This Packet Belongs to Student Name Topic 4

This Packet Belongs to ____________ (Student Name) Topic 4: Quadrilaterals and Coordinate Proof Unit 4 – Quadrilaterals and Coordinate Proof Module 9: Properties of Quadrilaterals 9. 1 9. 2 9. 3 9. 4 Properties of Parallelograms Conditions of Parallelograms Properties of Rectangles, Rhombi, and Squares Conditions of Rectangles, Rhombi, and Squares Module 10: Coordinate Proof Using Slope and Distance 10. 1 Slope and Parallel Lines 10. 2 Slope and Perpendicular Lines 10. 3 Coordinate Proof Using Distance with Segments and Triangles 10. 4 Coordinate Proof Using Distance with Quadrilaterals 10. 5 Perimeter and Area on the Coordinate Plane Please follow along with notes. At the end of quarter 2 there will be a BINDER CHECK to check Topic 1, 2, 3, 4, etc… 1

Module 9 Properties of Quadrilaterals Part 1: Parallelograms 2 2

Definition • A parallelogram is a quadrilateral whose opposite sides are parallel. • Its symbol is a small figure: 3 3

Naming a Parallelogram • A parallelogram is named using all four vertices. • You can start from any one vertex, but you must continue in a clockwise or counterclockwise direction. • For example, this can be either ABCD or ADCB. 4

Basic Properties • There are four basic properties of all parallelograms. – Opposite Sides – Opposite Angles – Consecutive Angles – Diagonals • These properties have to do with the angles, the sides and the diagonals. 5 5

Opposite Sides Theorem Opposite sides of a parallelogram are congruent. • That means that . • So, if AB = 7, then _____ = 7? CD 6 6

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Opposite Angles One pair of opposite angles is A and C. The other pair is B and D. Theorem Opposite angles of a parallelogram are congruent. • Complete: If m A = 75 and m B = 105 , then m C = ______ and m D = ______. 8 8

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Consecutive Angles • Each angle is consecutive to two other angles. A is consecutive with B and D. 10 10

Consecutive Angles in Parallelograms Theorem Consecutive angles in a parallelogram are supplementary. • Therefore, m A + m B = 180 and m A + m D = 180. • If m<C = 46 , then m B = _____? Consecutive INTERIOR Angles are Supplementary! 11

Diagonals • Diagonals are segments that join nonconsecutive vertices. • For example, in this diagram, the only two diagonals are . 12 12

Diagonal Property When the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal. • So, P is the midpoint of . • Therefore, they bisect each other; so and . • But, the diagonals are not congruent! 13 13

Diagonal Property Theorem The diagonals of a parallelogram bisect each other. 14 14

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Parallelogram Summary • By its definition, opposite sides are parallel. Other properties (theorems): • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • The diagonals bisect each other. 16 16

Examples • 1. Draw HKLP. PL • 2. Complete: HK = _______ and KL HP = ____. P • 3. m<K = m<______. P or <K • 4. m<L + m<______ = 180. 115 • 5. If m<P = 65 , then m<H = ____, 65 115 m<K = ______ and m<L =______. 17 17

Examples (cont’d) • 6. Draw in the diagonals. They intersect at M. 5 • 7. Complete: If HM = 5, then ML = ____. 14 • 8. If KM = 7, then KP = ____. 7. 5 • 9. If HL = 15, then ML = ____. 36 • 10. If m<HPK = 36 , then m<PKL = _____. 18 18

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Part 2 Tests for Parallelograms 21 21

Review: Properties of Parallelograms • • • Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other. 22 22

How can you tell if a quadrilateral is a parallelogram? • Defn: A quadrilateral is a parallelogram iff opposite sides are parallel. • Property If a quadrilateral is a parallelogram, then opposite sides are parallel. • Test If opposite sides of a quadrilateral are parallel, then it is a parallelogram. 23 23

Proving Quadrilaterals as Parallelograms Theorem 1: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. H E G F Theorem 2: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. 24

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Proving Quadrilaterals as Parallelograms (part 2) Theorem 3: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. G H then Quad. EFGH is a parallelogram. M Theorem 4: E F If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. then Quad. EFGH is a parallelogram. EM = GM and HM = FM 26

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5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are ||. [definition] 2. Show that both pairs of opposite sides are . 3. Show that one pair of opposite sides are both || and . 4. Show that both pairs of opposite angles are . 5. Show that the diagonals bisect each other. 28 28

Examples …… Example 1: Find the values of x and y that ensures the quadrilateral y+2 is a parallelogram. 6 x = 4 x + 8 2 y = y + 2 6 x 4 x+8 2 x = 8 y=2 2 y x=4 Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram. 2 x + 8 = 120 5 y + 120 = 180 (2 x + 8)° 5 y° 120° 2 x = 112 5 y = 60 x = 56 y = 12 29 29

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9. 1 -9. 2 Classwork PAGE 426 • GO ONLINE and complete 9. 1 -9. 2 hw. • Alternative: Honors: 9. 1: 3, 5 -6, 14, 17 -18, 23, 24 9. 2: 1, 5, 8, 11 -12, 18 -19 • Regular: 9. 1: 5 -6, 8, 17 -18 9. 2: 1, 5, 8, 12, 18 Reminders: q … 32 32

Part 3 Rectangles 33 33

Rectangles Definition: A rectangle is a quadrilateral with four right angles. Is a rectangle a parallelogram? Yes, since opposite angles are congruent. Thus a rectangle has all the properties of a parallelogram. • • • Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. 34 34

Properties of Rectangles Theorem: If a parallelogram is a rectangle, then its diagonals are congruent. Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles. A B E D C Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle. 35 35

Properties of Rectangles Parallelogram Properties: l Opposite sides are parallel. l Opposite sides are congruent. A l Opposite angles are congruent. l Consecutive angles are supplementary. l Diagonals bisect each other. Plus: D l All angles are right angles. l Diagonals are congruent. l B E C Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles 36 36

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Examples 1. If AE = 3 x +2 and BE = 29, find the value of x. x = 9 units 10. 5 units 2. If AC = 21, then BE = _______. 3. If m<1 = 4 x and m<4 = 2 x, find the value of x. x = 18 units 4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6. A m<1=50, m<3=40, m<4=80, m<5=100, m<6=40 B 1 2 3 4 D E 5 6 C 38 38

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Part 4 Rhombi and Squares 40 40

Rhombus Definition: A rhombus is a quadrilateral with four congruent sides. ≡ ≡ Is a rhombus a parallelogram? Yes, since opposite sides are congruent. Since a rhombus is a parallelogram the following are true: • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. 41 41

Rhombus Note: The four small triangles are congruent, by SSS. ≡ ≡ This means the diagonals form four angles that are congruent, and must measure 90 degrees each. So the diagonals are perpendicular. This also means the diagonals bisect each of the four angles of the rhombus So the diagonals bisect opposite angles. 42 42

Properties of a Rhombus Theorem: The diagonals of a rhombus are perpendicular. Theorem: Each diagonal of a rhombus bisects a pair of opposite angles. Note: The small triangles are RIGHT and CONGRUENT! 43 43

Your Turn: Rhombus Examples Given: ABCD is a rhombus. Complete the following. 1. 9 units If AB = 9, then AD = ______. 2. 65° If m<1 = 65, the m<2 = _____. 3. 90° m<3 = ______. 4. 100° If m<ADC = 80, the m<DAB = ______. 5. 10 If m<1 = 3 x -7 and m<2 = 2 x +3, then x = _____. 44 44

Properties of a Rhombus ≡ Since a rhombus is a parallelogram the following are true: . • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Plus: • All four sides are congruent. • Diagonals are perpendicular. • Diagonals bisect opposite angles. • Also remember: the small triangles are RIGHT and CONGRUENT! ≡ 45 45

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Square Definition: A square is a quadrilateral with four congruent angles and four congruent sides. Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals. • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Plus: • Four right angles. • Four congruent sides. • Diagonals are congruent. • Diagonals are perpendicular. • Diagonals bisect opposite angles. 52 52

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Squares – Examples…. . . Given: ABCD is a square. Complete the following. 10 units If AB = 10, then AD = _______ and DC = _______. 2. 5 units If CE = 5, then DE = _____. 3. 90° m<ABC = _____. 45° m<ACD = _____. 5. 90° m<AED = _____. 54 54

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9. 3 -9. 4 Classwork PAGE 452 • GO ONLINE and complete 9. 3 -9. 4 hw. • Alternative: Honors: 9. 3: 1 -2, 5 -8, 15 9. 4: 5 -13 odds, 22 • Regular: 9. 3: 1 -2, 5 -6, 10, 13 9. 4: 6, 9, 12, 15, 18, 22 Reminders: q Module 9 Quiz Next Class! q. MYA next week! 56 56

Module 10: Getting Ready 57 57

Definition of a Parallelogram Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B B(2, 6), C (5, 7) and D(3, 1). I need to show that both pairs of opposite sides are parallel by showing that their slopes are equal. 58 A C D

Definition of a Parallelogram Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B B(2, 6), C (5, 7) and D(3, 1). AB: m = 6 – 0 = 6 = 3 2– 0 2 CD: m = 1 – 7 = - 6 = 3 3– 5 -2 BC: m = 7 – 6 = 1 5– 2 3 AD: m = 1 – 0 = 1 3– 0 3 59 A C D ABCD is a Parallelogram by Definition

Both Pairs of Opposite Sides Congruent Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3, 1). C B I need to show that both pairs of opposite sides are congruent by using the distance formula to find their lengths. 60 A D

Both Pairs of Opposite Sides Congruent C B A D Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3, 1). AB = (2 – 0)2 + (6 – 0)2 CD = (3 – 5)2 + (1 – 7)2 = 4 + 36 = 40 AB CD BC = (5 – 2)2 + (7 – 6)2 = 9 + 1 = 10 = 4 + 36 = 40 AD = (3 – 0)2 + (1 – 0)2 = 9 + 1 = 10 ABCD is a Parallelogram because both pair of opposite sides are congruent. 61 61

One Pair of Opposite Sides Both Parallel and Congruent Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3, 1). C B I need to show that one pair of opposite sides is both parallel and congruent. ll (slope) and (distance) 62 A D

One Pair of Opposite Sides Both Parallel and Congruent B Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3, 1). BC: m = 7 – 6 = 1 5– 2 3 BC ll AD BC = (5 – 2)2 + (7 – 6)2 = 9 + 1 = 10 BC AD A C D AD: m = 1 – 0 = 1 3– 0 3 AD = (3 – 0)2 + (1 – 0)2 = 9 + 1 = 10 ABCD is a Parallelogram because one pair of opposite sides are parallel and congruent. 6363

Diagonals Bisect Each Other Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3, 1). B I need to show that each diagonal shares the midpoint SAME _____. 64 A C D

Diagonals Bisect Each Other C B Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3, 1). A The midpoint of AC is 0+5 , 0+7 2 2 5 , 7 2 2 The midpoint of BD is 2+3 , 6+1 2 2 5 , 7 2 2 ABCD is a Parallelogram because the diagonals share the same midpoint, thus bisecting each other. D 6565

Module 10: Coordinate Proof Using Slope and Distance 66 66

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Understanding Slope • Two (non-vertical) lines are parallel if and only if they have the same slope. (All vertical lines are parallel. ) 71 71

Understanding Slope • The slope of AB is: • The slope of CD is: • Since m 1=m 2, AB || CD 72 72

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10. 1 Classwork PAGE 501 • GO ONLINE and complete 10. 1 hw. • Alternative: Honors: 3, 4, 7 -10, 13, 17 -19, 22, 23 -26 • Regular: 2, 4, 7, 8, 16, 17, 22, 25 Reminders: q … 77 77

Module 10: Coordinate Proof Using Slope and Distance 78 78

Perpendicular Lines • (┴)Perpendicular Lines- 2 lines that intersect forming 4 right angles Right angle 79 79

Slopes of Lines • In a coordinate plane, 2 non vertical lines are iff the product of their slopes is -1. • This means, if 2 lines are their slopes are opposite reciprocals of each other; such as ½ and -2. • Vertical and horizontal lines are to each other. 80 80

Example • Line l passes through (0, 3) and (3, 1). • Line m passes through (0, 3) and (-4, -3). Are they ? Slope of line l = Opposite Reciprocals! Slope of line m = l m 81 81

Equation of a line in slope intercept form (y = mx+b) Now that we know how to find slope given any two points, we can generate an equation of the line connecting the two points. Example : points (3, 2) and (6, 9) 82 82

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10. 2 Classwork PAGE 515 • GO ONLINE and complete 10. 2 hw. • Alternative: Honors: 1, 2, 4 -6, 9 -18, 20, 22 • Regular: 2, 4, 6, 9 -15, 18, 20 Reminders: q … 85 85

Module 10: Coordinate Proof Using Slope and Distance 86 86

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10. 3 Classwork PAGE 531 • GO ONLINE and complete 10. 3 hw. • Alternative: Honors: 1, 3, 5, 8, 12, 18 • Regular: 3, 5, 8, 12, 18 Reminders: q … 94 94

Module 10: Coordinate Proof Using Slope and Distance 95 95

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10. 4 Classwork PAGE 543 • GO ONLINE and complete 10. 4 hw. • Alternative: Honors: 1, 3, 8, 11 -13, 17 -19, 26 • Regular: 3, 8, 11, 13, 17 -18, 19, 26 Reminders: q … 107

Module 10. 5: Perimeter and Area in the Coordinate Plane 108

Finding Perimeter and Area in the Coordinate Plane Concept: Distance in the Coordinate Plane EQ: How do we find area & perimeter in the coordinate plane? Vocabulary: distance formula, polygon, area, perimeter 109

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Area Formulas Perimeter for any polygon = sum of all sides. • A parallelogram includes shapes such as squares, rectangles, rhombi. • The length of the base and height are found using the distance formula. • The final answer must include the appropriate label (units², feet², inches², meters², centimeters², etc. ) 111

Guided practice, Example 1 Parallelogram ABCD has vertices A (-5, 4), B (3, 4), C (5, -1), and D (-3, -1). Calculate the perimeter and area of parallelogram ABCD. 112

Example 1, continued • 115

Example 1, continued • 116

Area of a triangle • 117

Guided Practice, Example 2 Triangle ABC has vertices A (2, 1), B (4, 5), and C (7, 1). Calculate the perimeter and area of triangle ABC. 118

Example 2, continued • 120

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10. 5 Classwork PAGE 559 • GO ONLINE and complete 10. 5 hw. • Alternative: Honors: 1, 2, 5, 7, 9 , 11, 13, 15 -18 • Regular: 1, 5, 7, 9 , 11, 15, 18 Reminders: q Topic 4 Review Next Class q. Topic 4 Test next week! 122