Geometry. Quadrilaterals. Slide 1 / 189. Slide 2 / 189. Slide 3 / 189. Table of Contents. New Jersey Center for Teaching and Learning

Transcription

1 New Jersey enter for Teaching and Learning Slide 1 / 189 Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. lick to go to website: Slide 2 / 189 Geometry Quadrilaterals Table of ontents Slide 3 / 189 ngles of Polygons Properties of Parallelograms Proving Quadrilaterals are Parallelograms onstructing Parallelograms Rhombi, Rectangles and Squares Trapezoids Kites Families of Quadrilaterals oordinate Proofs Proofs lick on a topic to go to that section.

2 Slide 4 / 189 ngles of Polygons Return to the Table of ontents Polygon Slide 5 / 189 polygon is a closed figure made of line segments connected end to end. Since it is made of line segments, there can be no curves. lso, it has only one inside regioin, so no two segments can cross each other. an you explain why the figure below is not a polygon? click to reveal is not a segment (it has a curve). There are two inside regions. Types of Polygons Slide 6 / 189 Polygons are named by their number of sides. Number of Sides 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

3 Slide 7 / 189 onvex polygons polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. interior oncave polygons Slide 8 / 189 polygon is concave if a line that contains a side of the polygon contains a point in the interior of the polygon. interior 1 The figure below is a polygon. Slide 9 / 189 True False

4 2 The figure below is a polygon. Slide 10 / 189 True False 3 Indentify the polygon. Slide 11 / 189 E F Pentagon Octagon Quadrilateral Hexagon ecagon Triangle 4 Is the polygon convex or concave? Slide 12 / 189 onvex oncave

5 5 Is the polygon convex or concave? Slide 13 / 189 onvex oncave Equilateral, Equiangular, Regular Slide 14 / 189 polygon is equilateral if all its sides are congruent. polygon is equiangular if all its angles are congruent. polygon is regular if it is equilateral and equiangular. 6 escribe the polygon. (hoose all that apply) Slide 15 / 189 Pentagon F onvex Octagon G oncave 4 60 o 4 Quadrilateral Hexagon H I Equilateral Equiangular 60 o 4 60 o E Triangle J Regular

6 7 escribe the polygon. (hoose all that apply) Slide 16 / 189 Pentagon F onvex Octagon G oncave Quadrilateral H Equilateral Hexagon I Equiangular E Triangle J Regular 8 escribe the polygon. (hoose all that apply) Slide 17 / 189 Pentagon F onvex Octagon G oncave Quadrilateral H Equilateral E Hexagon Triangle I J Equiangular Regular ngle Measures of Polygons Slide 18 / 189 bove are examples of a triangle, quadrilateral, pentagon and hexagon. In each polygon, diagonals are drawn from one vertex. What do you notice about the regions created by the diagonals? They are triangular click

7 omplete the table Slide 19 / 189 Polygon Number of Sides Number of Triangular Regions Sum of the Interior ngles triangle 3 1 1(180 o ) = 180 o quadrilateral 4 2 2(180 o ) = 360 o pentagon 5 3 3(180 o ) = 540 o hexagon 6 4 4(180 o ) = 720 o Slide 20 / 189 Given: Polygon EFG G F E lassify the polygon. How many triangular regions can be drawn in polygon EFG? What is the sum of the measures of the interior angles on EFG? Polygon Interior ngles Theorem Q1 Slide 21 / 189 The sum of the measures of the interior angles of a convex polygon with n sides is 180(n-2). Polygon Number of Sides Sum of the measures of the interior angles. hexagon 6 180(6-2) = 720 o omplete the table. heptagon 7 180(7-2) = 900 o octagon 8 180(8-2) = 1080 o nonagon 9 180(9-2)=1260 o decagon (10-2)=1440 o 11-gon (11-2) = 1620 o dodecagon (12-2) = 1800 o

8 Slide 22 / 189 Example: Find the value of each angle. L (3x+4) o M 146 o x o N P (2x+3) o (3x) o O The figure above is a pentagon. The sum of measures of the interior angles a pentagon is 540 o. Slide 23 / 189 m L + m M + m N + m O + m P = o 540 (3x+4) x + (3x) + (2x+3) = 540 (ombine Like Terms) 9x = x = x = 43 o o o m L=3(43)+4=133 m M=146 m N=x=43 o m O=3(43)=129 o m P=2(43)+3=89 click to reveal o o o o o o heck: =540 Polygon Interior ngles Theorem orollary Slide 24 / 189 The measures of each interior angle of a regular polygon is: 180(n-2) n regular polygon number of sides sum of interior angles measure of each angle triangle o 60 o omplete the table. quadrilateral o 90 o pentagon o 108 o hexagon o 120 o octagon o 135 o decagon o 144 o 15-gon o 156 o

9 9 What is the sum of the measures of the interior angles of the stop sign? Slide 25 / If the stop sign is a regular polygon. What is the measure of each interior angle? Slide 26 / What is the sum of the measures of the interior angles of a convex 20-gon? Slide 27 /

10 12 What is the measure of each interior angle of a regular 20-gon? Slide 28 / What is the measure of each interior angle of a regular 16-gon? Slide 29 / What is the value of x? Slide 30 / 189 (10x+8) o (8x) o (5x+15) o (9x-6) o (11x+16) o

11 Polygon Exterior ngle Theorem Q2 Slide 31 / 189 The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360 o. x y z In other words, x + y + z = 360 o Polygon Exterior ngle Theorem orollary Slide 32 / 189 The measure of each exterior angle of a regular polygon with n sides is 360 n a The polygon is a hexagon. n=6 a=360 6 a = 60 o 15 What is the sum of the measures of the exterior angles of a heptagon? Slide 33 /

12 16 If a heptagon is regular, what is the measure of each exterior angle? Slide 34 / What is the sum of the measures of the exterior angles of a pentagon? Slide 35 / If a pentagon is regular, what is the measure of each exterior angle? Slide 36 / 189

13 Slide 37 / 189 Example: The measure of each angle of a regular convex polygon is o 172. Find the number of sides of the polygon. We need to use 180(n-2) to find n. n 19 The measure of each angle of a regular convex o polygon is 174. Find the number of sides of the polygon. 64 Slide 38 / The measure of each angle of a regular convex o polygon is 162. Find the number of sides of the polygon. Slide 39 / 189

14 Slide 40 / 189 Properties of Parallelograms Return to the Table of ontents lick on the links below and complete the two labs before the Parallelogram lesson. Slide 41 / 189 Lab - Investigating Parallelograms Lab - Properties of Parallelograms Parallelograms Slide 42 / 189 Parallelogram is a quadrilateral whose both pairs of opposite sides are parallel. G F E In parallelogram EFG, G EF and E GF

15 Theorem Q3 Slide 43 / 189 If a quadrilateral is a parallelogram, then its opposite sides are congruent. If is a parallelogram, then = and = Theorem Q4 Slide 44 / 189 If a quadrilateral is a parallelogram, then its opposite angles are congruent. If is a parallelogram, then m = m and = m m Slide 45 / 189 Theorem Q5 If a quadrilateral is a parallelogram, then the consecutive angles are supplementary. x o y o y o x o If is a parallelogram, then x o + y o = 180 o

16 Slide 46 / 189 2y Example: 65 o w o is parallelogram. Find w, x, y, and z. 12 x-5 9 5z o 2y Slide 47 / o w o The opposite sides are congruent. 12 x-5 9 5z o Slide 48 / o 2y w o The opposite angles are congruent. 12 x-5 9 5z o

17 2y Slide 49 / 189 The consecutive angles are supplementary o w o x-5 9 5z o 21 EFG is a parallelogram. Find w. Slide 50 / o 15 2w z+12 G 3x-3 F y 2 E 22 EFG is a parallelogram. Find x. Slide 51 / o 15 2w z+12 G 3x-3 F y 2 E

18 23 EFG is a parallelogram. Find y. Slide 52 / o 15 2w z+12 G 3x-3 F y 2 E 24 EFG is a parallelogram. Find z. Slide 53 / o 15 2w z+12 G 3x-3 F y 2 E Theorem Q5 Slide 54 / 189 If a quadrilateral is a parallelogram, then the diagonals bisect each other. E If is a parallelogram, then E E and E E

19 Example: Slide 55 / 189 LMNP is a parallelogram. Find QN and MP. (The diagonals bisect each other) P L 6 4 Q N M Slide 56 / 189 Try this... ER is a parallelogram. Find x, y, and ER. x 4y E 8 S 10 R 25 In a parallelogram, the opposite sides are parallel. Slide 57 / 189 sometimes always never

20 26 MTH is a parallelogram. Find RT. Slide 58 / M R 12 H T 27 MTH is a parallelogram. Find R. Slide 59 / M 7 R 12 H T 28 MTH is a parallelogram. Find m H. Slide 60 / 189 M (3y+8) o 14 H 2x-4 98 o T

21 29 MTH is a parallelogram. Find x. Slide 61 / 189 M (3y+8) o 14 H 2x-4 98 o T 30 MTH is a parallelogram. Find y. Slide 62 / 189 M (3y+8) o 14 H 2x-4 98 o T Slide 63 / 189 Proving Quadrilaterals are Parallelograms Return to the Table of ontents

22 Theorem Q6 Slide 64 / 189 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. In quadrilateral, and, so is a parallelogram. Theorem Q7 Slide 65 / 189 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. In quadrilateral, and, so is a quadrilateral. Example Slide 66 / 189 Tell whether PQRS is a parallelogram. Explain. S 6 P 4 4 R 6 Q

23 Example Slide 67 / 189 Tell whether PQRS is a parallelogram. Explain. P Q S R 31 Tell whether the quadrilateral is a parallelogram. Slide 68 / 189 Yes No 136 o 2 78 o 32 Tell whether the quadrilateral is a parallelogram. Slide 69 / 189 Yes No

24 33 Tell whether the quadrilateral is a parallelogram. Slide 70 / 189 Yes No Slide 71 / 189 Theorem Q8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. 75 o 75 o 105 o In quadrilateral, o m + m =180 o and m + m =180, so is a parallelogram. Slide 72 / 189 Theorem Q9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. E In quadrilateral, E E and E E, so is a quadrilateral.

25 Slide 73 / 189 Theorem Q10 If one pair of sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. In quadrilateral, and, so is a parallelogram. 34 Tell whether the quadrilateral is a parallelogram. Slide 74 / 189 Yes No 35 Tell whether the quadrilateral is a parallelogram. Slide 75 / 189 Yes No 141 o 49 o 39 o

26 36 Tell whether the quadrilateral is a parallelogram. Slide 76 / 189 Yes No Tell whether the quadrilateral is a parallelogram. Slide 77 / 189 Yes No Example: Slide 78 / 189 o Three interior angles of a quadrilateral measure 67, 67 o and 113. Is this enough information to tell whether the quadrilateral is a parallelogram? Explain. o

27 Fill in the blank Slide 79 / 189 bisect congruent parallel perpendicular supplementary In a parallelogram... the opposite sides are and, the opposite angles are, the consecutive angles are and the diagonals each other. Fill in the blank Slide 80 / 189 bisect congruent parallel perpendicular supplementary 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. 38 Which theorem proves the quadrilateral is a parallelogram? 6(7-3) Slide 81 / 189 E F The opposite angle are congruent. The opposite sides are congruent. n angle in the quadrilateral is supplementary to its 3(2) consecutive angles. 3 The diagonals bisect each other. One pair of opposite sides are congruent and parallel. Not enough information.

28 39 Which theorem proves the quadrilateral is a parallelogram? Slide 82 / 189 E F The opposite angle are congruent. The opposite sides are congruent. n 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. 40 Which theorem proves the quadrilateral is a parallelogram? Slide 83 / 189 The opposite angle are congruent. 6 The opposite sides are congruent. 3(6-4) 6 E F n 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. Slide 84 / 189 onstructing Parallelograms Return to the Table of ontents

29 onstruct a Parallelogram Slide 85 / 189 To construct a parallelogram, there are 3 steps. onstruct a Parallelogram - Step 1 Slide 86 / 189 Step 1 - Use a ruler to draw a segment and its midpoint. onstruct a Parallelogram - Step 2 Slide 87 / 189 Step 2 - raw another segment such that the midpoints coincide.

30 onstruct a Parallelogram - Step 3 Slide 88 / 189 Why is this a parallelogram? Step 3 - onnect the endpoints of the segments. 3 steps to draw a parallelogram in a coordinate plane Slide 89 / units Step 1 - raw a horizontal segment in the plane. Find the length of the segment steps to draw a parallelogram in a coordinate plane Slide 90 / units units Step 2 - raw another horizontal line of the same length, anywhere in the plane

31 3 steps to draw a parallelogram in a coordinate plane Slide 91 / units Step 3 - onnect the endpoints units -6 Why is this a parallelogram? Slide 92 / Note: this method also works with vertical lines. 41 The opposite angles of a parallelogram are... Slide 93 / 189 bisect congruent parallel supplementary

32 42 The consecutive angles of a parallelogram are... Slide 94 / 189 bisect congruent parallel supplementary 43 The diagonals of a parallelogram each other. Slide 95 / 189 bisect congruent parallel supplementary 44 The opposite sides of a parallelogram are... Slide 96 / 189 bisect congruent parallel supplementary

33 Slide 97 / 189 Rhombi, Rectangles and Squares Return to the Table of ontents three special parallelograms Slide 98 / 189 Rhombus ll the same properties of a parallelogram apply to the rhombus, rectangle, and square. Square Rectangle Rhombus orollary Slide 99 / 189 quadrilateral is a rhombus if and only if it has four congruent sides. If is a quadrilateral with four congruent sides, then it is a rhombus.

34 45 What is the value of y that will make the quadrilateral a rhombus? Slide 100 / y 46 What is the value of y that will make the quadrilateral a rhombus? y+29 Slide 101 / 189 6y Theorem Q11 Slide 102 / 189 If a parallelogram is a rhombus, then its diagonals are perpendicular. If is a rhombus, then.

35 Theorem Q12 Slide 103 / 189 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If is a rhombus, then and Example Slide 104 / 189 EFGH is a rhombus. Find x, y, and z. E 2x-6 F 72 o z 5y G 10 H ll sides of a rhombus are congruent. EF = HG 2x-6 = x = x = 8 EG = HG 5y = y = 2 Slide 105 / 189 ecause the consecutive angles of parallelogram are supplementary, the consecutive angles of a rhombus are supplementary. m E + m F = 180 o 72 + m F = o m F = z = m F 2 o The diagonals of a rhombus bisect the opposite angles. 1 z = (108 ) 2 z = 54 o o

36 Try this... Slide 106 / 189 The quadrilateral is a rhombus. Find x, y, and z. z 86 o 8 3x y2 47 This is a rhombus. Find x. Slide 107 / 189 x o 48 This is a rhombus. Find x. Slide 108 / x-3 9

37 49 This is a rhombus. Find x. Slide 109 / o x 50 HJKL is a rhombus. Find the length of HJ. Slide 110 / 189 H J 6 M 16 L K Rectangle orollary Slide 111 / 189 quadrilateral is a rectangle if and only if it has four right angles.,, and are right angles. If a quadrilateral is a rectangle, then it has four right angles.

38 51 What value of y will make the quadrilateral a rectangle? 6y Slide 112 / Theorem Q13 Slide 113 / 189 If a quadrilateral is a rectangle, then its diagonals are congruent. If is a rectangle, then. Example Slide 114 / 189 RET is a rectangle. Find x and y. R 63 o 9y o E 2x-5 13 T

39 52 RSTU is a rectangle. Find z. R S Slide 115 / 189 U 8z T 53 RSTU is a rectangle. Find z. R S Slide 116 / 189 4z-9 7 U T Square orollary Slide 117 / 189 quadrilateral is a square if and only if it is a rhombus and a rectangle. square has all the properties of a rectangle and rhombus.

40 Example Slide 118 / 189 The quadrilateral is a square. Find x, y, and z. 6 z - 4 3y (5x) o Try this... Slide 119 / 189 The quadrilateral is a square. Find x, y, and z. 3y 8y z (x 2 + 9) o 54 The quadrilateral is a square. Find y. Slide 120 / y

41 55 The quadrilateral is a rhombus. Find x. Slide 121 / x 2x The quadrilateral is parallelogram. Find x. Slide 122 / o (4x) o 57 The quadrilateral is a rectangle. Find x. Slide 123 / x 3x + 7

42 Slide 124 / 189 Slide the description under the correct special parallelogram. rhombus rectangle square iagonals are iagonals bisect opposite <'s iagonals are Has 4 right <'s Has 4 sides Opposite sides are Slide 125 / 189 lick on the link below and complete the lab. Lab - Quadrilaterals in the oordinate Plane Slide 126 / 189 Trapezoids Return to the Table of ontents

43 trapezoid Slide 127 / 189 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. trapezoid also has two pairs of base angles. isosceles trapezoid Slide 128 / 189 n isosceles trapezoid is a trapezoid with congruent legs. Theorem Q14 Slide 129 / 189 If a trapezoid is isosceles, then each pair of base angles are congruent. is an isosceles trapezoid. < < and < <.

44 Theorem Q15 Slide 130 / 189 If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. In trapezoid,. is an isosceles trapezoid. Slide 131 / The quadrilateral is an isosceles trapezoid. Find x. Slide 132 / o (9x + 1) o

45 Theorem Q16 Slide 133 / 189 trapezoid is isosceles if and only if its diagonals are congruent. In trapeziod,. is isosceles. Example Slide 134 / 189 PQRS is a trapeziod. Find the m S and m R. P 112 o 147 o Q S (6w+2) o (3w) o R Slide 135 / 189 Option o The sum of the interior angles of a quadrilateral is 360. (6w+2) + (3w) = 360 9w = 360 9w = 99 w = 11 m S = 6w+2 = 6(11)+2 = 68 m R = 3w = 3(11) = 33 o o

46 Option Slide 136 / 189 The parallel lines in a trapezoid create pairs of consecutive interior angles. o m P + m S = 180 and m Q + m R = 180 o (6w+2) = 180 6w = 180 w = 11 OR (3w) = 180 3w = 33 w = 11 m S = 6w+2 = 6(11)+2 = 68 m R = 3w = 3(11) = 33 o o Try this... Slide 137 / 189 PQRS is an isosceles trapezoid. Find the m Q, m R and m S. P 123 o (9w-3) o Q S (4w+1) o R 60 The trapezoid is isosceles. Find x. Slide 138 / x + 3 2x + 2

47 61 The trapeziod is isosceles. Find x. Slide 139 / o x o 62 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals? Slide 140 / 189 Yes No H I K J midsegment of a trapezoid Slide 141 / 189 The midsegment of a trapezoid is a segment that joins the midpoints of the legs. lick on the link below and complete the lab. Lab - Midsegments of a Trapezoid

48 Theorem Q17 Slide 142 / 189 The midsegment is parallel to both the bases, and the length of the midsegment is half the sum of the bases. E F EF 1 EF = (+) 2 Example Slide 143 / 189 PQRS is a trapezoid. Find LM. Q 15 R L M P 7 S Example Slide 144 / 189 PQRS is a trapezoid. Find PS. Q 20 R L 14.5 M P S

49 Try this... Slide 145 / 189 PQRS is an trapezoid. ML is the midsegment. Find x, y, and z. S 10 P y M 5 R 14 x 7 L z Q 63 EF is the midsegment of trapezoid HIJK. Find x. Slide 146 / 189 H 6 I E x F K 15 J 64 EF is the midsegment of trapezoid HIJK. Find x. Slide 147 / 189 J 19 K F 10 x E I H

50 65 Which of the following is true of every trapezoid? hoose all that apply. Slide 148 / 189 Exactly 2 sides are congruent. Exactly one pair of sides are parallel. The diagonals are perpendicular. There are 2 pairs of base angles. Slide 149 / 189 Kites Return to the Table of ontents kites Slide 150 / 189 kite is a quadrilateral with two pairs of adjacent congruent sides. The opposite sides are not congruent. lick on the link below and complete the lab. Lab - Properties of Kites

51 Theorem Q18 Slide 151 / 189 If a quadrilateral is a kite, then it has one pair of congruent opposite angles. In kite, < < and < < Theorem Q18 Slide 152 / 189 If a quadrilateral is a kite, then it has one pair of congruent opposite angles. In kite, and Example Slide 153 / 189 LMNP is a kite. Find x. M (x 2 o -1) L o 72 o 48 N P

52 m L + m M +m N +m P = 360 (Remember M P) o Slide 154 / (x 2-1) + (x 2-1) + 48 = 360 2x = 360 2x 2 = 242 x 2 = 121 x = ±11 66 RE is a kite. RE is congruent to. Slide 155 / 189 E E R R 67 RE is a kite. is congruent to. Slide 156 / 189 E E R R

53 68 Find the value of z in the kite. Slide 157 / 189 z 5z-8 69 Find the value of x in the kite. Slide 158 / o (8x+4) o 44 o 70 Find the value of x. Slide 159 / o (3x 2 + 3) o 24 o

54 Theorem Q19 Slide 160 / 189 If a quadrilateral is a kite then the diagonals are perpendicular. In kite 71 Find the value of x in the kite. Slide 161 / 189 x 72 Find the value of y in the kite. Slide 162 / y

55 Slide 163 / 189 Families of Quadrilaterals Return to the Table of ontents In this unit, you have learned about several special quadrilaterals. Now you will study what links these figures. Slide 164 / 189 quadrilateral Every rhombus is a special kite kite rhombus parallelogram square rectangle trapezoid isosceles trapezoid Each quadrilateral shares the properties with the quadrilateral above it. omplete the chart by sliding the special quadrilateral next to its description. (There can be more than one answer). Slide 165 / 189 parallelogram rhombus rectangle square kite trapezoid isosceles trapezoid escription n equilateral quadrilateral n equiangular quadrilateral The diagonals are perpendicular The diagonals are congruent Has at least 1 pair of parallel sides Special Quadrilateral(s) nswer(s) rhombus & square rectangle & square rectangle, square & kite rhombus, square & isosceles trapezoid ll except kite

56 Slide 166 / 189 Parallelogram Trapezoid Kite Rhombus Square Rectangle Rhombus Isosceles Trapezoid QURILTERLS 73 rhombus is a square. Slide 167 / 189 always sometimes never 74 square is a rhombus. Slide 168 / 189 always sometimes never

57 75 rectangle is a rhombus. Slide 169 / 189 always sometimes never 76 trapezoid is isosceles. Slide 170 / 189 always sometimes never 77 kite is a quadrilateral. Slide 171 / 189 always sometimes never

58 78 parallelogram is a kite. Slide 172 / 189 always sometimes never Slide 173 / 189 oordinate Proofs Return to the Table of ontents Given: PQRS is a quadrilateral Prove: PQRS is a kite Slide 174 / P (-1,6) 6 4 (-4,3) S 2 Q(2,3) R(-1,-2)

59 kite has one unique property. The adjacent sides are congruent. Slide 175 / 189 P(-1,6) (-4,3) S Q(2,3) # # # R(-1,-2) SP = (6-3) + (-1-(-4)) PQ = (3-6) + (2-(-1)) = = (-3) # # # = = = # 18 = 18 # = 4.24 = 4.24 P(-1,6) Slide 176 / 189 (-4,3) S Q(2,3) # # # R(-1,-2) SR = (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 # # # # So, because SP=PQ and SR=RQ, PQRS is a kite. Given: JKLM is a parallelogram Prove: JKLM is a square Slide 177 / J (1,3) 2-10 (-3,0) M K 6(4,-1) L (0,-4) -8-10

60 J (1,3) Slide 178 / 189 (-3,0) M K (4,-1) 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 sides are congruent and perpendicular. # # # # # # # # MJ = (3-0) + (1-(-3)) JK = (-1-3) + (4-1) = = (-4) = = = 25 = 25 = 5 = 5 J (1,3) Slide 179 / 189 (-3,0) M K (4,-1) (-3) 4 L (0,-4) m MJ = = m JK = = MJ JK and MJ JK What else do you know? MJ LK and JK LM (Opposite sides are congruent) MJ LM and JK LK (Perpendicular Transversal Theorem) JKLM is a square Try this... Slide 180 / 189 Given: PQRS is a trapezoid Prove: LM is the midsegment P (2,2) 2 Q (5,1) (1,0) L (0,-2)-2S M (7,-2) -4-6 R (9,-5) -8-10

61 Slide 181 / 189 Proofs Return to the Table of ontents Given: TE M, <1 <2 Prove: TEM is a parallelogram. Slide 182 / 189 T 1 E M 2 T 1 E Slide 183 / 189 M 2 Option statements reasons 1) TE M, <1 <2 1) Given 2) EM EM 2) Reflexive Property 3) Triangle MTE Triangle EM 3) Side ngle Side 4) TM E 4) PT 5) TEM is a parallelogram 5) The opposite sides of a parallelogram are congruent

62 T 1 E Slide 184 / 189 M 2 Option TE We are given that TE M and 2 3. M, by the alternate interior angles converse. click So, TEM is a parallelogram because each click to pair reveal of opposite sides is parallel and congruent. Given: FGHJ is a parallelogram, Prove: FGHJ is a rectangle F is a right angle Slide 185 / 189 F G J H F G Slide 186 / 189 J H statements 1) FGHJ is a parallelogram and F is a right angle 2) J and G are right angles 3) H is a right angle 1) Given reasons 2) The consecutive angles of a parallelogram are supplementary 3) The opposite angles of a parallelogram are congruent 4) TEM is a rectangle 4) Rectangle orollary

63 Given: OL is a quadrilateral, m O=140 o, m =40 o, m L=60 o Prove: OL is a trapezoid Slide 187 / o O 60 o 40 o L 140 o O Slide 188 / o 40 o L statements 1) OL is a quadrilateral, m O=140,m L=40,m =60 1) Given reasons 2) m O + m L = 180 m L + m = 100 2) ngle ddition 3) O and are supplementary 3) efinition of Supplementary ngles 4) L and are not supplementary 4) efinition of Supplementary ngles 5) O is parallel to L 5) onsecutive Interior ngles onverse 6) L is not parallel to O 6) onsecutive Interior ngles onverse 7) OL is a trapezoid 7) efinition of a Trapezoid ( trapezoid has one pair of parallel sides) Try this... Given: F FE Prove: F E Slide 189 / 189 F E