This Packet Belongs to (Student Name) Topic 4 ...
Transcript of This Packet Belongs to (Student Name) Topic 4 ...
1
Topic 4: Quadrilaterals and Coordinate Proof
This Packet Belongs to ________________________(Student Name)
Unit 4 – Quadrilaterals and Coordinate ProofModule 9: Properties of Quadrilaterals
9.1 Properties of Parallelograms9.2 Conditions of Parallelograms9.3 Properties of Rectangles, Rhombi, and Squares9.4 Conditions of Rectangles, Rhombi, and Squares
1
Please follow along with notes.
Topic 1,2,3,4,etc…
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…
Module 10: Coordinate Proof Using Slope and Distance10.1 Slope and Parallel Lines
10.2 Slope and Perpendicular Lines10.3 Coordinate Proof Using Distance with Segments and Triangles10.4 Coordinate Proof Using Distance with Quadrilaterals10.5 Perimeter and Area on the Coordinate Plane
1
Module 9Properties of Quadrilaterals
Part 1:
Parallelograms2
2
Definition
• A parallelogram is a quadrilateral whose opposite sides are parallel.
• Its symbol is a small figure:
CB
A D
AB CD and BC AD
33
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. CB
A D4
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.
55
Opposite Sides
Theorem Opposite sides of a parallelogram are congruent.
• That means that .
• So, if AB = 7, then _____ = 7?
CB
A D
AB CD and BC AD
CD
66 7
2
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 = ______ .
CB
A D8
75° 105°
8 9
Consecutive Angles
• Each angle is consecutive to two other angles. A is consecutive with B and D.
CB
A D
1010
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 = _____?
CB
A D
Consecutive INTERIOR Angles are
Supplementary!
134°
11
Diagonals• Diagonals are segments that join non‐consecutive vertices.
• For example, in this diagram, the only two diagonals are .
AC and BD
CB
A D
1212
Diagonal PropertyWhen 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!
AC and BD
AP PC BP PD
P
CB
A D
AC BD
1313
3
Diagonal PropertyTheorem The diagonals of a parallelogram bisect each other.
P
CB
A D
1414
1515
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.
1616
Examples
• 1. Draw HKLP.
• 2. Complete: HK = _______ and HP = ________ .
• 3. m<K = m<______ .
• 4. m<L + m<______ = 180.• 5. If m<P = 65, then m<H = ____,
m<K = ______ and m<L =______ .
PL
KL
P
P or <K
11565 115
1717
Examples (cont’d)
• 6. Draw in the diagonals. They intersect at M.
• 7. Complete: If HM = 5, then ML = ____ .
• 8. If KM = 7, then KP = ____ .
• 9. If HL = 15, then ML = ____ .
• 10. If m<HPK = 36, then m<PKL = _____ .
5
14
7.5
36
1818
1919
4
20
Part 2
Tests for Parallelograms
2121
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.
2222
How can you tell if a quadrilateral is a parallelogram?
• Defn: A quadrilateral is a parallelogram iffopposite 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.
2323
Proving Quadrilaterals as Parallelograms
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram .
Theorem 1:
H G
E F
If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .
Theorem 2:
If EF GH; FG EH, then Quad. EFGH is a parallelogram.
If EF GH and EF || HG, then Quad. EFGH is a parallelogram.24 25
5
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 3:
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .
Theorem 4:
H G
EF
M
,If H F and E G
then Quad. EFGH is a parallelogram.
intIf M is themidpo of EG and FH
then Quad. EFGH is a parallelogram.EM = GM and HM = FM
Proving Quadrilaterals as Parallelograms (part 2)
2627
27
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 .
2828
Examples ……
Find the values of x and y that ensures the quadrilateral is a parallelogram.
Example 1:
6x4x+8
y+2
2y
6x = 4x + 8
2x = 8
x = 4
2y = y + 2
y = 2
Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.
120°
5y°(2x + 8)°2x + 8 = 120
2x = 112
x = 56
5y + 120 = 180
5y = 60
y = 12
292929
3030
3131
6
9.1‐9.2 ClassworkPAGE 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:
…32
3233
Part 3
Rectangles
33
34
Rectangles
• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• Diagonals bisect each other.
Definition: A rectangle is a quadrilateral with four right angles.
Is a rectangle a parallelogram?
Thus a rectangle has all the properties of a parallelogram.
Yes, since opposite angles are congruent.
3435
Properties of Rectangles
Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles.
If a parallelogram is a rectangle, then its diagonals are congruent.
E
D C
BA
Theorem:
Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.
35
36
Properties of Rectangles
E
D C
BA
Parallelogram Properties: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.Plus: All angles are right angles. Diagonals are congruent.
Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles
36 37
7
38
Examples
1. If AE = 3x +2 and BE = 29, find the value of x.
2. If AC = 21, then BE = _______.
3. If m<1 = 4x and m<4 = 2x, find the value of x.
4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.
m<1=50, m<3=40, m<4=80, m<5=100, m<6=40
10.5 units
x = 9 units
x = 18 units
6
54
321
E
D C
BA
3839
39
40
Part 4
Rhombi and
Squares40
41
Rhombus
Definition: A rhombus is a quadrilateral with four congruent sides.
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.
Is a rhombus a parallelogram?
Yes, since opposite sides are congruent.
41
42
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.
4243
Properties of a RhombusTheorem: 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
8
44
Your Turn: Rhombus Examples
Given: ABCD is a rhombus. Complete the following.
1. If AB = 9, then AD = ______.
2. If m<1 = 65, the m<2 = _____.
3. m<3 = ______.
4. If m<ADC = 80, the m<DAB = ______.
5. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.
54
3
21E
D C
BA9 units
65°
90°
100°
10
4445
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
4646 47
4848
4949
9
5050 51
52
Square
• 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.
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.
5253
53
54
Squares – Examples…...Given: ABCD is a square. Complete the following.
1. If AB = 10, then AD = _______ and DC = _______.
2. If CE = 5, then DE = _____.
3. m<ABC = _____.
4. m<ACD = _____.
5. m<AED = _____.
8 7 65
4321
E
D C
BA10 units 10 units
5 units
90°
45°
90°
5462
62
10
9.3‐9.4 ClassworkPAGE 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:
Module 9 Quiz Next Class!
MYA next week!63
63
Module 10:Getting Ready
6464
Definition of a ParallelogramUse 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) .
I need to show that both pairs of opposite sides are parallel by showing
that their slopes are equal.
A
B C
D
65
Definition of a ParallelogramUse 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: m = 6 – 0 = 6 = 32 – 0 2
CD: m = 1 – 7 = - 6 = 33 – 5 - 2
BC: m = 7 – 6 = 15 – 2 3
AD: m = 1 – 0 = 13 – 0 3
ll
ll
ABCD is a Parallelogramby Definition
A
B C
D
66
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) .
I need to show that both pairs of opposite sides are congruent by using the distance formula to
find their lengths.
A
B C
D
67
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) .AB = (2 – 0)2 + (6 – 0)2
= 4 + 36 = 40
ABCD is a Parallelogram because both pair of opposite sides are congruent.
CD = (3 – 5)2 + (1 – 7)2
= 4 + 36 = 40
AB CDBC = (5 – 2)2 + (7 – 6)2
= 9 + 1 = 10
AD = (3 – 0)2 + (1 – 0)2
= 9 + 1 = 10
A
B C
D
6868
11
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) .
I need to show that onepair of opposite sides is
both parallel and congruent.
One Pair of Opposite Sides Both Parallel and Congruent
ll (slope) and (distance)
A
B C
D
69
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) .
BC ADABCD is a Parallelogram because one pair of opposite sides are parallel and congruent.
BC ll ADBC = (5 – 2)2 + (7 – 6)2
= 9 + 1 = 10
AD = (3 – 0)2 + (1 – 0)2
= 9 + 1 = 10
BC: m = 7 – 6 = 15 – 2 3
AD: m = 1 – 0 = 13 – 0 3
A
B C
D
7070
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) .
I need to show that each diagonal shares the SAME _________.
Diagonals Bisect Each Other
A
B C
D
71
midpoint
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) .
ABCD is a Parallelogram because the diagonals share the same midpoint, thus bisecting each other.
Diagonals Bisect Each Other
The midpoint of AC is 0 + 5 , 0 + 72 2
5 , 72 2
The midpoint of BD is 2 + 3 , 6 + 12 2
5 , 72 2
A
B C
D
7272
Module 10:Coordinate Proof Using Slope and
Distance
7373
7474
12
7575
7676
7777
Understanding Slope
• Two (non-vertical) lines are parallel if and only if they have the same slope. (All vertical lines are parallel.)
4
2
-2
-4
-6
5
D: (4, -1)
C: (-2, -4)
B: (3, 3)
A: (-1, 1)
7878
Understanding Slope
• The slope of AB is:
• The slope of CD is:
• Since m1=m2, AB || CD
4
2
-2
-4
-6
5
D: (4, -1)
C: (-2, -4)
B: (3, 3)
A: (-1, 1) 1
3 1 2 1
3 1 4 2m
2
1 4 3 1
4 2 6 2m
7979 80
13
8181
8282
8383
10.1 ClassworkPAGE 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:
…
8484
Module 10:Coordinate Proof Using Slope and
Distance
8585
Perpendicular Lines
• (┴)Perpendicular Lines‐ 2 lines that intersect forming 4 right angles
Right angle
8686
14
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.
8787
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 =
Slope of line m =
l m
30
13
3
2-or
3
2
40
33
2
3or
4
6
Opposite Reciprocals!
8888
Equation of a line in slope intercept form (y = mx+b)
Now that we know how to find slope given any two points, we cangenerate an equation of the line connecting the two points.
Example : points (3,2) and (6,9)
8989
9090
9191
10.2 ClassworkPAGE 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:
…
9292
16
9999 100
10.3 ClassworkPAGE 531
• GO ONLINE and complete 10.3 hw.
• Alternative:Honors: 1, 3, 5, 8, 12, 18
• Regular: 3, 5, 8, 12,18
Reminders:
…
101101
Module 10:Coordinate Proof Using Slope and
Distance
102102
103103
104104
18
111111
112112
10.4 ClassworkPAGE 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:
…
114114
Module 10.5:Perimeter and Area in the Coordinate
Plane
115115
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
116116
117
19
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.) 118
118
Guided practice, Example 1Parallelogram ABCD has vertices A (‐5, 4), B (3, 4), C (5, ‐1), and D (‐3, ‐1).
Calculate the perimeter and area of parallelogramABCD.
119119
Example 1, continuedWe need to find the length of all four sides before we can find the area and the
perimeter. So we will use the distance formula:
, , , ,
The length of is 8 units The length of is 8 units
120120
Example 1, continuedWe need to find the length of all four sides before we can find the area and the
perimeter. So we will use the distance formula:
, , , ,
.
The length of is 5.39 units
.
The length of is 5.39 units
121121
Example 1, continued
• 8units 5.39units 8units 5.39units• Find the perimeter by adding up all the sides:
• . . . •Find the area by using the formula
– or is the base and they are the same length so 8
– The height can be found by drawing a perpendicularline straight up from D to side and down from B to side .
• You can do this by counting the units or using the distance formula
• Finding the distance from D to the point 3, 4 and the distance from B to the point where the perpendicular line touches at 4, 1
122122
Example 1, continued
Area of a Parallelogram =
Base = 8 units
Height = 5 units
Area of a parallelogram = ∗
123123
20
Area of a triangle
• The area of a triangle is found by using the formula:
Area = ∗ ∗
• The height of a triangle is the perpendicular distance from a vertex to the base of the triangle.
• Determining the lengths of the base and the height is necessary if these lengths are not stated in the problem.
• The final answer must include the appropriate label (units², feet², inches², meters², centimeters², etc.)
124124
Guided Practice, Example 2
Triangle ABC has vertices
A (2, 1), B (4, 5), and C (7, 1).
Calculate the perimeterand area of triangle ABC.
125125
Example 1, continuedWe need to find the length of all four sides before we can find the area and the
perimeter. So we will use the distance formula:
, , ,
.
The length of is 4.47 units The length of is 5 units.
126
5
The length of is 5 units.
126
Example 2, continued
• . • Find the perimeter by adding up all the sides:
• . .
•Find the area by using the formula
– is the base so 5– The height can be found by drawing a
perpendicularline straight down from B to side .
– Then find the distance from B to the point where the perpendicular line touches at 4,1
• You can do this by counting the units or
using the distance formula
127
Area of a Triangle = Base = 5 unitsHeight = the distance from , to
, = 4
Area of a triangle =
127
128128
10.5 ClassworkPAGE 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:
Topic 4 Review Next Class
Topic 4 Test next week!
131131