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

Size: px
Start display at page:

Transcription

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

2 Slide 4 / 189 ngles of Polygons Return to the Table of ontents Polygon Slide 5 / 189 polygon is a closed figure made of line segments connected end to end. Since it is made of line segments, there can be no curves. lso, it has only one inside regioin, so no two segments can cross each other. an you explain why the figure below is not a polygon? click to reveal is not a segment (it has a curve). There are two inside regions. Types of Polygons Slide 6 / 189 Polygons are named by their number of sides. Number of Sides Type of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon gon 12 dodecagon n n-gon

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

28 39 Which theorem proves the quadrilateral is a parallelogram? Slide 82 / 189 E F The opposite angle are congruent. The opposite sides are congruent. n angle in the quadrilateral is supplementary to its consecutive angles. The diagonals bisect each other. One pair of opposite sides are congruent and parallel. Not enough information. 40 Which theorem proves the quadrilateral is a parallelogram? Slide 83 / 189 The opposite angle are congruent. 6 The opposite sides are congruent. 3(6-4) 6 E F n angle in the quadrilateral is supplementary to its consecutive angles. The diagonals bisect each other. One pair of opposite sides are congruent and parallel. Not enough information. Slide 84 / 189 onstructing Parallelograms Return to the Table of ontents

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

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

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

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

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

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

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

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

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

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

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

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

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

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

43 trapezoid Slide 127 / 189 trapezoid is a quadrilateral with one pair of parallel sides. base leg base angles leg base The parallel sides are called bases. The nonparallel sides are called legs. trapezoid also has two pairs of base angles. isosceles trapezoid Slide 128 / 189 n isosceles trapezoid is a trapezoid with congruent legs. Theorem Q14 Slide 129 / 189 If a trapezoid is isosceles, then each pair of base angles are congruent. is an isosceles trapezoid. < < and < <.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

59 kite has one unique property. The adjacent sides are congruent. Slide 175 / 189 P(-1,6) (-4,3) S Q(2,3) # # # R(-1,-2) SP = (6-3) + (-1-(-4)) PQ = (3-6) + (2-(-1)) = = (-3) # # # = = = # 18 = 18 # = 4.24 = 4.24 P(-1,6) Slide 176 / 189 (-4,3) S Q(2,3) # # # R(-1,-2) SR = (3-(-2)) 2 +(-4-(-1)) 2 RQ = (-2-3) 2 + (-1-2) 2 = (-3) 2 = (-5) 2 + (-3) 2 = = = 34 # = 34 = 5.83 = 5.83 # # # # So, because SP=PQ and SR=RQ, PQRS is a kite. Given: JKLM is a parallelogram Prove: JKLM is a square Slide 177 / J (1,3) 2-10 (-3,0) M K 6(4,-1) L (0,-4) -8-10

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

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

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

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

Slide 1 / 343 Slide 2 / 343 Geometry Quadrilaterals 2015-10-27 www.njctl.org Table of ontents Polygons Properties of Parallelograms Proving Quadrilaterals are Parallelograms Rhombi, Rectangles and Squares

### Slide 1 / 343 Slide 2 / 343

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

### Geometry. Kites Families of Quadrilaterals Coordinate Proofs Proofs. Click on a topic to

Geometry Angles of Polygons Properties of Parallelograms Proving Quadrilaterals are Parallelograms Constructing Parallelograms Rhombi, Rectangles and Squares Kites Families of Quadrilaterals Coordinate

### Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review

Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon -

### 6.1 What is a Polygon?

6. What is a Polygon? Unit 6 Polygons and Quadrilaterals Regular polygon - Polygon Names: # sides Name 3 4 raw hexagon RPTOE 5 6 7 8 9 0 Name the vertices: Name the sides: Name the diagonals containing

### Maintaining Mathematical Proficiency

Name ate hapter 7 Maintaining Mathematical Proficiency Solve the equation by interpreting the expression in parentheses as a single quantity. 1. 5( 10 x) = 100 2. 6( x + 8) 12 = 48 3. ( x) ( x) 32 + 42

### U4 Polygon Notes January 11, 2017 Unit 4: Polygons

Unit 4: Polygons 180 Complimentary Opposite exterior Practice Makes Perfect! Example: Example: Practice Makes Perfect! Def: Midsegment of a triangle - a segment that connects the midpoints of two sides

### A closed plane figure with at least 3 sides The sides intersect only at their endpoints. Polygon ABCDEF

A closed plane figure with at least 3 sides The sides intersect only at their endpoints B C A D F E Polygon ABCDEF The diagonals of a polygon are the segments that connects one vertex of a polygon to another

### Geometry/Trigonometry Unit 5: Polygon Notes Period:

Geometry/Trigonometry Unit 5: Polygon Notes Name: Date: Period: # (1) Page 270 271 #8 14 Even, #15 20, #27-32 (2) Page 276 1 10, #11 25 Odd (3) Page 276 277 #12 30 Even (4) Page 283 #1-14 All (5) Page

### Dates, assignments, and quizzes subject to change without advance notice. Monday Tuesday Block Day Friday & 6-3.

Name: Period P UNIT 11: QURILTERLS N POLYONS I can define, identify and illustrate the following terms: Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid Isosceles trapezoid Kite oncave polygon

### Polygon notes

1.6-6.1 Polygon notes Polygon: Examples: Nonexamples: Named by the letters of the vertices written in order polygon will be: oncave - Or: onvex- Regular Polygon: 1.6-6.1 Polygon notes iagonal is a segment

### 1. Revision Description Reflect and Review Teasers Answers Recall of basics of triangles, polygons etc. Review Following are few examples of polygons:

1. Revision Recall of basics of triangles, polygons etc. The minimum number of line segments required to form a polygon is 3. 1) Name the polygon formed with 4 line segments of equal length. 1) Square

Any questions about the material so far? About the exercises? Here is a question for you. In the diagram on the board, DE is parallel to AC, DB = 4, AB = 9 and BE = 8. What is the length EC? Polygons Definitions:

### Geometry Unit 5 - Notes Polygons

Geometry Unit 5 - Notes Polygons Syllabus Objective: 5.1 - The student will differentiate among polygons by their attributes. Review terms: 1) segment 2) vertex 3) collinear 4) intersect Polygon- a plane

### Name: Period 1/4/11 1/20/11 GH

Name: Period 1/4/11 1/20/11 UNIT 10: QURILTERLS N POLYONS I can define, identify and illustrate the following terms: Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid Isosceles trapezoid Kite

### Unit 6 Polygons and Quadrilaterals

6.1 What is a Polygon? A closed plane figure formed by segments that intersect only at their endpoints Regular Polygon- a polygon that is both equiangular and equilateral Unit 6 Polygons and Quadrilaterals

### 8.1 Find Angle Measures in Polygons

VOCABULARY 8.1 Find Angle Measures in Polygons DIAGONAL Review: EQUILATERAL EQUIANGULAR REGULAR CLASSIFYING POLYGONS Polygon Interior Angle Theorem: The sum of the measures of the interior angles of a

### Polygons are named by the number of sides they have:

Unit 5 Lesson 1 Polygons and Angle Measures I. What is a polygon? (Page 322) A polygon is a figure that meets the following conditions: It is formed by or more segments called, such that no two sides with

### Polygon. Note: Each segment is called a side. Each endpoint is called a vertex.

Polygons Polygon A closed plane figure formed by 3 or more segments. Each segment intersects exactly 2 other segments at their endpoints. No 2 segments with a common endpoint are collinear. Note: Each

### Secondary Math II Honors. Unit 4 Notes. Polygons. Name: Per:

Secondary Math II Honors Unit 4 Notes Polygons Name: Per: Day 1: Interior and Exterior Angles of a Polygon Unit 4 Notes / Secondary 2 Honors Vocabulary: Polygon: Regular Polygon: Example(s): Discover the

### 14. How many sides does a regular polygon have, if the measure of an interior angle is 60?

State whether the figure is a polygon; if it is a polygon, state whether the polygon is convex or concave. HINT: No curves, no gaps, and no overlaps! 1. 2. 3. 4. Find the indicated measures of the polygon.

### B. Algebraic Properties Reflexive, symmetric, transitive, substitution, addition, subtraction, multiplication, division

. efinitions 1) cute angle ) cute triangle 3) djacent angles 4) lternate exterior angles 5) lternate interior angles 6) ltitude of a triangle 7) ngle ) ngle bisector of a triangle 9) ngles bisector 10)

### Vocabulary. Term Page Definition Clarifying Example base angle of a trapezoid. base of a trapezoid. concave (polygon) convex (polygon)

HPTER 6 Vocabulary The table contains important vocabulary terms from hapter 6. s you work through the chapter, fill in the page number, definition, and a clarifying example. Term Page efinition larifying

### 6.1: Date: Geometry. Polygon Number of Triangles Sum of Interior Angles

6.1: Date: Geometry Polygon Number of Triangles Sum of Interior Angles Triangle: # of sides: # of triangles: Quadrilateral: # of sides: # of triangles: Pentagon: # of sides: # of triangles: Hexagon: #

### Name Date Class. The Polygon Angle Sum Theorem states that the sum of the interior angle measures of a convex polygon with n sides is (n 2)180.

Name Date Class 6-1 Properties and Attributes of Polygons continued The Polygon Angle Sum Theorem states that the sum of the interior angle measures of a convex polygon with n sides is (n 2)180. Convex

### Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms

Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms Getting Ready: How will you know whether or not a figure is a parallelogram? By definition, a quadrilateral is a parallelogram if it has

### Polygon Interior Angles

Polygons can be named by the number of sides. A regular polygon has All other polygons are irregular. A concave polygon has All other polygons are convex, with all vertices facing outwards. Name each polygon

### Math Polygons

Math 310 9.2 Polygons Curve & Connected Idea The idea of a curve is something you could draw on paper without lifting your pencil. The idea of connected is that a set can t be split into two disjoint sets.

### Rectilinear Figures. Introduction

2 Rectilinear Figures Introduction If we put the sharp tip of a pencil on a sheet of paper and move from one point to the other, without lifting the pencil, then the shapes so formed are called plane curves.

### Unit 2: Triangles and Quadrilaterals Lesson 2.1 Apply Triangle Sum Properties Lesson 4.1 from textbook

Unit 2: Triangles and Quadrilaterals Lesson 2.1 pply Triangle Sum Properties Lesson 4.1 from textbook Objectives Classify angles by their sides as equilateral, isosceles, or scalene. Classify triangles

### Unit 3: Triangles and Polygons

Unit 3: Triangles and Polygons Background for Standard G.CO.9: Prove theorems about triangles. Objective: By the end of class, I should Example 1: Trapezoid on the coordinate plane below has the following

### 6-1 Study Guide and Intervention Angles of Polygons

6-1 Study Guide and Intervention Angles of Polygons Polygon Interior Angles Sum The segments that connect the nonconsecutive vertices of a polygon are called diagonals. Drawing all of the diagonals from

### Properties of Rhombuses, Rectangles, and Squares

6- Properties of Rhombuses, Rectangles, and Squares ontent Standards G.O. Prove theorems about parallelograms... rectangles are parallelograms with congruent diagonals. lso G.SRT.5 Objectives To define

### Examples: Identify the following as equilateral, equiangular or regular. Using Variables: S = 180(n 2)

Ch. 6 Notes 6.1: Polygon Angle-Sum Theorems Examples: Identify the following as equilateral, equiangular or regular. 1) 2) 3) S = 180(n 2) Using Variables: and Examples: Find the sum of the interior angles

### Ch. 7 Test. 1. Find the sum of the measures of the interior angles of the given figure.

h. 7 Test 1. Find the sum of the measures of the interior angles of the given figure. a. 900 c. 70 b. 10 d. 1080. The sum of the measures of the interior angles of a polygon is 10. lassify the polygon

8 Quadrilaterals 8. Find Angle Measures in Polygons 8. Use Properties of Parallelograms 8.3 Show that a Quadrilateral is a Parallelogram 8.4 Properties of Rhombuses, Rectangles, and Squares 8.5 Use Properties

### CC Geometry H Do Now: Complete the following: Quadrilaterals

im #26: What are the properties of parallelograms? Geometry H o Now: omplete the following: Quadrilaterals Kite iagonals are perpendicular One pair of opposite angles is congruent Two distinct pairs of

### 8 sides 17 sides. x = 72

GEOMETRY Chapter 7 Review Quadrilaterals Name: Hour: Date: SECTION 1: State whether each polygon is equilateral, equiangular, or regular. 1) 2) 3) equilateral regular equiangular SECTION 2: Calculate the

### 6-1 Properties and Attributes of Polygons

6-1 Properties and Attributes of Polygons Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up 1. A? is a three-sided polygon. triangle 2. A? is a four-sided polygon. quadrilateral Evaluate each expression

### CHAPTER 6. SECTION 6-1 Angles of Polygons POLYGON INTERIOR ANGLE SUM

HPTER 6 Quadrilaterals SETION 6-1 ngles of Polygons POLYGON INTERIOR NGLE SUM iagonal - a line segment that connects two nonconsecutive vertices. Polygon interior angle sum theorem (6.1) - The sum of the

### Geometry Ch 7 Quadrilaterals January 06, 2016

Theorem 17: Equal corresponding angles mean that lines are parallel. Corollary 1: Equal alternate interior angles mean that lines are parallel. Corollary 2: Supplementary interior angles on the same side

### RPDP Geometry Seminar Quarter 1 Handouts

RPDP Geometry Seminar Quarter 1 Handouts Geometry lassifying Triangles: State Standard 4.12.7 4.12.9 Syllabus Objectives: 5.11, 6.1, 6.4, 6.5 enchmarks: 2 nd Quarter - November Find the distance between:

### Definition: Convex polygon A convex polygon is a polygon in which the measure of each interior angle is less than 180º.

Definition: Convex polygon A convex polygon is a polygon in which the measure of each interior angle is less than 180º. Definition: Convex polygon A convex polygon is a polygon in which the measure of

### Geometry Lesson 1 Introduction to Geometry (Grades 9-12) Instruction 1-5 Definitions of Figures

efinitions of igures Quadrilaterals Quadrilaterals are closed four-sided figures. The interior angles of a quadrilateral always total 360. Quadrilaterals classified in two groups: Trapeziums and Trapezoids.

HTE 8 UILTEL In this chapter we address three ig IE: ) Using angle relationships in polygons. ) Using properties of parallelograms. 3) lassifying quadrilaterals by the properties. ection: Essential uestion

### Geometry: A Complete Course

Geometry: omplete ourse with Trigonometry) Module Progress Tests Written by: Larry. ollins Geometry: omplete ourse with Trigonometry) Module - Progress Tests opyright 2014 by VideotextInteractive Send

### Pre-AICE 2: Unit 5 Exam - Study Guide

Pre-AICE 2: Unit 5 Exam - Study Guide 1 Find the value of x. (The figure may not be drawn to scale.) A. 74 B. 108 C. 49 D. 51 2 Find the measure of an interior angle and an exterior angle of a regular

### Ch 5 Polygon Notebook Key

hapter 5: iscovering and Proving Polygon Properties Lesson 5.1 Polygon Sum onjecture & Lesson 5.2 xterior ngles of a Polygon Warm up: efinition: xterior angle is an angle that forms a linear pair with

### STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY. 3 rd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

### Problems #1. A convex pentagon has interior angles with measures (5x 12), (2x + 100), (4x + 16), (6x + 15), and (3x + 41). Find x.

1 Pre-AP Geometry Chapter 10 Test Review Standards/Goals: G.CO.11/ C.1.i.: I can use properties of special quadrilaterals in a proof. D.2.g.: I can identify and classify quadrilaterals, including parallelograms,

### STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for

### Assumption High School. Bell Work. Academic institution promoting High expectations resulting in Successful students

Bell Work Geometry 2016 2017 Day 36 Topic: Chapter 4 Congruent Figures Chapter 6 Polygons & Quads Chapter 4 Big Ideas Visualization Visualization can help you connect properties of real objects with two-dimensional

### Angle Unit Definitions

ngle Unit Definitions Name lock Date Term Definition Notes Sketch D djacent ngles Two coplanar angles with a coon side, a coon vertex, and no coon interior points. Must be named with 3 letters OR numbers

### Geometry Review for Test 3 January 13, 2016

Homework #7 Due Thursday, 14 January Ch 7 Review, pp. 292 295 #1 53 Test #3 Thurs, 14 Jan Emphasis on Ch 7 except Midsegment Theorem, plus review Betweenness of Rays Theorem Whole is Greater than Part

### Geometry Basics of Geometry Precise Definitions Unit CO.1 OBJECTIVE #: G.CO.1

OBJECTIVE #: G.CO.1 OBJECTIVE Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance

### UNIT 5 LEARNING TARGETS. HP/4 Highly Proficient WOW, Excellent. PR/3 Proficient Yes, Satisfactory. DP/1 Developing Proficiency Not yet, Insufficient

Geometry 1-2 Properties of Polygons My academic goal for this unit is... UNIT 5 Name: Teacher: Per: heck for Understanding Key: Understanding at start of the unit Understanding after practice Understanding

### 1/25 Warm Up Find the value of the indicated measure

1/25 Warm Up Find the value of the indicated measure. 1. 2. 3. 4. Lesson 7.1(2 Days) Angles of Polygons Essential Question: What is the sum of the measures of the interior angles of a polygon? What you

### Postulates, Theorems, and Corollaries. Chapter 1

Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. Post. 1-1-3 If two points lie in a

### Lesson 9: Coordinate Proof - Quadrilaterals Learning Targets

Lesson 9: Coordinate Proof - Quadrilaterals Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of the way along each median

### Geometry Reasons for Proofs Chapter 1

Geometry Reasons for Proofs Chapter 1 Lesson 1.1 Defined Terms: Undefined Terms: Point: Line: Plane: Space: Postulate 1: Postulate : terms that are explained using undefined and/or other defined terms

### Period: Date Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means

: Analytic Proofs of Theorems Previously Proved by Synthetic Means Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of

### Lesson 7.1. Angles of Polygons

Lesson 7.1 Angles of Polygons Essential Question: How can I find the sum of the measures of the interior angles of a polygon? Polygon A plane figure made of three or more segments (sides). Each side intersects

### Geometry Chapter 8 Test Review

Geometry Chapter 8 Test Review Short Answer 1. Find the sum of the measures of the interior angles of the indicated convex polygon. Decagon 2. Find the sum of the measures of the interior angles of the

### A calculator, scrap paper, and patty paper may be used. A compass and straightedge is required.

The Geometry and Honors Geometry Semester examination will have the following types of questions: Selected Response Student Produced Response (Grid-in) Short nswer calculator, scrap paper, and patty paper

Name: Quadrilaterals Polygons Basics Date: Objectives: SWBAT identify, name and describe polygons. SWBAT use the sum of the measures of the interior angles of a quadrilateral. A. The basics on POLYGONS

### ( ) A calculator may be used on the exam. The formulas below will be provided in the examination booklet.

The Geometry and Honors Geometry Semester examination will have the following types of questions: Selected Response Student Produced Response (Grid-in) Short nswer calculator may be used on the exam. The

### Unit 10 Study Guide: Plane Figures

Unit 10 Study Guide: Plane Figures *Be sure to watch all videos within each lesson* You can find geometric shapes in art. Whether determining the amount of leading or the amount of glass needed for a piece

### Geometry Third Quarter Study Guide

Geometry Third Quarter Study Guide 1. Write the if-then form, the converse, the inverse and the contrapositive for the given statement: All right angles are congruent. 2. Find the measures of angles A,

### Geometry Unit 6 Note Sheets Date Name of Lesson. 6.2 Parallelograms. 6.3 Tests for Parallelograms. 6.4 Rectangles. 6.5 Rhombi and Squares

Date Name of Lesson 6.2 Parallelograms 6.3 Tests for Parallelograms 6.4 Rectangles 6.5 Rhombi and Squares 6.6 Trapezoids and Kites 1 Quadrilaterals Properties Property Parallelogram Rectangle Rhombus Square

### 1.6 Classifying Polygons

www.ck12.org Chapter 1. Basics of Geometry 1.6 Classifying Polygons Learning Objectives Define triangle and polygon. Classify triangles by their sides and angles. Understand the difference between convex

Chapter 8 Quadrilaterals 8.1 Find Angle Measures in Polygons Objective: Find angle measures in polygons. Essential Question: How do you find a missing angle measure in a convex polygon? 1) Any convex polygon.

### Geometry Chapter 5 Review Sheet

Geometry hapter 5 Review Sheet Name: 1. List the 6 properties of the parallelogram. 2. List the 5 ways to prove that a quadrilateral is a parallelogram. 3. Name two properties of the rectangle that are

### INTUITIVE GEOMETRY SEMESTER 1 EXAM ITEM SPECIFICATION SHEET & KEY

INTUITIVE GEOMETRY SEMESTER EXM ITEM SPEIFITION SHEET & KEY onstructed Response # Objective Syllabus Objective NV State Standard istinguish among the properties of various quadrilaterals. 7. 4.. lassify

### Angles of Polygons Concept Summary

Vocabulary and oncept heck diagonal (p. 404) isosceles trapezoid (p. 439) kite (p. 438) median (p. 440) parallelogram (p. 411) rectangle (p. 424) rhombus (p. 431) square (p. 432) trapezoid (p. 439) complete

### GEOMETRY COORDINATE GEOMETRY Proofs

GEOMETRY COORDINATE GEOMETRY Proofs Name Period 1 Coordinate Proof Help Page Formulas Slope: Distance: To show segments are congruent: Use the distance formula to find the length of the sides and show

### equilateral regular irregular

polygon three polygon side common sides vertex nonconsecutive diagonal triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon dodecagon n-gon equiangular concave exterior equilateral

### Lines Plane A flat surface that has no thickness and extends forever.

Lines Plane A flat surface that has no thickness and extends forever. Point an exact location Line a straight path that has no thickness and extends forever in opposite directions Ray Part of a line that

### Angle Unit Definition Packet

ngle Unit Definition Packet Name lock Date Term Definition Notes Sketch djacent ngles Two angles with a coon, a coon you normay name and, and no coon interior points. 3 4 3 and 4 Vertical ngles Two angles

### Formal Geometry UNIT 6 - Quadrilaterals

Formal Geometry UNIT 6 - Quadrilaterals 14-Jan 15-Jan 16-Jan 17-Jan 18-Jan Day 1 Day Day 4 Kites and Day 3 Polygon Basics Trapezoids Proving Parallelograms Day 5 Homefun: Parallelograms Pg 48 431 #1 19,

### MATH 113 Section 8.2: Two-Dimensional Figures

MATH 113 Section 8.2: Two-Dimensional Figures Prof. Jonathan Duncan Walla Walla University Winter Quarter, 2008 Outline 1 Classifying Two-Dimensional Shapes 2 Polygons Triangles Quadrilaterals 3 Other

### Name Date Class. 6. In JKLM, what is the value of m K? A 15 B 57 A RS QT C QR ST

Name Date Class CHAPTER 6 Chapter Review #1 Form B Circle the best answer. 1. Which best describes the figure? 6. In JKLM, what is the value of m K? A regular convex heptagon B irregular convex heptagon

### 6-1. The Polygon Angle-Sum Theorems. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

6-1 The Polygon ngle-sum Theorems Vocabulary Review 1. Underline the correct word to complete the sentence. In a convex polygon, no point on the lines containing the sides of the polygon is in the interior

### Answer Key. 1.1 The Three Dimensions. Chapter 1 Basics of Geometry. CK-12 Geometry Honors Concepts 1. Answers

1.1 The Three Dimensions 1. Possible answer: You need only one number to describe the location of a point on a line. You need two numbers to describe the location of a point on a plane. 2. vary. Possible

### HIGH SCHOOL. Geometry. Soowook Lee

HIGH SHOOL Geometry Soowook Lee hapter 4 Quadrilaterals This chapter will cover basic quadrilaterals, including parallelograms, trapezoids, rhombi, rectangles, squares, kites, and cyclic quadrilaterals.

### arallelogram: quadrilateral with two pairs of sides. sides are parallel Opposite sides are Opposite angles are onsecutive angles are iagonals each oth

olygon: shape formed by three or more segments (never curved) called. Each side is attached to one other side at each endpoint. The sides only intersect at their. The endpoints of the sides (the corners

### Capter 6 Review Sheet. 1. Given the diagram, what postulate or theorem would be used to prove that AP = CP?

apter 6 Review Sheet Name: ate: 1. Given the diagram, what postulate or theorem would be used to prove that P = P? 4.. S. SSS.. SS 2. Given the diagram, what postulate or theorem would be used to prove

### 22. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

Chapter 4 Quadrilaterals 4.1 Properties of a Parallelogram Definitions 22. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. 23. An altitude of a parallelogram is the

### Squares and Rectangles

LESSON.1 Assignment Name Date Squares and Rectangles Properties of Squares and Rectangles 1. In quadrilateral VWXY, segments VX and WY bisect each other, and are perpendicular and congruent. Is this enough

### Cambridge Essentials Mathematics Core 9 GM1.1 Answers. 1 a

GM1.1 Answers 1 a b 2 Shape Name Regular Irregular Convex Concave A Decagon B Octagon C Pentagon D Quadrilateral E Heptagon F Hexagon G Quadrilateral H Triangle I Triangle J Hexagon Original Material Cambridge

### Points, lines, angles

Points, lines, angles Point Line Line segment Parallel Lines Perpendicular lines Vertex Angle Full Turn An exact location. A point does not have any parts. A straight length that extends infinitely in

### theorems & postulates & stuff (mr. ko)

theorems & postulates & stuff (mr. ko) postulates 1 ruler postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of

### Geometry Honors. Midterm Review

eometry Honors Midterm Review lass: ate: I: eometry Honors Midterm Review Multiple hoice Identify the choice that best completes the statement or answers the question. 1 What is the contrapositive of the

### MAT104: Fundamentals of Mathematics II Introductory Geometry Terminology Summary. Section 11-1: Basic Notions

MAT104: Fundamentals of Mathematics II Introductory Geometry Terminology Summary Section 11-1: Basic Notions Undefined Terms: Point; Line; Plane Collinear Points: points that lie on the same line Between[-ness]:

### Geometry First Semester Practice Final (cont)

49. Determine the width of the river, AE, if A. 6.6 yards. 10 yards C. 12.8 yards D. 15 yards Geometry First Semester Practice Final (cont) 50. In the similar triangles shown below, what is the value of

### Proving Theorems about Lines and Angles

Proving Theorems about Lines and Angles Angle Vocabulary Complementary- two angles whose sum is 90 degrees. Supplementary- two angles whose sum is 180 degrees. Congruent angles- two or more angles with

### Ch 1 Note Sheet L2 Key.doc 1.1 Building Blocks of Geometry

1.1 uilding locks of Geometry Read page 28. It s all about vocabulary and notation! To name something, trace the figure as you say the name, if you trace the figure you were trying to describe you re correct!