of the line segment to create your square. C 249

250 Construction of a Square Which properties of squares does this construction technique depend upon? C 250

251 Construction of a Square Try this! Construct a square using the segment below. 7) 251

252 Construction of a Square Try this! Construct a square using the segment below. 8) 252

253 Videos Demonstrating the Constructions of a Square using Dynamic Geometric Software Click here to see square video: compass & straightedge Click here to see square video: Menu options 253

254 Construction of an Isosceles Trapezoid 1. Draw two intersecting lines. 254

255 Construction of an Isosceles Trapezoid 2. Place a compass at the point of intersection and draw an arc where it intersects two of the lines one time each

256 Construction of an Isosceles Trapezoid 3. Keeping the compass at the point of intersection, either enlarge, or shorten the radius of the compass and draw an arc where it intersects the two lines on the other side (e.g. Since I drew my 1st 2 arcs in the upper half, these 2 arcs will be in the lower half)

257 Construction of an Isosceles Trapezoid 4. Connect the four points where the arcs intersect the lines. 257

258 Construction of an Isosceles Trapezoid Which properties of isosceles trapezoids does this construction technique depend upon? 258

259 Construction of an Isosceles Trapezoid Try this! Construct an isosceles trapezoid using the lines below. 9) 259

260 Construction of an Isosceles Trapezoid Try this! Construct an isosceles trapezoid using the lines below. 10) 260

261 Videos Demonstrating the Constructions of an Isosceles Trapezoid using Dynamic Geometric Software Click here to see isosceles trapezoid video: compass & straightedge Click here to see isosceles trapezoid video: Menu options 261

262 Construction of a Kite 1. Construct a segment of any length. 262

263 Construction of a Kite 2. Place your compass tip at any point and draw 2 arcs that intersect your segment

264 Construction of a Kite 3. Find the perpendicular bisector between the 2 red arcs a) Place your compass tip at one of the intersection points. Extend your pencil out more than half way between the 2 arcs. Draw an arc. 264

265 Construction of a Kite 3. Find the perpendicular bisector between the 2 red arcs b) Keeping the radius of your compass the same, place your compass tip at the other intersection point and draw an arc. 265

266 Construction of a Kite 3. Find the perpendicular bisector between the 2 red arcs. c) Connect the intersections of the 2 arcs to construct your perpendicular line. 266

267 Construction of a Kite 4. Place the tip of your compass at the intersection point of our perpendicular lines. Extend your compass to any length. Draw 2 arcs where the compass intersects the perpendicular line Note: If you wanted to, you could use the intersection points of the 2 arcs from step #3 as your extra points. 267

268 Construction of a Kite 5. Connect the intersection points of your perpendicular line and the arcs from step #4 with the initial vertices of the line segment to create your kite. 268

269 Construction of a Kite Which properties of a kite does this construction technique depend upon? 269

270 Construction of a Kite Try this! Construct a kite using the segment below. 11) 270

271 Construction of a Kite Try this! Construct a kite using the segment below. 12) 271

272 Videos Demonstrating the Constructions of a Kite using Dynamic Geometric Software Click here to see kite video: compass & straightedge Click here to see kite video: Menu options 272

273 Families of Quadrilaterals Return to the Table of Contents 273

274 parallelogram Complete the chart (There can be more than one answer). trapezoid Description An equilateral quadrilateral An equiangular quadrilateral The diagonals are perpendicular The diagonals are congruent Has at least 1 pair of parallel sides rhombus rectangle square isosceles trapezoid kite Special Quadrilateral(s) (s) rhombus & square click to reveal rectangle & square click to reveal rectangle, square & kite click to reveal rhombus, square & isosceles trapezoid click to reveal All except kite click to reveal 274

275 119 A rhombus is a square. A B C always sometimes never 275

276 120 A square is a rhombus. A B C always sometimes never 276

277 121 A rectangle is a rhombus. A B C always sometimes never 277

278 122 A trapezoid is isosceles. A B C always sometimes never 278

279 123 A kite is a quadrilateral. A B C always sometimes never 279

280 124 A parallelogram is a kite. A B C always sometimes never 280

281 Families of Quadrilaterals Sometimes, you will be given a type of quadrilateral and using its qualities find the value of a variable. After substituting the variable back into the algebraic expression(s), the quadrilateral might become a more specific type. Using what we know about the different types of quadrilaterals, let's try these next few problems. Math Practice 281

282 Quadrilaterals Quadrilateral ABCD is a parallelogram. A B x x + 5 D Determine the value of x. 25 C Since we know that ABCD is a parallelogram, we know that the opposite sides are parallel and congruent. click Using the congruence property, we can write an equation & solve it. click 282

283 A Quadrilaterals Quadrilateral ABCD is a parallelogram. B x 2-75 = 2x + 5 click x x + 5 x 2-2x - 80 = 0 click 25 (x - 10)(x + 8) = 0 click D C x = 10 and x = -8 click Do any of these answers not make sense? Yes, since the value of x cannot be negative, x = -8 can be ruled out. BC = 2(-8) + 5 = -11 which is not possible. click Is parallelogram ABCD a more specific type of of quadrilateral? Explain. Since ABCD is a parallelogram, AB = 25 because opposite sides are congruent. When substituting x = 10 back into the expressions, then AD = CD = BC = AB = 25. Therefore, ABCD is a rhombus. click 283

284 125 Quadrilateral EFGH is a parallelogram. Find the value of x. E F x x J 3x x + 8 H 32 G 284

285 126 Based on your answer to the previous slide, what type of quadrilateral is EFGH? Explain. When you are finished, choose the classification of quadrilateral EFGH from the choices below. E F A Rhombus x x J 3x x + 8 B Rectangle C Square H 32 G 285

286 127 Quadrilateral KLMN is a trapezoid. Find the value of x. K (7x + 30) (10x) L N (x 2-20) M 286

287 128 Based on your answer to the previous slide, what type of quadrilateral is KLMN? Explain. N When you are finished, determine whether trapezoid KLMN is isosceles. K (7x + 30) (10x) L (x 2-20) M A Isosceles Trapezoid B Not an Isosceles Trapezoid 287

288 Extension: Inscribed Quadrilaterals A quadrilateral is inscribed if all its vertices lie on a circle... inscribed quadrilateral. 288

289 Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. D E m D = 1/2 (mefg) [green arc] m F = 1/2 (medg) [red arc]. G F Since mefg + medg = 360, m D + m F = 1/2 (mefg) + 1/2 (medg) m D + m F = 1/2 (mefg + medg) m D + m F = 1/2(360 ) m D + m F = 180 m D + m F = 180 m E + m G =

290 Inscribed Quadrilaterals Example: Find the m G. Then, find the value of a. E. D (3a + 8) 121 (2a - 3) F G 290

291 129 What is m L? M A 25 J 115 B C L D K 291

292 130 What is m M? M A 23 J 115 B C L D K 292

293 131 What is the value of x? A 8 J B 10 C 17. D 19 (7x - 6) (5y - 4) M (8y + 15) (3x - 4) L K 293

294 132 What is the value of y? M A 13 B C 7 D 6.08 J. (7x - 6) (5y - 5) (8y + 16) (3x - 4) L K 294

295 Can special quadrilaterals be inscribed into a circle? Throughout this unit, we have been talking about the different quadrilaterals, their properties, and how to construct them. - Parallelogram - Rectangle - Rhombus - Square - Trapezoid - Kite Can any of these quadrilaterals be inscribed into a circle? Math Practice Find out by completing the lab below. Lab: Inscribed Quadrilaterals 295

296 133 Complete the sentence below. A trapezoid can be inscribed into a circle. A Always B Sometimes C Never 296

297 134 Complete the sentence below. A rectangle can be inscribed into a circle. A Always B Sometimes C Never 297

298 135 Complete the sentence below. A rhombus can be inscribed into a circle. A Always B Sometimes C Never 298

299 136 Complete the sentence below. A kite can be inscribed into a circle. A Always B Sometimes C Never 299

300 Proofs Math Practice Return to the Table of Contents 300

301 Proofs Given: TE MA, 1 2 Prove: TEAM is a parallelogram. T 1 E M 2 A 301

302 Proofs T E 1 Option A M 2 A statements reasons 1) TE MA, 1 2 2) EM EM 3) MTE EAM 4) TM AE 5) TEAM is a parallelogram 302

303 Proofs T 1 E Option B M 2 A We are given that TE MA and 2 3. TE AM, by the alternate interior angles converse. click to reveal So, TEAM is a parallelogram because one pair of opposite sides is parallel and congruent. click to reveal 303

304 Proofs Given: FGHJ is a parallelogram, F is a right angle Prove: FGHJ is a rectangle F G J H 304

305 Proofs Given: FGHJ is a parallelogram, F is a right angle Prove: FGHJ is a rectangle F G J H statements 1) FGHJ is a parallelogram and F is a right angle 2) J and G are right angles 1) Given reasons 3) H is a right angle 4) TEAM is a rectangle 305

306 Proofs Given: COLD is a quadrilateral, m O = 140, m D = 40, m L = 60 Prove: COLD is a trapezoid C O 140 D L 306

307 statements 1) COLD is a quadrilateral, m O = 140, m L = 40, m D = 60 2) m O + m L = 180 m L + m D = 100 Proofs Given: COLD is a quadrilateral, m O = 140, m D = 40, m L = 60 Prove: COLD is a trapezoid 1) Given D 60 o C reasons 140 o 3) Definition of Supplementary Angles O 40 o L 4) L and D are not supplementary 5) Consecutive Interior Angles Converse 6) CL OD 7) COLD is a trapezoid 307

308 Proofs Try this... Given: ΔFCD ΔFED Prove: FD CE C Hint F D E 308

309 Coordinate Proofs Return to the Table of Contents 309

310 Quadrilateral Coordinate Proofs Recall that Coordinate Proofs place figures on the Cartesian Plane to make use of the coordinates of key features of the figure to help prove something. Usually the distance formula, midpoint formula and slope formula are used in a coordinate proof. We'll provide a few examples and then have you do some proofs. Math Practice 310

311 Coordinate Proofs Given: PQRS is a quadrilateral Prove: PQRS is a kite y 10 8 P(-1,6) 6 4 S(-4,3) Q(2,3) R(-1,-2) x 311

312 Coordinate Proofs A kite has one unique property. The adjacent sides are congruent. P(-1,6) S(-4,3) Q(2,3) PS = (6-3) 2 + (-1 - (-4)) 2 PQ = (3-6) 2 + (2 - (-1)) 2 = = (-3) = = = 18 = 18 = 4.24 = 4.24 click click R(-1,-2) 312

313 P(-1,6) Coordinate Proofs S(-4,3) Q(2,3) R(-1,-2) RS = (3 - (-2)) 2 +(-4 - (-1)) 2 RQ = (-2-3) 2 + (-1-2) 2 = (-3) 2 = (-5) 2 + (-3) 2 = = = 34 = 34 = 5.83 = 5.83 click click Because PS = PQ and RS = RQ, PQRS is a kite. click 313

314 Coordinate Proofs Given: JKLM is a parallelogram Prove: JKLM is a square y 10 M(-3, 0) J(1, 3) L(0, -4) K(4, -1) x 314

315 Coordinate Proofs M(-3, 0) J(1, 3) L(0, -4) Since JKLM is a parallelogram, we know the opposite sides are parallel and congruent. We also know that a square is a rectangle and a rhombus. We need to prove the adjacent sides are congruent and perpendicular. Let's start by proving the adjacent sides are congruent. JM = (3-0) 2 + (1 - (-3)) 2 JK = (-1-3) 2 + (4-1) 2 = = (-4) = = = 25 = 25 = 5 = 5 click click K(4, -1) 315

316 Coordinate Proofs M(-3, 0) J(1, 3) Now, let's prove that the adjacent sides are perpendicular. m JM = 3-0 = 3 1- (-3) 4 click L(0, -4) K(4, -1) MJ JK and JM JK What else do you know? m JK = -1-3 = MJ LK and JK LM (Opposite sides are congruent) MJ LM and JK LK (Perpendicular Transversal Theorem) Therefore, JKLM is a square click click click 316

317 Try this... Coordinate Proofs Given: PQRS is a trapezoid Prove: LM is the midsegment y Hint 4 2 L(1, 0) S(0, -2) -4-6 P(2, 2) Q(5, 1) x M(7, -2) R(9, -5)

318 Quadrilateral Coordinate Proofs Sometimes, shapes might be placed in a coordinate plane, but without numbered values. Instead, algebraic expressions will be used for the coordinates. You can still use the distance formula, midpoint formula and slope formula in this type of coordinate proof. We'll provide a few examples and then have you do some proofs. Math Practice 318

319 Quadrilateral Coordinate Proofs Given: EFGH is a parallelogram Prove: EFGH is a rectangle y E (q, r) F(p, r) x H(q, u) G(p, u) 319

320 y Quadrilateral Coordinate Proofs E (q, r) F(p, r) x H(q, u) G(p, u) Since EFGH is a parallelogram, we know the opposite sides are parallel and congruent. Since we need to prove that EFGH is a rectangle, we can either prove that adjacent sides are perpendicular or diagonals are congruent. Let's choose the 1st option: proving the adjacent sides are perpendicular. continued

321 y Quadrilateral Coordinate Proofs E (q, r) F(p, r) x H(q, u) G(p, u) m EF = r - r = 0 = 0 p - q p - q click m FG = r - u = r - u = undef. p - p 0 EF FG and both pairs of opposite sides are What else do you know? click EF GH and FG EH (Opposite sides are congruent) EF EH and FG GH (Perpendicular Transversal Theorem) Therefore, EFGH is a rectangle click click 321

322 Quadrilateral Coordinate Proofs E (q, r) F(p, r) y x H(q, u) G(p, u) Since EFGH is a parallelogram, we know the opposite sides are parallel and congruent. Since we need to prove that EFGH is a rectangle, we can either prove that adjacent sides are perpendicular or diagonals are congruent. This problem can also be done using the 2nd option: proving that the diagonals are congruent 322

323 Quadrilateral Coordinate Proofs E (q, r) F(p, r) y x H(q, u) G(p, u) EG = (p - q) 2 + (u - r) 2 HF = (p - q) 2 + (r - u) 2 = (p - q) 2 + (-(u - r)) 2 = (p - q) 2 + (u - r) 2 click click We proved that EG HF (Diagonals are congruent) We also know that EF GH and FG EH (Opp. sides are congruent) Therefore, EFGH is a rectangle click 323

324 Quadrilateral Coordinate Proofs Given: JKLM is a parallelogram Prove: JL and KM bisect each other y K (0, 2b) J (a, 2b) U L(-a, 0) M(0, 0) x 324

325 Quadrilateral Coordinate Proofs y K (0, 2b) J (a, 2b) U L(-a, 0) M(0, 0) x Since JKLM is a parallelogram, we know the opposite sides are parallel and congruent and the diagonals bisect each other. If we can prove the last part, that the diagonals bisect each other using one of our formulas, we will achieve our goal. Which formula would prove that the diagonals bisect each other? 325

326 Quadrilateral Coordinate Proofs y K (0, 2b) J (a, 2b) U L(-a, 0) M(0, 0) x Midpoint of KM: U U(0, b) click b + 0 2, 2 Midpoint of JL: U U(0, b) click -a + a 0 + 2b 2, 2 Because the midpoint U is the same for both of the diagonals, it proves that KM & JL bisect each other. click 326

327 137 If Quadrilateral STUV is a rectangle, what would be the coordinates for point U? A U (e, d) y B U (d + e, e) C U (d, e) V(0, e) U(?,?) D U (d, e - d) S(0, 0) T(d, 0) x 327

328 138 If Quadrilateral OPQR is a parallelogram, what would be the coordinates for point Q? A Q (b, a) y Q(?,?) B Q (b + c, a) P(0, a) C Q (b + c, a + c) D Q (b, a + c) O(0, 0) R(b, c) x 328

329 139 If Quadrilateral ABCD is an isosceles trapezoid, what would be the coordinates for point B? A B (-2f, 2h) y B B (-2g, 2h) C B (-2g, -2h) B(?,?) C(2g, 2h) D B (2f, 2h) A(-2f, 0) D(2f, 0) x 329

330 140 Since ABCD is an isosceles trapezoid, BD and AC are congruent. Use the coordinates to verify that BD and AC are congruent. When you are finished, type in the number "1". B(?,?) y C(2g, 2h) A(-2f, 0) D(2f, 0) x 330

331 PARCC Released Questions The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing unit 10, you should be able to answer these questions. Good Luck! Return to the Table of Contents 331

332 The figure shows parallelogram ABCD with AE = 16. B C 16 E A D not drawn to scale Part A 141 Let BE = x 2-48 and let DE = 2x. What are the lengths of BE and DE? Justify your answer. PARCC Released Question - PBA - Calculator Section - #7 332

333 The figure shows parallelogram ABCD with AE = 16. B C 16 E Part B A 142 What conclusion can be made regarding the specific classification of parallelogram ABCD? Justify your answer. Once you are finished with the question, type in the word to describe the classification of parallelogram ABCD. D not drawn to scale PARCC Released Question - PBA - Calculator Section - #7 333

334 The figure shows parallelogram PQRS on a coordinate plane. Diagonals SQ and PR intersect at point T. y P(2b, 2c) Q(?,?) S(0, 0) T R(2a, 0) x Part A 143 Find the coordinates of point Q in terms of a, b, and c. Enter only your answer. PARCC Released Question - PBA - Calculator Section - #8 334

335 Part B 144 Since PQRS is a parallelogram, SQ and PR bisect each other. Use the coordinates to verify that SQ and PR bisect each other. Which formula(s) could be used to answer this question? A Distance Formula B Slope C Midpoint Formula D A and C only E A, B and C PARCC Released Question - PBA - Calculator Section - #8 335

336 Part B Continued 145 Since PQRS is a parallelogram, SQ and PR bisect each other. Use the coordinates to verify that SQ and PR bisect each other. When you are finished, type in the word "Done". PARCC Released Question - PBA - Calculator Section - #8 336

337 One method that can be used to prove that the diagonals of a parallelogram bisect each other is shown in the given partial proof. P Q? T S R Given: Quadrilateral PQRS is a parallelogram. Prove: PT = RT, ST = QT 337

338 Part A 146 Which reason justifies the statement for step 3 in the proof? A When two parallel lines are intersected by a transversal, same side interior angles are congruent. B When two parallel lines are intersected by a transversal, alternate interior angles are congruent. C When two parallel lines are intersected by a transversal, same side interior angles are supplementary. D When two parallel lines are intersected by a transversal, alternate interior angles are supplementary. PARCC Released Question - EOY - Calculator Section - #19 338

339 Part B 147 Which statement is justified by the reason for step 4 in the proof? A PQ RS B PQ SP C PT TR D SQ PR PARCC Released Question - EOY - Calculator Section - #19 339

340 Part C 148 Which reason justifies the statement for step 5 in the proof? A side-side-side triangle congruence B side-angle-side triangle congruence C angle-side-angle triangle congruence D angle-angle-side triangle congruence PARCC Released Question - EOY - Calculator Section - #19 340

341 Part D Another method of proving diagonals of a parallelogram bisect each other uses a coordinate grid. 341