Geometry. Slide 1 / 343. Slide 2 / 343. Slide 3 / 343. Quadrilaterals. Table of Contents

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1 Slide 1 / 343 Slide 2 / 343 Geometry Quadrilaterals Table of ontents Polygons Properties of Parallelograms Proving Quadrilaterals are Parallelograms Rhombi, Rectangles and Squares Trapezoids Kites onstructing Quadrilaterals Families of Quadrilaterals Proofs oordinate Proofs PR Released Questions & Standards lick on a topic to go to that section. Slide 3 / 343

2 Slide 4 / 343 Polygons Return to the Table of ontents Polygons Slide 5 / 343 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." Would a circle be considered a polygon? Why or why not? Polygons Slide 6 / 343 Simple Polygons omplex Polygons This course will only consider simple polygons, which we will refer to as "polygons."

3 Types of Polygons Slide 7 / 343 Polygons are named by their number of sides, angles or vertices; which are all equal in number. Number of Sides, ngles 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 onvex or oncave Polygons Slide 8 / 343 onvex Polygons interior ll interior angles are less than 180º. No side, if extended, will cross into the interior. interior onvex or oncave Polygons Slide 9 / 343 oncave Polygons t least one interior angle is at least 180º. t least two sides, if extended, will cross into the interior. interior way to remember the difference is that concave shapes "cave in". interior

4 1 The figure below is a concave polygon. Slide 10 / 343 True False 2 Indentify the polygon. Slide 11 / 343 E F Pentagon Octagon Quadrilateral Hexagon ecagon Triangle 3 Is the polygon convex or concave? onvex oncave Slide 12 / 343

5 4 Is the polygon convex or concave? Slide 13 / 343 onvex oncave Equilateral Polygons Slide 14 / 343 polygon is equilateral if all its sides are congruent. Equiangular Polygons Slide 15 / 343 polygon is equiangular if all its angles are congruent.

6 Regular Polygons Slide 16 / 343 polygon is regular if it is equilateral and equiangular. 5 escribe the polygon. (hoose all that apply) Pentagon Octagon Quadrilateral Hexagon 60 o E Triangle 4 4 F onvex G oncave H Equilateral I Equiangular 60 o 60 o J Regular 4 Slide 17 / escribe the polygon. (hoose all that apply) Pentagon Octagon Quadrilateral Hexagon E Triangle F onvex G oncave H Equilateral I Equiangular J Regular Slide 18 / 343

7 7 escribe the polygon. (hoose all that apply) Pentagon Octagon Quadrilateral Hexagon E Triangle F onvex G oncave H Equilateral I Equiangular J Regular Slide 19 / 343 ngle Measures of Polygons Slide 20 / 343 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. ngle Measures of Polygons Slide 21 / 343 Every polygon is comprised of a number of triangles. The sum of the interior angles of each triangle is 180. 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?

8 ngle Measures of Polygons Slide 22 / 343 Every polygon can be thought of as consisting of some number of triangles. quadrilateral is comprised of two triangles What is the sum of the interior angles of a quadrilateral? ngle Measures of Polygons Slide 23 / 343 Every polygon can be thought of as consisting of some number of triangles. pentagon is comprised of three triangles. 180º 180º What is the sum of the interior angles of a pentagon? 180º ngle Measures of Polygons Slide 24 / 343 Every polygon can be thought of as consisting of some number of triangles. n octagon is comprised of six triangles. 180º 180º 180º 180º 180º What is the sum of the interior angles of an octagon? 180º

9 ngle Measures of Polygons Slide 25 / 343 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 n Math Practice ngle Measures of Polygons Slide 26 / 343 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 ngles º º º 6 4 n 8 What is the sum of the interior angles of this polygon? Slide 27 / 343

10 9 How many triangular regions can be drawn in this polygon? Slide 28 / What is the sum of the interior angles of this polygon? Slide 29 / How many triangular regions can be drawn in this polygon? Slide 30 / 343

11 12 What is the sum of the interior angles of this polygon? Slide 31 / Which of the following road signs has interior angles that add up to 360? Slide 32 / 343 Yield Sign No Passing Sign National Forest Sign Stop Sign E No rossing Sign Polygon Interior ngles Theorem Q1 The sum of the measures of the interior angles of a convex polygon with n sides is 180(n-2). Polygon omplete the table. Number of Sides Sum of the measures of the interior angles. hexagon 6 180(6-2) = 720 Slide 33 / 343 heptagon 7 180(7-2) = 900 octagon 8 180(8-2) = 1080 nonagon 9 180(9-2) = 1260 decagon (10-2) = gon (11-2) = 1620 dodecagon (12-2) = 1800

12 14 What is the measurement of each angle in the diagram below? Slide 34 / 343 L (3x+4) M 146 x N P (2x+3) (3x) O Interior ngles of Equiangular Polygons Slide 35 / 343 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 = Sum of the ngles Number of ngles m = (n - 2)180º n Interior ngles of Regular Polygons Slide 36 / 343 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-gon n (n-2)(180 ) (n-2)(180 )/n

13 15 If the stop sign is a regular octagon, what is the measure of each interior angle? Slide 37 / What is the sum of the measures of the interior angles of a convex 20-gon? Slide 38 / What is the measure of each interior angle of a regular 20-gon? Slide 39 /

14 18 What is the measure of each interior angle of a regular 16-gon? Slide 40 / What is the value of x? Slide 41 / 343 (10x+8) (8x) (5x+15) (9x-6) (11x+16) Polygon Exterior ngle Theorem The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360. Slide 42 / In this case, m 1 + m 2 + m 3 + m 4 + m 5 = 360

15 Proof of Exterior ngles Theorem Slide 43 / 343 Given: ny 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, "the sum of the interior angles" becomes "Ʃ interior s." Proof of Exterior ngles Theorem Given: ny polygon has n interior angles, where n is the number of its sides, interior angles, or vertices Prove: Ʃ external s = 360 Slide 44 / 343 This proof is based on the fact that each exterior angle is supplementary to an interior angle. nd, 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 180 Polygon Exterior ngle Theorem Slide 45 / 343 External angles are formed by extending one side of a polygon. Take a look at 1 in the diagram. The external and internal angles at a vertex form a linear pair. What is the sum of their measures? 1 internal 1 external

16 Polygon Exterior ngle Theorem Slide 46 / 343 ny 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 180. Polygon Exterior ngle Theorem Slide 47 / 343 Each linear pair includes an external and an internal. dding 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 180 Polygon Exterior ngle Theorem Slide 48 / 343 Ʃ external s + Ʃ internal s = n x 180 ut 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:

17 Polygon Exterior ngle Theorem Slide 49 / 343 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. Proof of Exterior ngles Theorem Slide 50 / 343 Given: ny 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 ny 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 efinition 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 Proof of Exterior ngles Theorem Slide 51 / 343 Given: ny 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 ngle Theorem 8 n x 180 = sum of exterior s + (n - 2) x 180 Substitution Property 9 The sum of the exterior s = n x [(n - 2) x 180 ] 10 The sum of the exterior s = 180 [n - (n - 2)] 11 The sum of the exterior s = 180 (n - n + 2) The sum of the exterior s = (2) = 360 Subtraction Property Subtraction Property istributive Property Simplifying

18 Polygon Exterior ngle Theorem orollary Slide 52 / 343 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. ll 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. Polygon Exterior ngle Theorem orollary Slide 53 / 343 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 = What is the sum of the measures of the exterior angles of a heptagon? Slide 54 /

19 21 If a heptagon is regular, what is the measure of each exterior angle? Slide 55 / What is the sum of the measures of the exterior angles of a pentagon? Slide 56 / What is the measure of each exterior angle of a regular pentagon? Slide 57 / 343

20 24 The measure of each interior angle of a regular convex polygon is 172. How many sides does the polygon have? Slide 58 / 343 nswer 25 The measure of each interior angle of a regular convex polygon is 174. Find the number of sides of the polygon. Slide 59 / The measure of each interior angle of a regular convex polygon is 162. Find the number of sides of the polygon. Slide 60 / 343

21 Slide 61 / 343 Properties of Parallelograms Return to the Table of ontents lick on the link below and complete the two labs before the Parallelogram lesson. Slide 62 / 343 Lab - Investigating Parallelograms Lab - Quadrilaterals Party Parallelograms Slide 63 / 343 y definition, the opposite sides of a parallelogram are parallel. E G F In parallelogram EFG, G? E?

22 Theorem Slide 64 / 343 The opposite sides a parallelogram are congruent. If is a parallelogram, then and 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. Slide 65 / 343 onstruct diagonal as shown and consider it a transversal to sides and, which we know are. What does that tell about the pairs of angles: & and & Why? Proof of Theorem Slide 66 / 343 The opposite sides a parallelogram are congruent. Given that and Name a pair of congruent Δs For what reason are they congruent?

23 Proof of Theorem The opposite sides a parallelogram are congruent. Slide 67 / 343 Given that Δ Δ Which pairs of sides are must be congruent? Why? 27 The opposite sides of a parallelogram are Slide 68 / 343 ongruent Parallel oth a and b Theorem Slide 69 / 343 The opposite angles of a parallelogram are congruent. If is a parallelogram, then m = m and m = m

24 Proof of Theorem Slide 70 / 343 The opposite angles of a parallelogram are congruent. s we did earlier, construct diagonal. We earlier proved Δ Δ What does this tell you about and? Why? 28 Δ Δ. Prove your answer. True False Slide 71 / ased on the previous question, m =? Slide 72 / 343 Prove your answer. m m m None of the above

25 Theorem Slide 73 / 343 The consecutive angles of a parallelogram are supplementary. x y y x If is a parallelogram, then x o + y o = 180 o Proof of Theorem Slide 74 / 343 The consecutive angles of a parallelogram are supplementary. x y y x onsider and to be transversals of and, which are. What are these pairs of angles called: & and &? What is the relationship of each pair of angles like that? Proof of Theorem Slide 75 / 343 The consecutive angles of a parallelogram are supplementary. x y y x Repeat the same process, but this time consider and to be transversals of and, which are. What are these pairs of angles called: & and &? What is the relationship of each pair of angles like that?

26 30 is a parallelogram. Find the value of x. Slide 76 / y w 12 x-5 9 (5z) 31 is a parallelogram. Find the value of y. 2y 65 w Slide 77 / x-5 9 (5z) 32 is a parallelogram. Find the value of z. 2y 65 w Slide 78 / x-5 9 (5z)

27 33 is a parallelogram. Find the value of w. 2y 65 w Slide 79 / x-5 9 (5z) 34 EFG is a parallelogram. Find the value of w. Slide 80 / (2w) (z + 12) G 3x - 3 F y 2 E 35 EFG is a parallelogram. Find the value of x. Slide 81 / (2w) (z + 12) G 3x - 3 F y 2 E

28 36 EFG is a parallelogram. Find the value of y. Slide 82 / (2w) (z + 12) G 3x - 3 F y 2 E 37 EFG is a parallelogram. Find the value of z. Slide 83 / (2w) (z + 12) G 3x - 3 F y 2 E Theorem Slide 84 / 343 The diagonals of a parallelogram bisect each other. E If is a parallelogram, then E E and E E

29 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. Slide 85 / 343 E Steps to follow: 1) Identify pairs of triangles in the above diagram that you need to prove congruent to prove this theorem. 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 (PT). 38 LMNP is a parallelogram. The diagonals bisect each other. Find QN Slide 86 / 343 L 4 M P 6 Q N 39 LMNP is a parallelogram. Find MP Slide 87 / 343 L 4 M P 6 Q N

30 40 ER is a parallelogram. Find the value of x. Slide 88 / 343 x 4y E 8 S 10 R 41 ER is a parallelogram. Find the value of y. Slide 89 / 343 x 4y E 8 S 10 R 42 ER is a parallelogram. Find ER. Slide 90 / 343 x 4y E 8 S 10 R

31 43 In a parallelogram, the opposite sides are parallel. Slide 91 / 343 sometimes always never 44 The opposite angles of a parallelogram are. Slide 92 / 343 bisect congruent parallel supplementary 45 The consecutive angles of a parallelogram are. Slide 93 / 343 bisect congruent parallel supplementary

32 46 The diagonals of a parallelogram each other. Slide 94 / 343 bisect congruent parallel supplementary 47 The opposite sides of a parallelogram are. Slide 95 / 343 bisect congruent parallel supplementary 48 MTH is a parallelogram. H = 12 and MR = 7. Find RT. M 6 Slide 96 / R 12 H T

33 49 MTH is a parallelogram. Find R. 6 M R 12 Slide 97 / 343 H T 50 MTH is a parallelogram. Find m H. Slide 98 / 343 M 14 (3y + 8) H 2x T 51 MTH is a parallelogram. Find the value of x. Slide 99 / 343 M x 2-24 (3y + 8) H 2x 98 T

34 52 MTH is a parallelogram. Find the value of y. M 14 (3y + 8) Slide 100 / 343 H 2x T Slide 101 / 343 Proving Quadrilaterals are Parallelograms Return to the Table of ontents Theorem Slide 102 / 343 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. In quadrilateral, and, so is a parallelogram.

35 Theorem Slide 103 / 343 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. In quadrilateral, and, so is a parallelogram. 53 PQRS is a Parallelogram. Explain your reasoning. Slide 104 / 343 True False S 6 P 4 4 R 6 Q 54 Is PQRS a parallelogram? Explain your reasoning. Slide 105 / 343 Yes No P Q S R

36 55 Is this quadrilateral definitely a parallelogram? Slide 106 / 343 Yes No Is this quadrilateral definitely a parallelogram? Slide 107 / 343 Yes No Is this quadrilateral definitely a parallelogram? Slide 108 / 343 Yes No

37 Theorem Slide 109 / 343 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram In quadrilateral, m + m = 180 and m + m = 180, so is a parallelogram. Theorem Slide 110 / 343 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. Theorem Slide 111 / 343 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.

38 58 Is this quadrilateral definitely a parallelogram? Slide 112 / 343 Yes No 59 Is this quadrilateral definitely a parallelogram? Slide 113 / 343 Yes No Is this quadrilateral definitely a parallelogram? Slide 114 / 343 Yes No

39 61 Is this quadrilateral definitely a parallelogram? Slide 115 / 343 Yes No 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. Slide 116 / 343 Yes No Fill in the blank Slide 117 / 343 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.

40 Fill in the blank Slide 118 / 343 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. 63 Which theorem proves the quadrilateral is a parallelogram? 6(7-3) Slide 119 / 343 The opposite angles are congruent. The opposite sides are congruent. 3(2) 3 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. 64 Which theorem proves this quadrilateral is a parallelogram? Slide 120 / 343 E F The opposite angles 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.

41 65 Which theorem proves the quadrilateral is a parallelogram? 6 Slide 121 / 343 The opposite angles are congruent. 3(6-4) 6 The opposite sides are congruent. 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 122 / 343 Rhombi, Rectangles and Squares Return to the Table of ontents Three Special Parallelograms ll the same properties of a parallelogram apply to the rhombus, rectangle, and square. Slide 123 / 343 Rhombus Rectangle Square

42 Rhombus Slide 124 / 343 rhombus is a quadrilateral with four congruent sides. If, then is a rhombus. 66 What is the value of y that will make the quadrilateral a rhombus? Slide 125 / y 67 What is the value of y that will make this parallelogram a rhombus? y+29 Slide 126 / 343 6y

43 Theorem The diagonals of a rhombus are perpendicular. Slide 127 / 343 If is a rhombus, then. Theorem The diagonals of a rhombus are perpendicular. Slide 128 / 343 O Since is a parallelogram, its diagonals bisect each other. O O and O O. Theorem Slide 129 / 343 The diagonals of a rhombus are perpendicular. O y definition, the sides of a rhombus are of equal measure, which means that = = =. Then by SSS, ΔO ΔO ΔO ΔO nd due to corresponding parts of Δs being (or PT) O O O O If the adjacent angles formed by intersecting lines are congruent, then the lines are perpendicular.

44 Theorem Slide 130 / 343 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If is a rhombus, then and Theorem If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Slide 131 / 343 We proved earlier by SSS, that ΔO ΔO ΔO ΔO O orresponding parts of Δs are ; so O O O O N O O O O So each diagonal bisects a pair of opposite angles 68 EFGH is a rhombus. Find the value of x. Slide 132 / 343 E 72 2x - 6 z F 5y G 10 H

45 69 EFGH is a rhombus. Find the value of y. Slide 133 / 343 E 72 2x - 6 z F 5y G 10 H 70 EFGH is a rhombus. Find the value of z. Slide 134 / 343 E 72 2x - 6 z F 5y G 10 H 71 This quadrilateral is a rhombus. Find the value of x. Slide 135 / 343 z x + 2 1y 2 2

46 72 This quadrilateral is a rhombus. Find the value of y. Slide 136 / 343 z x + 2 1y This quadrilateral is a rhombus. Find the value of z. Slide 137 / 343 z x + 2 1y This is a rhombus. Find the value of x. Slide 138 / 343 x

47 75 This is a rhombus. Find the value of x. Slide 139 / x This is a rhombus. Find the value of x. Slide 140 / x 77 HJKL is a rhombus. Find the length of HJ. Slide 141 / 343 H J 6 M 16 L K

48 Rectangles rectangle is a quadrilateral with only right angles. Slide 142 / 343 If is a rectangle, then,, and are right angles. If,, and are right angles, then is a rectangle. 78 What value of y will make the quadrilateral a rectangle? Slide 143 / 343 (6y) 12 Theorem Slide 144 / 343 The diagonals of a rectangle are congruent. M If is a rectangle, then. lso, since a rectangle is a parallelogram, M M M M

49 Theorem Slide 145 / 343 The diagonals of a rectangle are congruent. Since is a rectangle (and therefore a parallelogram) ecause of the reflexive property,. Since is a rectangle m = m = 90º Then Δ Δ by SS nd, since they are corresponding parts of Δs 79 RET is a rectangle. Find the value of x. Slide 146 / 343 R (9y) 63 E 2x T 80 RET is a rectangle. Find the value of y. Slide 147 / 343 R 63 (9y) E 2x T

50 81 RSTU is a rectangle. Find the value of z. Slide 148 / 343 R S U (8z) T 82 RSTU is a rectangle. Find the value of z. Slide 149 / 343 R S 4z U T Square Slide 150 / 343 square is a quadrilateral with equal sides and equal angles. It is, therefore both a rhombus and a rectangle. square has all the properties of a rectangle and rhombus.

51 83 This quadrilateral is a square. Find the value of x. Slide 151 / z - 4 (5x) (3y) 84 This quadrilateral is a square. Find the value of y. Slide 152 / z - 4 (5x) (3y) 85 This quadrilateral is a square. Find the value of z. Slide 153 / z - 4 (5x) (3y)

52 86 This quadrilateral is a square. Find the value of x. Slide 154 / 343 y y (12z) (x 2 + 9) 87 This quadrilateral is a square. Find the value of y. Slide 155 / 343 y y (12z) (x 2 + 9) 88 This quadrilateral is a square. Find the value of z. Slide 156 / 343 y y (12z) (x 2 + 9)

53 89 This quadrilateral is a square. Find the value of y. Slide 157 / 343 (18y) 90 This quadrilateral is a rhombus. Find the value of x. Slide 158 / 343 4x 2x The quadrilateral is parallelogram. Find the value of x. Slide 159 / (4x)

54 92 The quadrilateral is a rectangle. Find the value of x. Slide 160 / x 3x + 7 Fill in the Table with the orrect efinition Slide 161 / 343 rhombus rectangle square Opposite sides are iagonals are iagonals bisect opposite 's Has 4 right 's Has 4 sides iagonals are Slide 162 / 343 Lab - Quadrilaterals in the oordinate Plane

55 Slide 163 / 343 Trapezoids Return to the Table of ontents Trapezoid Slide 164 / 343 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 165 / 343 n isosceles trapezoid is a trapezoid with congruent legs.

56 Theorem Slide 166 / 343 If a trapezoid is isosceles, then each pair of base angles are congruent. If is an isosceles trapezoid, then and. Theorem If a trapezoid is isosceles, then each pair of base angles are congruent. Slide 167 / 343 Given: is an isosceles trapezoid Prove: and E F Theorem If a trapezoid is isosceles, then each pair of base angles are congruent. Slide 168 / 343 Given: is an isosceles trapezoid Prove: and. E F continued...

57 E F Statement Theorem Given: is an isosceles trapezoid Prove: and. Reason 1 is an isosceles trapezoid Given efinition of an Isosceles 2, Trapezoid raw segments from and 3 perpendicular to and. These Geometric onstruction segments are E and F E, EF, FE, F, F and efinition of E are right angles Perpendicular Lines E EF FE F F E m E = m EF = m FE = m F = m F = m E = 90 ll right angles are efinition of congruent and right angles continued... Slide 169 / 343 Theorem 7 FE is a rectangle efinition of a rectangle 8 E F Opposite sides of a rectangle are 9 ΔE ΔF HL 10 E F PT 11 m E = m F efinition of angles 12 m E + 90 = m F + 90 ddition Property of = m E + m E = m F + m F m E + m E = m m F + m F = m Substitution Property of = ngle ddition Postulate 15 m = m Substitution Property of = 16, or efinition of angles Slide 170 / 343 Theorem Slide 171 / 343 If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. If in trapezoid,. Then, is an isosceles trapezoid.

58 Theorem If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. Slide 172 / 343 ase 1 - Longer ase s are Given: In trapezoid, Prove: is an isosceles trapezoid E F continued... Theorem Statement Reason 1 In trapezoid, Given 2 efinition of an Isosceles Trapezoid raw segments from and 3 perpendicular to and. These Geometric onstruction segments are E and F. 4 E, EF, FE, F, F and efinition of E are right angles Perpendicular Lines E EF FE F 5 F E ll right angles are 6 FE is a rectangle efinition of a rectangle 7 E F Opposite sides of a rectangle are 8 ΔE ΔF S 9 PT 10 is an isosceles trapezoid efinition of an isosceles trapezoid Slide 173 / 343 Theorem If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. Slide 174 / 343 ase 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.

59 F If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. Theorem E ase 2 - s of shorter base are Slide 175 / 343 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. an you now see how to extend the approach we used for the first case to this second case? Theorem Slide 176 / 343 If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. F E ase 2 - s of shorter base are Given: In trapezoid, Prove: is an isosceles trapezoid Steps to follow: 1) Include the constructions of our new perpendicular segments E & F 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 (PT). 93 The quadrilateral is an isosceles trapezoid. Find the value of x x Slide 177 / 343

60 94 The quadrilateral is an isosceles trapezoid. Find the value of x (9x + 1) Slide 178 / 343 Theorem Slide 179 / 343 trapezoid is isosceles if and only if its diagonals are congruent. In trapezoid, If. Then, is isosceles. If is isosceles Then,. Theorem trapezoid is isosceles if and only if its diagonals are congruent. Statement Reason 1 Trapezoid is isosceles Given 2,,, Given: Trapezoid is isosceles Prove: efinition of an Isosceles Trapezoid 4 Reflexive Property of 5 Δ Δ SS 6 PT Teacher's Note Slide 180 / 343

61 95 PQRS is a trapezoid. Find the m S. Slide 181 / 343 P Q S (6w + 2) (3w) R nswer 96 PQRS is a trapezoid. Find the m R. Slide 182 / 343 P Q S (6w + 2) (3w) R nswer 97 PQRS is an isosceles trapezoid. Find the m Q. Slide 183 / 343 P 123 Q (9w - 3) S (4w + 1) R

62 98 PQRS is an isosceles trapezoid. Find the m R. Slide 184 / 343 P 123 Q (9w - 3) S (4w + 1) R 99 PQRS is an isosceles trapezoid. Find the m S. Slide 185 / 343 P 123 Q (9w - 3) S (4w + 1) R 100 The trapezoid is isosceles. Find the value of x. Slide 186 / x + 3 2x + 2

63 101 The trapezoid is isosceles. Find the value of x. Slide 187 / x 102 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals? Yes No H I Slide 188 / 343 K J Midsegment of a Trapezoid Slide 189 / 343 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

64 Theorem Slide 190 / 343 The midsegment of a trapezoid is parallel to both the bases. E F EF Proof of Theorem The midsegment of a trapezoid is parallel to both the bases. It's given that E is the midpoint of ; F is the midpoint of ; and E F. Since, they have the same slope and therefore the distance between them, along any perpendicular line, will be the same. ny 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. Slide 191 / 343 That means that EF, where EF is the midsegment of the trapezoid. Theorem Slide 192 / 343 The length of the midsegment is half the sum of the bases. E F EF = 0.5( + )

65 E 2x Proof of Theorem The length of the midsegment is half the sum of the bases. x G H I J y 2y F Given: E is the midpoint of F is the midpoint of EF Prove: EF = 0.5( + ) Slide 193 / 343 continued... Proof of Theorem Statement Reason E is the midpoint of 1 F is the midpoint of Given EF 2 EG H, FI J orresponding angles are 3 EG H, FI J Reflexive Property of 4 EG ~ H, FI ~ J ngle-ngle ~ H H J J orresponding sides of ~ s 5 = = = = E G EG, F I IF are proportional The midpoints E and F divide and into 2 congruent 6 = 2E, = 2F segments, so and are twice the size of E and F respectfully. = 2 = 2 7 ivision property of = E, F The scale factor in both of our 8 efinition of scale factor similar triangles is 2. continued... Slide 194 / 343 Proof of Theorem Slide 195 / 343 Statement Reason If EG = x, then H = 2x. 9 If IF = y, then H = 2y. Substitution Property of = 10 EF = + x + y Segment ddition Postulate 112EF = 2 + 2x + 2y Multiplication Property of = 12 = + 2x + 2y Segment ddition Postulate 13 2x + 2y = - Subtraction Property of = 14 2EF = Substitution Property of = 15 2EF = + ombine Like Terms Multiplication Property of = OR 16 EF = 0.5( + ) ivision Property of = and istributive Property of =

66 103 PQRS is a trapezoid. Find LM. Slide 196 / 343 Q 15 R L M P 7 S 104 PQRS is a trapezoid. Find PS. Slide 197 / 343 Q 20 R L 14.5 M P S 105 PQRS is an trapezoid. ML is the midsegment. Find the value of x. S 10 P y 7 M 14 L 5 x z R Q Slide 198 / 343

67 106 PQRS is an trapezoid. ML is the midsegment. Find the value of y. Slide 199 / 343 S y M 5 R x P 7 L z Q 107 PQRS is an trapezoid. ML is the midsegment. Find the value of z. Slide 200 / 343 S y M 5 R x P 7 L z Q 108 EF is the midsegment of trapezoid HIJK. Find the value of x. H 6 I Slide 201 / 343 E x F K 15 J

68 109 EF is the midsegment of trapezoid HIJK. Find the value of x. Slide 202 / 343 J 19 K F I 10 x H E 110 Which of the following is true of every trapezoid? hoose all that apply. Slide 203 / 343 Exactly 2 sides are congruent. Exactly one pair of sides are parallel. The diagonals are perpendicular. There are 2 pairs of base angles. Slide 204 / 343 Kites Return to the Table of ontents

69 Kites Slide 205 / 343 kite is a quadrilateral with two pairs of adjacent congruent sides whose opposite sides are not congruent. Lab - Properties of Kites Theorem Slide 206 / 343 If a quadrilateral is a kite, then it has one pair of congruent opposite angles. In kite, and Proof of Theorem Slide 207 / 343 If a quadrilateral is a kite, then it has one pair of congruent opposite angles. raw line and use the Reflexive Property to say that. Δ Δ by SSS. since corresponding parts of Δs are. m m since. In kite, m = m and m m

70 111 LMNP is a kite. Find the value of x. M Slide 208 / 343 (x 2-1) L N P 112 RE is a kite. RE. Slide 209 / 343 E E R R 113 RE is a kite.. Slide 210 / 343 E R R E

71 114 Find the value of z in the kite. Slide 211 / 343 z 5z Find the value of x in the kite. Slide 212 / (8x + 4) Find the value of x. Slide 213 / (3x 2-2) 24

72 Theorem Slide 214 / 343 The diagonals of a kite are perpendicular. In kite Proof of Theorem The diagonals of a kite are perpendicular. Slide 215 / 343 Given that and and E E by reflexive property E Δ Δ by SSS E E by PT ΔE ΔE by SS E E by PT In kite E is perpendicular to since intersecting lines that create adjacent angles are perpendicular 117 Find the value of x in the kite. Slide 216 / 343 x

73 118 Find the value of y in the kite. Slide 217 / 343 (12y) Slide 218 / 343 onstructing Quadrilaterals Return to the Table of ontents onstruction of a Parallelogram Slide 219 / raw two intersecting lines.

74 onstruction of a Parallelogram Slide 220 / Place a compass at the point of intersection and draw an arc where it intersects one line in two places. onstruction of a Parallelogram Slide 221 / 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. onstruction of a Parallelogram Slide 222 / onnect the four points where the arcs intersect the lines.

75 onstruction of a Parallelogram Slide 223 / 343 Which property of parallelograms does this construction technique depend upon? onstruction of a Parallelogram Slide 224 / 343 Try this! onstruct a parallelogram using the lines below. 1) Slide 225 / 343

76 Videos emonstrating the onstructions of a Parallelogram using ynamic Geometric Software Slide 226 / 343 lick here to see parallelogram video: compass & straightedge lick here to see parallelogram video: Menu options onstruction of a Rectangle Slide 227 / raw two intersecting lines. onstruction of a Rectangle Slide 228 / 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.

77 onstruction of a Rectangle Slide 229 / onnect the four points where the arcs intersect the lines. onstruction of a Rectangle Slide 230 / 343 Which properties of rectangles does this construction technique depend upon? onstruction of a Rectangle Slide 231 / 343 Try this! onstruct a rectangle using the lines below. 3)

78 onstruction of a Rectangle Slide 232 / 343 Try this! onstruct a rectangle using the lines below. 4) Videos emonstrating the onstructions of a Rectangle using ynamic Geometric Software Slide 233 / 343 lick here to see rectangle video: compass & straightedge lick here to see rectangle video: Menu options onstruction of a Rhombus Slide 234 / onstruct a segment of any length.

79 onstruction of a Rhombus Slide 235 / 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 and open it so the pencil is more than halfway, it does not matter how far, to point onstruction of a Rhombus Slide 236 / 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 and draw an arc so that it intersects the one you just drew onstruction of a Rhombus 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points. Slide 237 / 343 c. raw a line through the arc intersections. The intersection point created with the two perpendicular lines is the center of your rhombus.

80 onstruction of a Rhombus 3. Place the tip of your compass on your center point (midpoint). Extend your compass to any length. raw 2 arcs where the compass intersects the perpendicular bisector. Slide 238 / 343 Note: If you wanted to, you could use the intersection points of the 2 arcs from step 2 as your extra points. onstruction of a Rhombus 4. onnect the intersection points of the 2 arcs and the perpendicular bisector from step 3 with the endpoints of your original segment. Slide 239 / 343 onstruction of a Rhombus Which properties of rhombii (pl. for rhombus) does this construction technique depend upon? Slide 240 / 343

81 Slide 241 / 343 Videos emonstrating the onstructions of a Rhombus using ynamic Geometric Software lick here to see rhombus video: compass & straightedge lick here to see rhombus video: Menu options onstruction of a Rhombus Slide 242 / 343 Try this! onstruct a rhombus using the segment below. 5) onstruction of a Rhombus Slide 243 / 343 Try this! onstruct a rhombus using the segment below. 6)

82 onstruction of a Square Slide 244 / onstruct a segment of any length. onstruction of a Square Slide 245 / 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 and open it so the pencil is more than halfway, it does not matter how far, to point. raw an arc. onstruction of a Square Slide 246 / 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 and draw an arc so that it intersects the one you just drew

83 onstruction of a Square 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points. Slide 247 / 343 c. raw a line through the arc intersections. The intersection point created with the two perpendicular lines is the center of your circle (and future square). onstruction of a Square 3. Place your compass tip on point and the pencil point on either or and construct your circle. Slide 248 / 343 onstruction of a Square 4. onnect the intersection points of your perpendicular bisector and the circle with the initial vertices of the line segment to create your square. Slide 249 / 343

84 onstruction of a Square Slide 250 / 343 Which properties of squares does this construction technique depend upon? onstruction of a Square Slide 251 / 343 Try this! onstruct a square using the segment below. 7) onstruction of a Square Slide 252 / 343 Try this! onstruct a square using the segment below. 8)

85 Videos emonstrating the onstructions of a Square using ynamic Geometric Software Slide 253 / 343 lick here to see square video: compass & straightedge lick here to see square video: Menu options onstruction of an Isosceles Trapezoid Slide 254 / raw two intersecting lines. onstruction of an Isosceles Trapezoid Slide 255 / Place a compass at the point of intersection and draw an arc where it intersects two of the lines one time each.

86 onstruction of an Isosceles Trapezoid Slide 256 / 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). onstruction of an Isosceles Trapezoid Slide 257 / onnect the four points where the arcs intersect the lines. onstruction of an Isosceles Trapezoid Slide 258 / 343 Which properties of isosceles trapezoids does this construction technique depend upon?

87 onstruction of an Isosceles Trapezoid Slide 259 / 343 Try this! onstruct an isosceles trapezoid using the lines below. 9) Slide 260 / 343 Videos emonstrating the onstructions of an Isosceles Trapezoid using ynamic Geometric Software Slide 261 / 343 lick here to see isosceles trapezoid video: compass & straightedge lick here to see isosceles trapezoid video: Menu options

88 onstruction of a Kite Slide 262 / onstruct a segment of any length. onstruction of a Kite Slide 263 / Place your compass tip at any point and draw 2 arcs that intersect your segment. onstruction of a Kite Slide 264 / 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. raw an arc.

89 onstruction of a Kite Slide 265 / 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. onstruction of a Kite Slide 266 / Find the perpendicular bisector between the 2 red arcs. c) onnect the intersections of the 2 arcs to construct your perpendicular line. onstruction of a Kite Slide 267 / Place the tip of your compass at the intersection point of our perpendicular lines. Extend your compass to any length. raw 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.

90 onstruction of a Kite Slide 268 / onnect 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. onstruction of a Kite Slide 269 / 343 Which properties of a kite does this construction technique depend upon? Slide 270 / 343

91 Slide 271 / 343 Slide 272 / 343 Videos emonstrating the onstructions of a Kite using ynamic Geometric Software lick here to see kite video: compass & straightedge lick here to see kite video: Menu options Slide 273 / 343 Families of Quadrilaterals Return to the Table of ontents

92 omplete the chart (There can be more than one answer). Slide 274 / 343 parallelogram rectangle rhombus square trapezoid isosceles trapezoid kite 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 click to reveal rectangle & square click to reveal rectangle, square & kite click to reveal rhombus, square & isosceles trapezoid click to reveal ll except kite click to reveal 119 rhombus is a square. Slide 275 / 343 always sometimes never 120 square is a rhombus. Slide 276 / 343 always sometimes never

93 121 rectangle is a rhombus. Slide 277 / 343 always sometimes never 122 trapezoid is isosceles. Slide 278 / 343 always sometimes never 123 kite is a quadrilateral. Slide 279 / 343 always sometimes never

94 124 parallelogram is a kite. Slide 280 / 343 always sometimes never Families of Quadrilaterals Slide 281 / 343 Sometimes, you will be given a type of quadrilateral and using its qualities find the value of a variable. fter 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. Quadrilaterals Slide 282 / 343 Quadrilateral is a parallelogram. x x + 5 etermine the value of x. 25 Since we know that 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

95 Quadrilateral is a parallelogram. x 2-75 = 2x + 5 x x + 5 Quadrilaterals click x 2-2x - 80 = 0 click Slide 283 / (x - 10)(x + 8) = 0 click x = 10 and x = -8 click o any of these answers not make sense? Yes, since the value of x cannot be negative, x = -8 can be ruled out. = 2(-8) + 5 = -11 which is not possible. click Is parallelogram a more specific type of of quadrilateral? Explain. Since is a parallelogram, = 25 because opposite sides are congruent. When substituting x = 10 back into the expressions, then = = = = 25. Therefore, is a rhombus. click 125 Quadrilateral EFGH is a parallelogram. Find the value of x. E F Slide 284 / 343 x x J 3x x + 8 H 32 G 126 ased 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 Rhombus Slide 285 / 343 x x J 3x x + 8 Rectangle Square H 32 G

96 127 Quadrilateral KLMN is a trapezoid. Find the value of x. Slide 286 / 343 K (7x + 30) L (10x) N (x 2-20) M 128 ased on your answer to the previous slide, what type of quadrilateral is KLMN? Explain. When you are finished, determine whether trapezoid KLMN is isosceles. K L (7x + 30) (10x) Isosceles Trapezoid Not an Isosceles Trapezoid Slide 287 / 343 N (x 2-20) M Extension: Inscribed Quadrilaterals Slide 288 / 343 quadrilateral is inscribed if all its vertices lie on a circle... inscribed quadrilateral.

97 Inscribed Quadrilaterals Slide 289 / 343 If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. E m = 1/2 (mefg) [green arc] m F = 1/2 (meg) [red arc]. G F Since mefg + meg = 360, m + m F = 1/2 (mefg) + 1/2 (meg) m + m F = 1/2 (mefg + meg) m + m F = 1/2(360 ) m + m F = 180 m + m F = 180 m E + m G = 180 Inscribed Quadrilaterals Slide 290 / 343 Example: Find the m G. Then, find the value of a. (3a + 8) 121 E. (2a - 3) F G 129 What is m L? M Slide 291 / J L K

98 130 What is m M? M Slide 292 / J L K 131 What is the value of x? M 8 10 J (7x - 6) (5y - 4) (8y + 15) (3x - 4) L Slide 293 / 343 K 132 What is the value of y? 13 J (7x - 6) M (8y + 16) Slide 294 / (5y - 5) (3x - 4) L K

99 an 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 an any of these quadrilaterals be inscribed into a circle? Slide 295 / 343 Find out by completing the lab below. Lab: Inscribed Quadrilaterals 133 omplete the sentence below. trapezoid can be inscribed into a circle. Slide 296 / 343 lways Sometimes Never 134 omplete the sentence below. rectangle can be inscribed into a circle. Slide 297 / 343 lways Sometimes Never

100 135 omplete the sentence below. rhombus can be inscribed into a circle. Slide 298 / 343 lways Sometimes Never 136 omplete the sentence below. kite can be inscribed into a circle. Slide 299 / 343 lways Sometimes Never Slide 300 / 343 Proofs Return to the Table of ontents

101 Proofs Slide 301 / 343 Given: TE M, 1 2 Prove: TEM is a parallelogram. T 1 E M 2 Proofs T E 1 Slide 302 / 343 Option M 2 statements reasons 1) TE M, 1 2 1) Given 2) EM EM 2) Reflexive Property 3) MTE EM 3) Side ngle Side 4) TM E 4) PT 5) TEM is a parallelogram 5) oth pairs of opposite sides of are congruent, so the quadrilateral is a parallelogram. Proofs Slide 303 / 343 T 1 E Option M 2 We are given that TE M and 2 3. TE M, by the alternate interior angles converse. click to reveal So, TEM is a parallelogram because click to one reveal pair of opposite sides is parallel and congruent.

102 Proofs Slide 304 / 343 Given: FGHJ is a parallelogram, F is a right angle Prove: FGHJ is a rectangle F G J H Proofs Slide 305 / 343 Given: FGHJ is a parallelogram, F is a right angle Prove: FGHJ is a rectangle F G 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 4) TEM is a rectangle 1) Given J reasons 2) The consecutive angles of a parallelogram are supplementary 3) The opposite angles of a parallelogram are congruent 4) If a parallelogram has 4 right angles, then it's a rectangle. H Proofs Slide 306 / 343 Given: OL is a quadrilateral, m O = 140, m = 40, m L = 60 Prove: OL is a trapezoid O L

103 statements 1) OL is a quadrilateral, m O = 140, m L = 40, m = 60 2) m O + m L = 180 m L + m = 100 Proofs Given: OL is a quadrilateral, m O = 140, m = 40, m L = 60 Prove: OL is a trapezoid 1) Given 60 o reasons 2) ngle ddition Postulate 3) O and are supplementary 3) efinition of Supplementary ngles 4) L and are not supplementary 4) efinition of Supplementary ngles 140 o O 40 o L Slide 307 / 343 5) O L 5) onsecutive Interior ngles onverse 6) L O 6) onsecutive Interior ngles onverse 7) OL is a trapezoid 7) efinition of a Trapezoid ( trapezoid has exactly one pair of parallel sides) Proofs Slide 308 / 343 Try this... Given: ΔF ΔFE Prove: F E F E Slide 309 / 343 oordinate Proofs Return to the Table of ontents

104 Quadrilateral oordinate Proofs Slide 310 / 343 Recall that oordinate Proofs place figures on the artesian 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. oordinate Proofs Slide 311 / 343 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 oordinate Proofs Slide 312 / 343 kite has one unique property. The adjacent sides are congruent. P(-1,6) S(-4,3) Q(2,3) R(-1,-2) PS = (6-3) 2 + (-1 - (-4)) 2 PQ = (3-6) 2 + (2 - (-1)) 2 = = (-3) = = = 18 = 18 = 4.24 = 4.24 click click

105 P(-1,6) oordinate Proofs Slide 313 / 343 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 ecause PS = PQ and RS = RQ, PQRS is a kite. click oordinate Proofs Slide 314 / 343 Given: JKLM is a parallelogram Prove: JKLM is a square y M(-3, 0) J(1, 3) L(0, -4) K(4, -1) x oordinate Proofs M(-3, 0) J(1, 3) Slide 315 / 343 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)

106 oordinate Proofs M(-3, 0) J(1, 3) Slide 316 / 343 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? click 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 Try this... oordinate Proofs Slide 317 / 343 Given: PQRS is a trapezoid Prove: LM is the midsegment y P(2, 2) 2 Q(5, 1) L(1, 0) S(0, -2) x M(7, -2) R(9, -5) Quadrilateral oordinate Proofs Slide 318 / 343 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.

107 Quadrilateral oordinate Proofs Slide 319 / 343 Given: EFGH is a parallelogram Prove: EFGH is a rectangle y E (q, r) F(p, r) x H(q, u) G(p, u) Quadrilateral oordinate Proofs E (q, r) F(p, r) y Slide 320 / 343 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... Quadrilateral oordinate Proofs E (q, r) F(p, r) y Slide 321 / 343 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

108 Quadrilateral oordinate Proofs E (q, r) F(p, r) y Slide 322 / 343 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 Quadrilateral oordinate Proofs E (q, r) F(p, r) y Slide 323 / 343 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 (iagonals are congruent) We also know that EF GH and FG EH (Opp. sides are congruent) Therefore, EFGH is a rectangle click Quadrilateral oordinate Proofs Slide 324 / 343 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

109 Quadrilateral oordinate Proofs y K (0, 2b) J (a, 2b) Slide 325 / 343 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? Quadrilateral oordinate Proofs y K (0, 2b) J (a, 2b) Slide 326 / 343 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 ecause the midpoint U is the same for both of the diagonals, it proves that KM & JL bisect each other. click 137 If Quadrilateral STUV is a rectangle, what would be the coordinates for point U? U (e, d) U (d + e, e) y V(0, e) U(?,?) Slide 327 / 343 U (d, e) U (d, e - d) S(0, 0) T(d, 0) x

110 138 If Quadrilateral OPQR is a parallelogram, what would be the coordinates for point Q? Q (b, a) Q (b + c, a) P(0, a) y Q(?,?) Slide 328 / 343 Q (b + c, a + c) Q (b, a + c) O(0, 0) R(b, c) x 139 If Quadrilateral is an isosceles trapezoid, what would be the coordinates for point? Slide 329 / 343 (-2f, 2h) y (-2g, 2h) (-2g, -2h) (2f, 2h) (?,?) (2g, 2h) x (-2f, 0) (2f, 0) 140 Since is an isosceles trapezoid, and are congruent. Use the coordinates to verify that and are congruent. When you are finished, type in the number "1". Slide 330 / 343 y (?,?) (2g, 2h) x (-2f, 0) (2f, 0)

111 PR Released Questions Slide 331 / 343 The remaining slides in this presentation contain questions from the PR Sample Test. fter finishing unit 10, you should be able to answer these questions. Good Luck! Return to the Table of ontents The figure shows parallelogram with E = 16. Slide 332 / E not drawn to scale Part 141 Let E = x 2-48 and let E = 2x. What are the lengths of E and E? Justify your answer. PR Released Question - P - alculator Section - 7 The figure shows parallelogram with E = 16. Slide 333 / 343 Students type their answers here 16 E not drawn to scale Part 142 What conclusion can be made regarding the specific classification of parallelogram? Justify your answer. Once you are finished with the question, type in the word to describe the classification of parallelogram. PR Released Question - P - alculator Section - 7

112 The figure shows parallelogram PQRS on a coordinate plane. iagonals SQ and PR intersect at point T. Students type their answers here y Slide 334 / 343 P(2b, 2c) Q(?,?) T S(0, 0) R(2a, 0) x Part 143 Find the coordinates of point Q in terms of a, b, and c. Enter only your answer. PR Released Question - P - alculator Section - 8 Part 144 Since PQRS is a parallelogram, SQ and PR bisect each other. Use the coordinates to verify that SQ and PR bisect each other. Slide 335 / 343 Which formula(s) could be used to answer this question? istance Formula Slope Midpoint Formula and only E, and PR Released Question - P - alculator Section - 8 Part ontinued Slide 336 / Since PQRS is a parallelogram, SQ and PR bisect each other. Use the coordinates to verify that SQ and PR bisect each other. Students type their answers here When you are finished, type in the word "one". PR Released Question - P - alculator Section - 8

113 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? Slide 337 / 343 T S R Given: Quadrilateral PQRS is a parallelogram. Prove: PT = RT, ST = QT Part 146 Which reason justifies the statement for step 3 in the proof? When two parallel lines are intersected by a transversal, same side interior angles are congruent. When two parallel lines are intersected by a transversal, alternate interior angles are congruent. When two parallel lines are intersected by a transversal, same side interior angles are supplementary. When two parallel lines are intersected by a transversal, alternate interior angles are supplementary. Slide 338 / 343 PR Released Question - EOY - alculator Section - 19 Part 147 Which statement is justified by the reason for step 4 in the proof? PQ RS Slide 339 / 343 PQ SP PT TR SQ PR PR Released Question - EOY - alculator Section - 19

114 Part 148 Which reason justifies the statement for step 5 in the proof? side-side-side triangle congruence side-angle-side triangle congruence Slide 340 / 343 angle-side-angle triangle congruence angle-angle-side triangle congruence PR Released Question - EOY - alculator Section - 19 Part Slide 341 / 343 nother method of proving diagonals of a parallelogram bisect each other uses a coordinate grid. 149 What could be shown about the diagonals of parallelogram PQRS to complete the proof? PR and SQ have the same length. PR is a perpendicular bisector of SQ. PR and SQ have the same midpoint. ngles formed by the intersection of PR and SQ each measure 90. Slide 342 / 343 PR Released Question - EOY - alculator Section - 19