Slide 1 / 343 Slide 2 / 343 Geometry Quadrilaterals 2015-10-27 www.njctl.org Slide 3 / 343 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 4 / 343 Polygons Return to the Table of ontents Slide 5 / 343 Polygons polygon is a closed figure made of line segments connected end to end at vertices. Slide 6 / 343 Polygons Simple Polygons omplex Polygons 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? This course will only consider simple polygons, which we will refer to as "polygons."
Slide 7 / 343 Types of Polygons Polygons are named by their number of sides, angles or vertices; which are all equal in number. Slide 8 / 343 onvex or oncave Polygons 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 11 11-gon 12 dodecagon n n-gon onvex Polygons ll interior angles are less than 180º. No side, if extended, will cross into the interior. interior interior oncave Polygons Slide 9 / 343 onvex or oncave Polygons Slide 10 / 343 1 The figure below is a concave polygon. True False 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 2 Indentify the polygon. Pentagon Octagon Quadrilateral Hexagon ecagon F Triangle Slide 11 / 343 Slide 12 / 343 3 Is the polygon convex or concave? onvex oncave
Slide 13 / 343 4 Is the polygon convex or concave? Slide 14 / 343 quilateral Polygons onvex oncave polygon is equilateral if all its sides are congruent. Slide 15 / 343 quiangular Polygons polygon is equiangular if all its angles are congruent. Slide 16 / 343 Regular Polygons polygon is regular if it is equilateral and equiangular. Slide 17 / 343 5 escribe the polygon. (hoose all that apply) Pentagon Octagon Quadrilateral Hexagon 60 o Triangle 4 4 F onvex G oncave H quilateral I quiangular 60 o 60 o J Regular 4 Slide 18 / 343 6 escribe the polygon. (hoose all that apply) Pentagon Octagon Quadrilateral Hexagon Triangle F onvex G oncave H quilateral I quiangular J Regular
Slide 19 / 343 7 escribe the polygon. (hoose all that apply) Pentagon Octagon Quadrilateral Hexagon Triangle F onvex G oncave H quilateral I quiangular J Regular Slide 20 / 343 ngle 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. Slide 21 / 343 ngle Measures of Polygons very 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. Slide 22 / 343 ngle Measures of Polygons very 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 this polygon? 180 180 What is the sum of the interior angles of a quadrilateral? Slide 23 / 343 ngle Measures of Polygons Slide 24 / 343 ngle Measures of Polygons very polygon can be thought of as consisting of some number of triangles. pentagon is comprised of three triangles. very polygon can be thought of as consisting of some number of triangles. n octagon is comprised of six triangles. 180º 180º What is the sum of the interior angles of a pentagon? 180º 180º 180º What is the sum of the interior angles of an octagon? 180º 180º 180º 180º
Slide 25 / 343 ngle 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 4 2 5 3 6 4 n Math Practice Slide 26 / 343 ngle 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. Number of sides Using that information, complete this table. Number of triangular regions Sum of Interior ngles 3 1 180º 4 2 360º 5 3 540º 6 4 n Math Practice Slide 27 / 343 8 What is the sum of the interior angles of this polygon? Slide 28 / 343 9 How many triangular regions can be drawn in this polygon? Slide 29 / 343 10 What is the sum of the interior angles of this polygon? Slide 30 / 343 11 How many triangular regions can be drawn in this polygon?
Slide 31 / 343 12 What is the sum of the interior angles of this polygon? Slide 32 / 343 13 Which of the following road signs has interior angles that add up to 360? Yield Sign No Passing Sign National Forest Sign Stop Sign No rossing Sign Slide 33 / 343 Polygon Interior ngles 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 34 / 343 14 What is the measurement of each angle in the diagram below? L (3x+4) M 146 x N heptagon 7 180(7-2) = 900 octagon 8 180(8-2) = 1080 P (2x+3) (3x) nonagon 9 180(9-2) = 1260 O decagon 10 180(10-2) = 1440 11-gon 11 180(11-2) = 1620 dodecagon 12 180(12-2) = 1800 Slide 35 / 343 Slide 36 / 343 Interior ngles of quiangular Polygons Interior ngles of Regular Polygons Regular polygons are equiangular: their angles are equal. The measure of each interior angle of an regular polygon is given by: m = (n-2)180º n 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 ngles Number of ngles (n - 2)180º n regular polygon number of sides sum of interior angles measure of each angle triangle 3 180 60 quadrilateral 4 360 90 pentagon 5 540 108 hexagon 6 720 120 octagon 8 1080 135 decagon 10 1440 144 n-gon n (n-2)(180 ) (n-2)(180 )/n
Slide 37 / 343 15 If the stop sign is a regular octagon, what is the measure of each interior angle? Slide 38 / 343 16 What is the sum of the measures of the interior angles of a convex 20-gon? 2880 3060 3240 3420 Slide 39 / 343 17 What is the measure of each interior angle of a regular 20-gon? 162 3240 180 60 Slide 40 / 343 18 What is the measure of each interior angle of a regular 16-gon? 2520 2880 3240 157.5 19 What is the value of x? (10x+8) (8x) Slide 41 / 343 Slide 42 / 343 Polygon xterior ngle The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360. 5 (5x+15) 1 4 (9x-6) (11x+16) 2 3 In this case, m 1 + m 2 + m 3 + m 4 + m 5 = 360
Slide 43 / 343 Proof of xterior ngles 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 Slide 44 / 343 Proof of xterior ngles Given: ny polygon has n interior angles, where n is the number of its sides, interior angles, or vertices Prove: Ʃ external s = 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." 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 Slide 45 / 343 Polygon xterior ngle Slide 46 / 343 Polygon xterior ngle xternal 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 2 5 3 4 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. Slide 47 / 343 ach 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: Polygon xterior ngle Ʃ external s + Ʃ internal s = n x 180 Slide 48 / 343 Polygon xterior ngle 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:
Slide 49 / 343 Polygon xterior ngle Ʃ external s + [n x 180-2 x 180 ] = n x 180 Subtracting n x 180-2 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. Statement Slide 50 / 343 Reason 1 ny polygon has n interior s Given 2 3 4 Proof of xterior ngles 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 ach interior and exterior are a linear pair ach 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 Statement Slide 51 / 343 Reason 7 The sum of interior s is (n - 2) x 180 Interior ngle 8 9 Proof of xterior ngles 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 n x 180 = sum of exterior s + (n - 2) x 180 The sum of the exterior s = n x 180 - [(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 = 12 180 (2) = 360 Substitution Property Subtraction Property Subtraction Property istributive Property Simplifying Slide 53 / 343 Polygon xterior ngle orollary The measure of each exterior angle of a regular polygon with n sides is 360º/n Slide 52 / 343 Polygon xterior ngle orollary 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. Slide 54 / 343 20 What is the sum of the measures of the exterior angles of a heptagon? a This is a regular hexagon. n = 6 a = 360 6 a = 60 180 360 540 720
Slide 55 / 343 21 If a heptagon is regular, what is the measure of each exterior angle? Slide 56 / 343 22 What is the sum of the measures of the exterior angles of a pentagon? 72 60 51.43 45 Slide 57 / 343 23 What is the measure of each exterior angle of a regular pentagon? Slide 58 / 343 24 The measure of each interior angle of a regular convex polygon is 172. How many sides does the polygon have? nswer 25 Slide 59 / 343 The measure of each interior angle of a regular convex polygon is 174. Find the number of sides of the polygon. Slide 60 / 343 26 The measure of each interior angle of a regular convex polygon is 162. Find the number of sides of the polygon. 64 62 58 60
Slide 61 / 343 Slide 62 / 343 lick on the link below and complete the two labs before the Parallelogram lesson. Properties of Parallelograms Lab - Investigating Parallelograms Lab - Quadrilaterals Party Return to the Table of ontents Slide 63 / 343 Parallelograms y definition, the opposite sides of a parallelogram are parallel. Slide 64 / 343 The opposite sides a parallelogram are congruent. G F In parallelogram FG, G? If is a parallelogram, then and? Slide 65 / 343 Proof of 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 66 / 343 Proof of The opposite sides a parallelogram are congruent. 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 & Given that and Name a pair of congruent Δs For what reason are they congruent? Why?
Slide 67 / 343 Proof of The opposite sides a parallelogram are congruent. Slide 68 / 343 27 The opposite sides of a parallelogram are ongruent Parallel oth a and b Given that Δ Δ Which pairs of sides are must be congruent? Why? Slide 69 / 343 The opposite angles of a parallelogram are congruent. Slide 70 / 343 Proof of The opposite angles of a parallelogram are congruent. If is a parallelogram, then m = m and m = m s we did earlier, construct diagonal. We earlier proved Δ Δ What does this tell you about and? Why? Slide 71 / 343 Slide 72 / 343 28 Δ Δ. Prove your answer. True 29 ased on the previous question, m =? Prove your answer. m False m m None of the above
Slide 73 / 343 The consecutive angles of a parallelogram are supplementary. y x x If is a parallelogram, then x o + y o = 180 o y Slide 74 / 343 Proof of 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? Slide 75 / 343 Proof of The consecutive angles of a parallelogram are supplementary. x y Slide 76 / 343 30 is a parallelogram. Find the value of x. 2y 65 w y x 12 x-5 Repeat the same process, but this time consider and to be transversals of and, which are. 9 (5z) What are these pairs of angles called: & and &? What is the relationship of each pair of angles like that? Slide 77 / 343 31 is a parallelogram. Find the value of y. 2y 65 w Slide 78 / 343 32 is a parallelogram. Find the value of z. 2y 65 w 12 x-5 12 x-5 9 (5z) 9 (5z)
Slide 79 / 343 33 is a parallelogram. Find the value of w. 2y 65 w 12 x-5 (5z) 9 Slide 80 / 343 34 FG is a parallelogram. Find the value of w. G 15 70 (2w) (z + 12) 3x - 3 21 F y 2 Slide 81 / 343 35 FG is a parallelogram. Find the value of x. Slide 82 / 343 36 FG is a parallelogram. Find the value of y. 21 70 15 (2w) (z + 12) G 3x - 3 F y 2 21 70 15 (2w) (z + 12) G 3x - 3 F y 2 G 15 70 Slide 83 / 343 37 FG is a parallelogram. Find the value of z. (2w) (z + 12) 3x - 3 21 F y 2 Slide 84 / 343 The diagonals of a parallelogram bisect each other. If is a parallelogram, then and
Slide 85 / 343 Proof of 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 86 / 343 38 LMNP is a parallelogram. The diagonals bisect each other. Find QN L 4 M Steps to follow: 1) Identify pairs of triangles in the above diagram that you need to prove congruent to prove this theorem. P 6 Q N 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). Slide 87 / 343 39 LMNP is a parallelogram. Find MP Slide 88 / 343 40 R is a parallelogram. Find the value of x. L 4 M x 4y P 6 Q N R 8 S 10 Slide 89 / 343 41 R is a parallelogram. Find the value of y. Slide 90 / 343 42 R is a parallelogram. Find R. x 4y x 4y 8 S 10 8 S 10 R R
Slide 91 / 343 Slide 92 / 343 43 In a parallelogram, the opposite sides are parallel. 44 The opposite angles of a parallelogram are. sometimes bisect always congruent never parallel supplementary Slide 93 / 343 Slide 94 / 343 45 The consecutive angles of a parallelogram are. 46 The diagonals of a parallelogram each other. bisect bisect congruent congruent parallel parallel supplementary supplementary Slide 95 / 343 Slide 96 / 343 47 The opposite sides of a parallelogram are. bisect congruent parallel supplementary 48 MTH is a parallelogram. H = 12 and MR = 7. Find RT. M 6 7 8 9 7 R 12 H T
Slide 97 / 343 Slide 98 / 343 49 MTH is a parallelogram. Find R. 6 M 7 8 7 9 R 12 50 MTH is a parallelogram. Find m H. M 14 (3y + 8) H T H 2x - 4 98 T Slide 99 / 343 51 MTH is a parallelogram. Find the value of x. M x 2-24 (3y + 8) Slide 100 / 343 52 MTH is a parallelogram. Find the value of y. M 14 (3y + 8) H 2x 98 T H 2x - 4 98 T Slide 101 / 343 Proving Quadrilaterals are Parallelograms In quadrilateral, and, so is a parallelogram. Slide 102 / 343 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Return to the Table of ontents
Slide 103 / 343 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Slide 104 / 343 53 PQRS is a Parallelogram. xplain your reasoning. True P 6 False S 4 In quadrilateral, and, so is a parallelogram. 4 R 6 Q Slide 105 / 343 54 Is PQRS a parallelogram? xplain your reasoning. Slide 106 / 343 55 Is this quadrilateral definitely a parallelogram? Yes Yes No P Q No 136 2 78 S R Slide 107 / 343 56 Is this quadrilateral definitely a parallelogram? Slide 108 / 343 57 Is this quadrilateral definitely a parallelogram? Yes No 4.99 Yes No 3 3 5
Slide 109 / 343 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Slide 110 / 343 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 75 105 75 In quadrilateral, m + m = 180 and m + m = 180, so is a parallelogram. In quadrilateral, and, so is a quadrilateral. Slide 111 / 343 If one pair of sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. Slide 112 / 343 58 Is this quadrilateral definitely a parallelogram? Yes No In quadrilateral, and, so is a parallelogram. Slide 113 / 343 Slide 114 / 343 59 Is this quadrilateral definitely a parallelogram? 60 Is this quadrilateral definitely a parallelogram? Yes No 141 49 Yes No 19 8 9.5 39 8
Slide 115 / 343 61 Is this quadrilateral definitely a parallelogram? Yes No 62 Yes Slide 116 / 343 Three interior angles of a quadrilateral measure 67, 67 and 113. Is this enough information to tell whether the quadrilateral is a parallelogram? xplain. No Slide 117 / 343 Fill in the blank Slide 118 / 343 Fill in the blank parallel perpendicular supplementary parallel perpendicular supplementary bisect congruent bisect congruent In a parallelogram... the opposite sides are and, the opposite angles are, the consecutive angles are, and the diagonals each other. 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. Slide 119 / 343 63 Which theorem proves the quadrilateral is a parallelogram? 6(7-3) Slide 120 / 343 64 Which theorem proves this quadrilateral is a parallelogram? The opposite angles are congruent. 3(2) 3 The opposite sides are congruent. The opposite angles are congruent. 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. F 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.
Slide 121 / 343 65 Which theorem proves the quadrilateral is a parallelogram? The opposite angles are congruent. The opposite sides are congruent. 3(6-4) 6 6 Slide 122 / 343 Rhombi, Rectangles and Squares 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. Return to the Table of ontents Slide 123 / 343 Three Special Parallelograms ll the same properties of a parallelogram apply to the rhombus, rectangle, and square. Slide 124 / 343 Rhombus rhombus is a quadrilateral with four congruent sides. If, then is a rhombus. Rhombus Rectangle Square Slide 125 / 343 66 What is the value of y that will make the quadrilateral a rhombus? 7.25 12 20 25 12 3 5 y Slide 126 / 343 67 What is the value of y that will make this parallelogram a rhombus? 7.25 12 20 25 2y+29 6y
Slide 127 / 343 The diagonals of a rhombus are perpendicular. Slide 128 / 343 The diagonals of a rhombus are perpendicular. If is a rhombus, then. O Since is a parallelogram, its diagonals bisect each other. O O and O O. Slide 129 / 343 Slide 130 / 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 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If is a rhombus, then and 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. Slide 131 / 343 If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. We proved earlier by SSS, that ΔO ΔO ΔO ΔO Slide 132 / 343 68 FGH is a rhombus. Find the value of x. 2x - 6 F 72 z 5y O orresponding parts of Δs are ; so O O O O N G 10 H O O O O So each diagonal bisects a pair of opposite angles
Slide 133 / 343 69 FGH is a rhombus. Find the value of y. Slide 134 / 343 70 FGH is a rhombus. Find the value of z. 72 2x - 6 z F 72 2x - 6 z F 5y 5y G 10 H G 10 H Slide 135 / 343 71 This quadrilateral is a rhombus. Find the value of x. Slide 136 / 343 72 This quadrilateral is a rhombus. Find the value of y. 8 z 86 3x + 2 8 z 86 3x + 2 1y 2 2 1y 2 2 Slide 137 / 343 73 This quadrilateral is a rhombus. Find the value of z. Slide 138 / 343 74 This is a rhombus. Find the value of x. 8 z 86 3x + 2 x 1y 2 2
Slide 139 / 343 75 This is a rhombus. Find the value of x. Slide 140 / 343 76 This is a rhombus. Find the value of x. 1 3 x - 3 9 126 x Slide 141 / 343 77 HJKL is a rhombus. Find the length of HJ. H J Slide 142 / 343 Rectangles rectangle is a quadrilateral with only right angles. 6 M 16 L K If is a rectangle, then,, and are right angles. If,, and are right angles, then is a rectangle. Slide 143 / 343 78 What value of y will make the quadrilateral a rectangle? (6y) Slide 144 / 343 The diagonals of a rectangle are congruent. 12 M If is a rectangle, then. lso, since a rectangle is a parallelogram, M M M M
Slide 145 / 343 Slide 146 / 343 79 RT is a rectangle. Find the value of x. The diagonals of a rectangle are congruent. Since is a rectangle (and therefore a parallelogram) R (9y) 63 ecause of the reflexive property,. 2x - 5 13 Since is a rectangle m = m = 90º T Then Δ Δ by SS nd, since they are corresponding parts of Δs Slide 147 / 343 80 RT is a rectangle. Find the value of y. Slide 148 / 343 81 RSTU is a rectangle. Find the value of z. R 63 (9y) R S T 2x - 5 13 U (8z) T Slide 149 / 343 82 RSTU is a rectangle. Find the value of z. Slide 150 / 343 Square R 4z - 9 S square is a quadrilateral with equal sides and equal angles. It is, therefore both a rhombus and a rectangle. 7 square has all the properties of a rectangle and rhombus. U T
Slide 151 / 343 83 This quadrilateral is a square. Find the value of x. Slide 152 / 343 84 This quadrilateral is a square. Find the value of y. z - 4 6 z - 4 6 (5x) (3y) (5x) (3y) Slide 153 / 343 85 This quadrilateral is a square. Find the value of z. Slide 154 / 343 86 This quadrilateral is a square. Find the value of x. z - 4 (3y) (5x) 6 y 2 (12z) 10-3y (x 2 + 9) Slide 155 / 343 87 This quadrilateral is a square. Find the value of y. Slide 156 / 343 88 This quadrilateral is a square. Find the value of z. y 2 10-3y y 2 10-3y (12z) (12z) (x 2 + 9) (x 2 + 9)
Slide 157 / 343 Slide 158 / 343 89 This quadrilateral is a square. Find the value of y. 90 This quadrilateral is a rhombus. Find the value of x. (18y) 4x 2x + 6 Slide 159 / 343 Slide 160 / 343 91 The quadrilateral is parallelogram. Find the value of x. 92 The quadrilateral is a rectangle. Find the value of x. 112 10x (4x) 3x + 7 Slide 161 / 343 Slide 162 / 343 Fill in the Table with the orrect efinition rhombus rectangle square Lab - Quadrilaterals in the oordinate Plane Opposite sides are iagonals are iagonals bisect opposite 's Has 4 right 's Has 4 sides iagonals are
Slide 163 / 343 Slide 164 / 343 Trapezoid trapezoid is a quadrilateral with one pair of parallel sides. base Trapezoids leg base angles leg base Return to the Table of ontents The parallel sides are called bases. The nonparallel sides are called legs. trapezoid also has two pairs of base angles. Slide 165 / 343 Isosceles Trapezoid n isosceles trapezoid is a trapezoid with congruent legs. Slide 166 / 343 If a trapezoid is isosceles, then each pair of base angles are congruent. If is an isosceles trapezoid, then and. Slide 167 / 343 If a trapezoid is isosceles, then each pair of base angles are congruent. Slide 168 / 343 If a trapezoid is isosceles, then each pair of base angles are congruent. Given: is an isosceles trapezoid Prove: and Given: is an isosceles trapezoid Prove: and. F F continued...
F Statement Slide 169 / 343 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 and F. 4 5 6, F, F, F, F and efinition of are right angles Perpendicular Lines F F F F m = m F = m F = m F = m F = m = 90 Slide 171 / 343 ll right angles are efinition of congruent and right angles continued... If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. Slide 170 / 343 7 F is a rectangle efinition of a rectangle 8 F Opposite sides of a rectangle are 9 Δ ΔF HL 10 F PT 11 m = m F efinition of angles 12 m + 90 = m F + 90 ddition Property of = 13 14 m + m = m F + m F m + m = m m F + m F = m Substitution Property of = ngle ddition Postulate 15 m = m Substitution Property of = 16, or efinition of angles Slide 172 / 343 If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. ase 1 - Longer ase s are Given: In trapezoid, Prove: is an isosceles trapezoid If in trapezoid,. Then, is an isosceles trapezoid. F continued... Slide 173 / 343 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 and F. 4, F, F, F, F and efinition of are right angles Perpendicular Lines F F F 5 F ll right angles are 6 F is a rectangle efinition of a rectangle 7 F Opposite sides of a rectangle are 8 Δ ΔF S 9 PT 10 is an isosceles trapezoid efinition of an isosceles trapezoid Slide 174 / 343 If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. 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.
F Slide 175 / 343 If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. 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, as shown. an you now see how to extend the approach we used for the first case to this second case? Slide 177 / 343 93 The quadrilateral is an isosceles trapezoid. Find the value of x. 6 3 5 7 9 11 7x - 10 14 Slide 176 / 343 If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles. F 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 & 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). Slide 178 / 343 94 The quadrilateral is an isosceles trapezoid. Find the value of x. 3 5 7 9 64 (9x + 1) Slide 179 / 343 trapezoid is isosceles if and only if its diagonals are congruent. In trapezoid, If. Then, is isosceles. If is isosceles Then,. Slide 180 / 343 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 181 / 343 95 PQRS is a trapezoid. Find the m S. P Q 112 147 Slide 182 / 343 96 PQRS is a trapezoid. Find the m R. P Q 112 147 S (6w + 2) (3w) R nswer S (6w + 2) (3w) R nswer Slide 183 / 343 97 PQRS is an isosceles trapezoid. Find the m Q. Slide 184 / 343 98 PQRS is an isosceles trapezoid. Find the m R. P 123 Q (9w - 3) P 123 Q (9w - 3) S (4w + 1) R S (4w + 1) R Slide 185 / 343 Slide 186 / 343 99 PQRS is an isosceles trapezoid. Find the m S. 100 The trapezoid is isosceles. Find the value of x. S P 123 (4w + 1) Q (9w - 3) R 4 9 6x + 3 2x + 2
Slide 187 / 343 101 The trapezoid is isosceles. Find the value of x. 137 Slide 188 / 343 102 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals? Yes No H I x K J Slide 189 / 343 Midsegment of a Trapezoid Slide 190 / 343 The midsegment of a trapezoid is a segment that joins the midpoints of the legs The midsegment of a trapezoid is parallel to both the bases. F F lick on the link below and complete the lab. Lab - Midsegments of a Trapezoid Slide 191 / 343 Proof of The midsegment of a trapezoid is parallel to both the bases. It's given that is the midpoint of ; F is the midpoint of ; and 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. That means that F, where F is the midsegment of the trapezoid. Slide 192 / 343 The length of the midsegment is half the sum of the bases. F F = 0.5( + )
2x x G Slide 193 / 343 Proof of The length of the midsegment is half the sum of the bases. H I J y 2y F Slide 195 / 343 Proof of Given: is the midpoint of F is the midpoint of F Prove: F = 0.5( + ) continued... Slide 194 / 343 Proof of Statement Reason is the midpoint of 1 F is the midpoint of Given F 2 G H, FI J orresponding angles are 3 G H, FI J Reflexive Property of 4 G ~ H, FI ~ J ngle-ngle ~ H H J J orresponding sides of ~ s 5 = = = = G G, F I IF are proportional The midpoints and F divide and into 2 congruent 6 = 2, = 2F segments, so and are twice the size of and F respectfully. = 2 = 2 7 ivision property of =, F The scale factor in both of our 8 efinition of scale factor similar triangles is 2. continued... Slide 196 / 343 103 PQRS is a trapezoid. Find LM. Statement Reason If G = x, then H = 2x. 9 If IF = y, then H = 2y. Substitution Property of = 10 F = + x + y Segment ddition Postulate 112F = 2 + 2x + 2y Multiplication Property of = 12 = + 2x + 2y Segment ddition Postulate 13 2x + 2y = - Subtraction Property of = Q L P 15 7 S M R 14 2F = 2 + - Substitution Property of = 15 2F = + ombine Like Terms Multiplication Property of = OR 16 F = 0.5( + ) ivision Property of = and istributive Property of = Slide 197 / 343 Slide 198 / 343 104 PQRS is a trapezoid. Find PS. Q 20 14.5 L P S M R 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
106 Slide 199 / 343 PQRS is an trapezoid. ML is the midsegment. Find the value of y. 107 Slide 200 / 343 PQRS is an trapezoid. ML is the midsegment. Find the value of z. S y M 5 R 10 14 x P 7 L z Q S y M 5 R 10 14 x P 7 L z Q Slide 201 / 343 108 F is the midsegment of trapezoid HIJK. Find the value of x. H 6 I Slide 202 / 343 109 F is the midsegment of trapezoid HIJK. Find the value of x. J 19 K x F F 10 K 15 J I x H Slide 203 / 343 Slide 204 / 343 110 Which of the following is true of every trapezoid? hoose all that apply. xactly 2 sides are congruent. xactly one pair of sides are parallel. The diagonals are perpendicular. There are 2 pairs of base angles. Kites Return to the Table of ontents
Slide 205 / 343 Kites kite is a quadrilateral with two pairs of adjacent congruent sides whose opposite sides are not congruent. Slide 206 / 343 If a quadrilateral is a kite, then it has one pair of congruent opposite angles. In kite, and Lab - Properties of Kites Slide 207 / 343 Slide 208 / 343 Proof of 111 LMNP is a kite. Find the value of x. 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. L 72 M (x 2-1) 48 N Δ Δ by SSS. since corresponding parts of Δs are. m m since. P In kite, m = m and m m Slide 209 / 343 Slide 210 / 343 112 R is a kite. R. R R 113 R is a kite.. R R
(8x + 4) Slide 211 / 343 114 Find the value of z in the kite. Slide 212 / 343 115 Find the value of x in the kite. z 5z - 8 68 44 116 Find the value of x. Slide 213 / 343 Slide 214 / 343 The diagonals of a kite are perpendicular. 36 (3x 2-2) In kite 24 Slide 215 / 343 Proof of The diagonals of a kite are perpendicular. Slide 216 / 343 117 Find the value of x in the kite. Given that and and by reflexive property x Δ Δ by SSS by PT Δ Δ by SS by PT In kite is perpendicular to since intersecting lines that create adjacent angles are perpendicular
Slide 217 / 343 Slide 218 / 343 118 Find the value of y in the kite. onstructing Quadrilaterals (12y) Return to the Table of ontents 1. raw two intersecting lines. Slide 219 / 343 onstruction of a Parallelogram Slide 220 / 343 onstruction of a Parallelogram 2. Place a compass at the point of intersection and draw an arc where it intersects one line in two places. Slide 221 / 343 onstruction 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. Slide 222 / 343 onstruction of a Parallelogram 4. onnect the four points where the arcs intersect the lines.
Slide 223 / 343 onstruction of a Parallelogram Which property of parallelograms does this construction technique depend upon? Slide 224 / 343 Try this! onstruct a parallelogram using the lines below. 1) onstruction of a Parallelogram Slide 225 / 343 Slide 226 / 343 Videos emonstrating the onstructions of a Parallelogram using ynamic Geometric Software lick here to see parallelogram video: compass & straightedge lick here to see parallelogram video: Menu options 1. raw two intersecting lines. Slide 227 / 343 onstruction of a Rectangle Slide 228 / 343 onstruction 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.
Slide 229 / 343 onstruction of a Rectangle 3. onnect the four points where the arcs intersect the lines. Slide 230 / 343 onstruction of a Rectangle Which properties of rectangles does this construction technique depend upon? Slide 231 / 343 onstruction of a Rectangle Try this! onstruct a rectangle using the lines below. 3) Slide 232 / 343 onstruction of a Rectangle Try this! onstruct a rectangle using the lines below. 4) Slide 233 / 343 Slide 234 / 343 Videos emonstrating the onstructions of a Rectangle using ynamic Geometric Software onstruction of a Rhombus 1. onstruct a segment of any length. lick here to see rectangle video: compass & straightedge lick here to see rectangle video: Menu options
Slide 235 / 343 onstruction of a Rhombus 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points. Slide 236 / 343 onstruction 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 and open it so the pencil is more than halfway, it does not matter how far, to point Slide 237 / 343 onstruction of a Rhombus 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points. Slide 238 / 343 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 3. Place the tip of your compass on your center point (midpoint). xtend your compass to any length. raw 2 arcs where the compass intersects the perpendicular bisector. c. raw a line through the arc intersections. The intersection point created with the two perpendicular lines is the center of your rhombus. Note: If you wanted to, you could use the intersection points of the 2 arcs from step 2 as your extra points. Slide 239 / 343 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 240 / 343 onstruction of a Rhombus Which properties of rhombii (pl. for rhombus) does this construction technique depend upon?
Slide 241 / 343 Videos emonstrating the onstructions of a Rhombus using ynamic Geometric Software lick here to see rhombus video: compass & straightedge Slide 242 / 343 onstruction of a Rhombus Try this! onstruct a rhombus using the segment below. 5) lick here to see rhombus video: Menu options Slide 243 / 343 onstruction of a Rhombus Try this! onstruct a rhombus using the segment below. Slide 244 / 343 onstruction of a Square 1. onstruct a segment of any length. 6) Slide 245 / 343 onstruction of a Square 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points. Slide 246 / 343 onstruction 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 and open it so the pencil is more than halfway, it does not matter how far, to point. raw an arc. 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
Slide 247 / 343 onstruction of a Square 2. Find the perpendicular bisector and midpoint of a line segment by constructing the locus between two points. Slide 248 / 343 onstruction of a Square 3. Place your compass tip on point and the pencil point on either or and construct your circle. 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). Slide 249 / 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 250 / 343 onstruction of a Square Which properties of squares does this construction technique depend upon? 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) onstruction of a Square
Slide 253 / 343 Videos emonstrating the onstructions of a Square using ynamic Geometric Software Slide 254 / 343 onstruction of an Isosceles Trapezoid 1. raw two intersecting lines. lick here to see square video: compass & straightedge lick here to see square video: Menu options Slide 255 / 343 onstruction 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. Slide 256 / 343 onstruction 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). Slide 257 / 343 onstruction of an Isosceles Trapezoid 4. onnect the four points where the arcs intersect the lines. Slide 258 / 343 onstruction of an Isosceles Trapezoid Which properties of isosceles trapezoids does this construction technique depend upon?
Slide 259 / 343 Slide 260 / 343 onstruction of an Isosceles Trapezoid Try this! onstruct an isosceles trapezoid using the lines below. 9) Slide 261 / 343 Videos emonstrating the onstructions of an Isosceles Trapezoid using ynamic Geometric Software Slide 262 / 343 onstruction of a Kite 1. onstruct a segment of any length. lick here to see isosceles trapezoid video: compass & straightedge lick here to see isosceles trapezoid video: Menu options Slide 263 / 343 onstruction of a Kite Slide 264 / 343 onstruction of a Kite 2. Place your compass tip at any point and draw 2 arcs that intersect your segment. 3. Find the perpendicular bisector between the 2 red arcs. a) Place your compass tip at one of the intersection points. xtend your pencil out more than half way between the 2 arcs. raw an arc.
Slide 265 / 343 onstruction of a Kite Slide 266 / 343 onstruction of a Kite 3. Find the perpendicular bisector between the 2 red arcs. 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. c) onnect the intersections of the 2 arcs to construct your perpendicular line. Slide 267 / 343 onstruction of a Kite 4. Place the tip of your compass at the intersection point of our perpendicular lines. xtend your compass to any length. raw 2 arcs where the compass intersects the perpendicular line. Slide 268 / 343 onstruction of a Kite 5. 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. Note: If you wanted to, you could use the intersection points of the 2 arcs from step 3 as your extra points. Slide 269 / 343 Slide 270 / 343 onstruction of a Kite Which properties of a kite does this construction technique depend upon?
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 Slide 274 / 343 omplete the chart (There can be more than one answer). 119 rhombus is a square. Families of Quadrilaterals Slide 275 / 343 Return to the Table of ontents 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 120 square is a rhombus. Special Quadrilateral(s) Slide 276 / 343 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 always always sometimes sometimes never never
121 rectangle is a rhombus. Slide 277 / 343 122 trapezoid is isosceles. Slide 278 / 343 always always sometimes sometimes never never 123 kite is a quadrilateral. Slide 279 / 343 124 parallelogram is a kite. Slide 280 / 343 always always sometimes sometimes never never Slide 281 / 343 Families of Quadrilaterals 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. Slide 282 / 343 Quadrilaterals Quadrilateral is a parallelogram. x 2-75 2x + 5 25 etermine the value of x. 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
x 2-75 2x + 5 Slide 283 / 343 Quadrilaterals Quadrilateral is a parallelogram. x 2-75 = 2x + 5 click x 2-2x - 80 = 0 click Slide 284 / 343 125 Quadrilateral FGH is a parallelogram. Find the value of x. F 25 (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? xplain. Since is a parallelogram, = 25 because opposite sides are congruent. When substituting x = 10 back into the expressions, then = = = = 25. Therefore, is a rhombus. x 2-32 H 2x + 6.63 32 J 3x - 1.37 G 3x + 8 click Slide 285 / 343 126 ased on your answer to the previous slide, what type of quadrilateral is FGH? xplain. When you are finished, choose the classification of quadrilateral FGH from the choices below. F Rhombus Slide 286 / 343 127 Quadrilateral KLMN is a trapezoid. Find the value of x. K L (7x + 30) (10x) x 2-32 2x + 6.63 J 3x - 1.37 3x + 8 Rectangle Square N (x 2-20) M H 32 G N Slide 287 / 343 128 ased on your answer to the previous slide, what type of quadrilateral is KLMN? xplain. When you are finished, determine whether trapezoid KLMN is isosceles. K L (7x + 30) (10x) (x 2-20) M Isosceles Trapezoid Not an Isosceles Trapezoid Slide 288 / 343 xtension: Inscribed Quadrilaterals quadrilateral is inscribed if all its vertices lie on a circle.. inscribed quadrilateral..
Slide 289 / 343 Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. Slide 290 / 343 Inscribed Quadrilaterals xample: Find the m G. Then, find the value of a.. G F m = 1/2 (mfg) [green arc] m F = 1/2 (mg) [red arc] Since mfg + mg = 360, m + m F = 1/2 (mfg) + 1/2 (mg) m + m F = 1/2 (mfg + mg) m + m F = 1/2(360 ) m + m F = 180 (3a + 8). G 121 (2a - 3) F m + m F = 180 m + m G = 180 Slide 291 / 343 Slide 292 / 343 129 What is m L? M 130 What is m M? M 25 J 115 23 J 115 65 115 155. 63 L 63 117 153. 63 L K K Slide 293 / 343 Slide 294 / 343 131 What is the value of x? M 8 10 J (7x - 6) (8y + 15) 132 What is the value of y? 13 J 14.69 (7x - 6) M (8y + 16) 17 19. (5y - 4) (3x - 4) L 7 6.08. (5y - 5) (3x - 4) L K K
Slide 295 / 343 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 296 / 343 133 omplete the sentence below. trapezoid can be inscribed into a circle. lways Sometimes Never Find out by completing the lab below. Lab: Inscribed Quadrilaterals Slide 297 / 343 134 omplete the sentence below. rectangle can be inscribed into a circle. Slide 298 / 343 135 omplete the sentence below. rhombus can be inscribed into a circle. lways Sometimes Never lways Sometimes Never Slide 299 / 343 Slide 300 / 343 136 omplete the sentence below. kite can be inscribed into a circle. lways Sometimes Never Proofs Return to the Table of ontents
Slide 301 / 343 Proofs Slide 302 / 343 Proofs T 1 Given: T M, 1 2 Prove: TM is a parallelogram. T 1 Option statements M 2 reasons 1) T M, 1 2 1) Given M 2 2) M M 2) Reflexive Property 3) MT M 3) Side ngle Side 4) TM 4) PT 5) TM is a parallelogram 5) oth pairs of opposite sides of are congruent, so the quadrilateral is a parallelogram. Slide 303 / 343 Slide 304 / 343 Proofs Proofs Option M T 2 1 Given: FGHJ is a parallelogram, F is a right angle Prove: FGHJ is a rectangle F G We are given that T M and 2 3. T M, by the alternate interior angles converse. click to reveal So, TM is a parallelogram because click to one reveal pair of opposite sides is parallel and congruent. J H Slide 305 / 343 Proofs Given: FGHJ is a parallelogram, F is a right angle Prove: FGHJ is a rectangle F G Slide 306 / 343 Proofs Given: OL is a quadrilateral, m O = 140, m = 40, m L = 60 Prove: OL is a trapezoid O 140 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 4) TM is a rectangle 1) Given 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. 60 40 L
Slide 307 / 343 Slide 308 / 343 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 140 o O 40 o L Try this... Given: ΔF ΔF Prove: F F Proofs 4) L and are not supplementary 4) efinition of Supplementary ngles 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) Slide 309 / 343 Slide 310 / 343 Quadrilateral oordinate Proofs oordinate Proofs 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. Return to the Table of ontents Slide 311 / 343 oordinate Proofs Slide 312 / 343 oordinate Proofs Given: PQRS is a quadrilateral Prove: PQRS is a kite y 10 kite has one unique property. The adjacent sides are congruent. P(-1,6) S(-4,3) Q(2,3) -10-8 -6-4 -2 0-2 8 P(-1,6) 6 4 S(-4,3) Q(2,3) 2 R(-1,-2) -4-6 -8-10 2 4 6 8 10 x R(-1,-2) PS = (6-3) 2 + (-1 - (-4)) 2 PQ = (3-6) 2 + (2 - (-1)) 2 = 3 2 + 3 2 = (-3) 2 + 3 2 = 9 + 9 = 9 + 9 = 18 = 18 = 4.24 = 4.24 click click
P(-1,6) Slide 313 / 343 oordinate Proofs S(-4,3) Q(2,3) R(-1,-2) RS = (3 - (-2)) 2 +(-4 - (-1)) 2 RQ = (-2-3) 2 + (-1-2) 2 = 5 2 + (-3) 2 = (-5) 2 + (-3) 2 = 25 + 9 = 25 + 9 = 34 = 34 = 5.83 = 5.83 click click ecause PS = PQ and RS = RQ, PQRS is a kite. click Given: JKLM is a parallelogram Prove: JKLM is a square Slide 314 / 343 oordinate Proofs M(-3, 0) y 10-10 -8-6 -4-2 0-2 8 6 4 2-4 -6-8 -10 J(1, 3) 2 4 6 8 10 L(0, -4) K(4, -1) x oordinate Proofs Slide 315 / 343 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 = 3 2 + 4 2 = (-4) 2 + 3 2 = 9 + 16 = 9 + 16 = 25 = 25 = 5 = 5 click click K(4, -1) m JM = 3-0 = 3 1- (-3) 4 Slide 316 / 343 M(-3, 0) J(1, 3) L(0, -4) K(4, -1) Now, let's prove that the adjacent sides are perpendicular. click MJ JK and JM JK What else do you know? click m JK = -1-3 = -4 4-1 3 MJ LK and JK LM (Opposite sides are congruent) MJ LM and JK LK (Perpendicular Transversal ) Therefore, JKLM is a square click oordinate Proofs click Try this... Slide 317 / 343 oordinate Proofs Slide 318 / 343 Quadrilateral oordinate Proofs Given: PQRS is a trapezoid Prove: LM is the midsegment y 10-10 -8-6 -4-2 0-2 8 6 4 P(2, 2) 2 Q(5, 1) L(1, 0) S(0, -2) -4-6 2 4 6 8 10 x M(7, -2) R(9, -5) 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. -8-10
Given: FGH is a parallelogram Prove: FGH is a rectangle Slide 319 / 343 Quadrilateral oordinate Proofs y Slide 320 / 343 Quadrilateral oordinate Proofs (q, r) F(p, r) y x (q, r) F(p, r) H(q, u) G(p, u) x Since FGH is a parallelogram, we know the opposite sides are parallel and congruent. Since we need to prove that FGH is a rectangle, we can either prove that adjacent sides are perpendicular or diagonals are congruent. H(q, u) G(p, u) Let's choose the 1st option: proving the adjacent sides are perpendicular. continued... Slide 321 / 343 Quadrilateral oordinate Proofs (q, r) F(p, r) y Slide 322 / 343 Quadrilateral oordinate Proofs (q, r) F(p, r) y x x H(q, u) G(p, u) H(q, u) G(p, u) m F = r - r = 0 = 0 p - q p - q click m FG = r - u = r - u = undef. p - p 0 F FG and both pairs of opposite sides are What else do you know? click F GH and FG H (Opposite sides are congruent) F H and FG GH (Perpendicular Transversal ) Therefore, FGH is a rectangle click click Since FGH is a parallelogram, we know the opposite sides are parallel and congruent. Since we need to prove that FGH 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 Slide 323 / 343 Slide 324 / 343 Quadrilateral oordinate Proofs (q, r) F(p, r) H(q, u) y G(p, u) x Quadrilateral oordinate Proofs Given: JKLM is a parallelogram Prove: JL and KM bisect each other y K (0, 2b) J (a, 2b) U G = (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 L(-a, 0) M(0, 0) x click click We proved that G HF (iagonals are congruent) We also know that F GH and FG H (Opp. sides are congruent) Therefore, FGH is a rectangle click
Slide 325 / 343 Slide 326 / 343 Quadrilateral oordinate Proofs U y K (0, 2b) J (a, 2b) Quadrilateral oordinate Proofs U y K (0, 2b) J (a, 2b) L(-a, 0) M(0, 0) x 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? Midpoint of KM: U U(0, b) click 0 + 0 2b + 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 Slide 327 / 343 137 If Quadrilateral STUV is a rectangle, what would be the coordinates for point U? U (e, d) U (d + e, e) U (d, e) y V(0, e) U(?,?) Slide 328 / 343 138 If Quadrilateral OPQR is a parallelogram, what would be the coordinates for point Q? Q (b, a) Q (b + c, a) Q (b + c, a + c) P(0, a) y Q(?,?) U (d, e - d) S(0, 0) T(d, 0) x Q (b, a + c) O(0, 0) R(b, c) x Slide 329 / 343 Slide 330 / 343 139 If Quadrilateral is an isosceles trapezoid, what would be the coordinates for point? (-2f, 2h) (-2g, 2h) (-2g, -2h) (2f, 2h) (?,?) y (2g, 2h) 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". (?,?) y (2g, 2h) x (-2f, 0) (2f, 0) x (-2f, 0) (2f, 0)
Slide 331 / 343 PR Released Questions Slide 332 / 343 The figure shows parallelogram with = 16. 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! Part 16 not drawn to scale Return to the Table of ontents 141 Let = x 2-48 and let = 2x. What are the lengths of and? Justify your answer. PR Released Question - P - alculator Section - 7 Slide 333 / 343 Slide 334 / 343 The figure shows parallelogram with = 16. Students type their answers here 16 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 Slide 335 / 343 The figure shows parallelogram PQRS on a coordinate plane. iagonals SQ and PR intersect at point T. Part Students type their answers here y P(2b, 2c) Q(?,?) S(0, 0) R(2a, 0) 143 Find the coordinates of point Q in terms of a, b, and c. nter only your answer. PR Released Question - P - alculator Section - 8 T Slide 336 / 343 x 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. Which formula(s) could be used to answer this question? istance Formula Slope Midpoint Formula and only, and Part ontinued 145 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 PR Released Question - P - alculator Section - 8
Slide 337 / 343 Slide 338 / 343 One method that can be used to prove that the diagonals of a parallelogram bisect each other is shown in the given partial proof. S P T R Q? 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. Given: Quadrilateral PQRS is a parallelogram. Prove: PT = RT, ST = QT PR Released Question - OY - alculator Section - 19 Slide 339 / 343 Slide 340 / 343 Part 147 Which statement is justified by the reason for step 4 in the proof? PQ RS PQ SP PT TR SQ PR Part 148 Which reason justifies the statement for step 5 in the proof? side-side-side triangle congruence side-angle-side triangle congruence angle-side-angle triangle congruence angle-angle-side triangle congruence PR Released Question - OY - alculator Section - 19 Slide 341 / 343 PR Released Question - OY - alculator Section - 19 Slide 342 / 343 Part 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. PR Released Question - OY - alculator Section - 19