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Applied Geometry Unit 10 Quadrilaterals
Name: ____________________________
The Quadrilateral Family Tree!
Quadrilaterals
- Four angles
- All four-sided shapes
- Sum of the angles is 360°
Trapezoid
Kite
Parallelogram
- Only 2 parallel sides
- Opposite sides parallel
- Opposite sides congruent
- Opposite angles congruent
- Diagonals bisect each other
- Consecutive angles supp.
- Diagonals form 2 congruent
triangles
- 2 pairs adjacent sides congruent
- Diagonals perpendicular
- Made of 2 non-congruent isosceles
triangles that share the same base
Rhombus
Rectangle
- A parallelogram
- Four right angles
- Diagonals are congruent
- A parallelogram
- Equilateral sides
- Diagonals bisect angles
- Diagonals perpendicular
Isosceles
Trapezoid
- Non-parallel sides are congruent
- Diagonals are congruent
- Base angles are congruent
- Legs are congruent
Square
- All properties of a rectangle
- All properties of a rhombus
1
Coordinate Geometry Reference Sheet!
Important Points:
1.
2.
3.
4.
5.
6.
7.
Draw and label the figure on the coordinate plane.
Determine which formulas (Distance, midpoint, or slope) you need to answer the question.
Write out the formulas.
You MUST use distance, midpoint, or slope formulas to receive credit for the problem.
Substitute the numbers into the formulas to show your work.
Be organized and neat when showing your work. Label.
Write a concluding statement (sentence) at the end of the proof justifying why the work you
have shown answers the question.
Graphing Instructions:
1.
2.
3.
4.
Always use graph paper.
Always label your axes, scale, equations (if any), and the coordinates of the points plotted.
Always use a straightedge.
Always use pencil.
Formulas:
Name:
Formula:
Slope
m
y2  y1
x2  x1
What it finds:
How its used in proofs:
The slope of a line
1. To prove two lines parallel
(Show 2 equal slopes)
2. To prove two lines perpendicular
(Show 2 slopes that are negative
reciprocals)
Distance
Midpoint
d
 x2  x1    y2  y 1 
2
x x y y 
midpt   1 2 , 1 2 
2 
 2
1. To prove two lines congruent
(Show 2 equal distances)
2
The length of a line
segment
The midpoint of a line
segment
2. To prove lines are not congruent
(Show 2 unequal distances)
1. To prove two line segments
bisect each other
(Show they have the same
midpoint, 2 equal midpoints)
2
Using Coordinate Geometry - HOW to prove a…
1.
2.
3.
 is isosceles
 is Equilateral
 is a right 
a.) Show 2  sides (3 distances)
a.) Show 3 sides  (3 distances)
a.) Find the lengths of all three sides (distance 3x) and show a2+b2=c2 … OR
b.) Find the slopes of the 2 sides that look  and show they are negative
reciprocals forming a right angle* (slopes 2x – easiest of these!)
4.
quad is a
a.) Both pairs opposite sides  (distance 4x)
b.) Both pairs opposite sides (slopes 4x)
c.) one pair of opposite sides  and (distance 2x & slopes 2x)
d.) diagonals bisect each other* (midpoint 2x – for diags – easiest of these!)
5.
quad is a rectangle
a.) Both pair of opp sides  and diagonals  (6 distances 4 sides and 2
diagonals)
b.) It has 4 right angles* (4 slopes)
b.) Show it is a
(do ONE of the above) AND diagonals  (distances 2x)
c.) Show it is a
(do ONE of the above) AND has 1 right angle (slopes 2x)
6.
quad is a rhombus
a.) it has 4  sides* (distances 4x – easiest of these!)
b.) if it is a
(do ONE of the above) AND diagonals are  (slopes 2x)
c.) if it is a
(do ONE of he above) AND 2 adjacent sides  (distances 2x)
7.
quad is a square
a.) All 4 sides  and diagonals  (6 distance - 4 sides & 2 diagonals)
b.) if it has 4 right angles and diagonals  (6 slopes – 4 sides and 2
diagonals)
c.) It’s a parallelogram and diagonals are both  and  (see parallelogram + 2
distances + 2 slopes)
a.) Show ONLY one pair of opposite sides are // and the other pair of opp
sides are not //. (4 slopes)
8.
.
9.
.
quad is a trapezoid
quad is isos. trap.
a.) if only one pair of opposite sides are
are  (slopes 4x & distance 2x)
AND the 2 non-parallel sides
** For all proofs above, give a sentence to justify your work (from the list above!)
***Include phrases like:
“AB // CD because their slopes are =”
“AB  CD because their slopes are negative reciprocals. ”
3
Using Statement – Reason Tables – How to Prove…
To Prove a Quadrilateral is a parallelogram
Prove any ONE of the following:
•
Both pairs of opposite sides are //.
•
Both pairs of opposite sides are congruent.
•
One pair of opposite sides are both congruent and parallel.
•
The diagonals bisect each other.
•
Both pairs of opposite angles are congruent.
To Prove a Quadrilateral is a Rectangle
Prove any ONE of the following:
•
The Quadrilateral is a parallelogram with one right angle.
•
The quadrilateral is equiangular.
•
The quadrilateral is a parallelogram with congruent diagonals.
To Prove a Quadrilateral is a Rhombus
Prove any ONE of the following
•
The quadrilateral is a parallelogram with 2 congruent consecutive sides.
•
The quadrilateral is equilateral
•
The quadrilateral is a parallelogram whose diagonals are perpendicular.
•
The quadrilateral is a parallelogram, and a diagonal bisects the angles
whose vertices it joins.
To Prove a quadrilateral is a Square
•
The quadrilateral is a rectangle with 2 consecutive sides congruent.
•
The quadrilateral is a rhombus one of whose angles is a right angle.
4
Your Reasons for Proofs…
49.
↔ 2 pr opp sides
50.
↔ 2 pr opp sides 
51.
↔ 2 pr opp s 
52.
→ consec s supp
53..
↔ diag bisect each other
54. 1 pr opp sides
&  →
55. RH ↔
w/ all sides 
56. RH ↔
w/  diag
57. RH ↔
w/ diag bisect s
58. RH ↔
w/ 2 consec sides 
59. RE ↔ all rt s
60. RE ↔
w/  diag
61. RE ↔
w/ ≥ 1 rt 
62. SQ ↔ RE & RH
63. TR ↔ quad w/ only 1 pr opp sides
64. isos TR ↔ TR w/ non
sides 
65. isos TR ↔ TR w/ base s 
66. isos TR ↔ TR w/ diag 
67. kite → 2 pr consec sides  & opp sides not 
68. kite →  diag
69. kite → only 1 pr opp s 
5
Applied Geometry Unit 10 Quadrilaterals CWU9.1
Properties and Angle Measures of Polygons
HW: Pg 356 #7-11, 15, 18-23
Name ____________________________
Three Important Formulas:
1.
Polygon Interior Angles Theorem:
Interior angles of polygon add up to…
(n  2) 180
(Where n is the number of sides)
**Note: to find each interior angle of a regular polygon, divide the sum by n.
2.
Interior Angles of a Quadrilateral:
Add up to 360°
3.
Polygon Exterior Angles Theorem:
Exterior angles of polygon add up to 360°
Polygon
Number of Sides
Number of Triangles
Sum of Interior Angles
6
Example!
Find the sum of the measures of the interior angles of a convex octagon.
Practice!
Find the sum of the measures of the interior angles of the indicated convex polygon.
1.
Decagon
2.
13-gon
3.
18-gon
4.
25-gon
5.
34-gon
The sum of the measures of the interior angles of a convex polygon is given.
Classify the polygon by the number of sides.
6.
1260°
7.
3240°
8.
7560°
9.
What is the value of x in the diagram shown?
a.)
b.)
c.)
d.)
How do you find a missing
exterior angle measure in a
convex polygon?
7
Applied Geometry Unit 10 Quadrilaterals CW 9.2
Properties of Parallelograms
HW: Pg 363-364 #1-4, 9, 4-16, 25-27
Name:_______________________
Review the Quadrilateral Family Tree
Consecutive angles supplementary
Opposite angles congruent
8
Opposite Sides Congruent
Consecutive Angles Supplementary
Diagonals Bisect Each Other
Which statement is not always true about a parallelogram?
(1) The diagonals are congruent.
(2) The opposite sides are congruent.
(3) The opposite angles are congruent.
(4) The opposite sides are parallel.
9
Vertical Angles Congruent
Linear Pair Supplementary
// lines -> congruent alt int angles
To Prove a Parallelogram:
- May need to prove two triangles congruent
- May need to use facts about parallel lines
- MUST SHOW EITHER:
1.
Both pairs opp sides //.
2.
Both pairs opp sides 
3.
One pair opp sides both  and //
4.
The diagonals bisect each other
5.
Both pairs opp angles 
Memorize
new proof
reasons
49-54
tonight!
10
D
C
Proof Example:
Given:
Prove:
2
3
Quadrilateral ABCD
1  2
BD bisects AC
ABCD is a parallelogram
Statements
4
A
1
E
B
Reasons
11
Applied Geometry Unit 10 Quadrilaterals CW 9.3
Proofs with Parallelograms
HW: Worksheet A
Name:_________________________
Vertical Angles Congruent
Opposite Angles Congruent
Consecutive Angles Supplementary
Example 1:
Given:
ABCD, FG bisects DB
Prove: DB bisects FG
Statements
Reasons
12
Example 2:
Given: DE  AC,BF  AC
AE  CF,DE  BF
Prove: ABCD is a
Statements
Reasons
13
Refer to the reference sheet on doing coordinate geometry proofs to help you with
the problem below.
Example 3:
The vertices of quadrilateral ABCD are given.
Graph it and use slopes to show that it is a
parallelogram.
A(1,2), B(2,5), C(5,7) D(4,4)
14
Applied Geometry Unit 10 Quadrilaterals CW 9.4
More Proofs with Parallelograms
HW: Worksheet B
Name: _____________________
Opposite Angles & Sides Congruent
Consecutive Angles Supplementary
15
Continue practicing proofs on
parallelograms today!!!
Example 1:
Given: 1  2,PQ  RS
Prove: PQRS is a
Statements
Reasons
16
Example 2:
Quadrilateral ABCD has vertices A  2, 2  , B(1, 4) , C  2,8 , and D  1,6  .
Use midpoints to prove that ABCD is a parallelogram.
Example 3:
Quadrilateral ABCD has vertices A  2, 1 , B(1,3) , C  6,5 , and D  7,1 .
Use distances to prove that ABCD is a parallelogram.
17
Example 4:
Quadrilateral MILK has vertices M  4,10 , I ( 6, 6) , L  2,14 , and K  0,10  .
Prove that MILK is a parallelogram.
18
Applied Geometry Unit 10 Quadrilaterals CW 9.5
Properties of a Rhombus, Rectangle, and Square
HW: Finish Notes
Name: __________________________
***Review the Quadrilateral Family Tree on Page 1***
Practice:
19
Rhombus
The diagonals are perpendicular
The diagonals bisect the angles
20
Rectangle
The diagonals are congruent
The diagonals bisect each other (from Parallelogram)
21
Square
Diagonals are congruent (from Rectangle)
Diagonals bisect each other (from Parallelogram)
Diagonals are perpendicular (from Rhombus)
Diagonals bisect the angles (from Rhombus)
All right angles (from Rectangle)
22
Coordinate Proof Example:
Quadrilateral FGHJ has vertices F  2,5 , G (4,1) , H  2, 3 , and J  0,1 .
Show that FGHJ is a rhombus.
Additional Practice:
Determine whether the following statements are always, sometimes, or never true.
Explain your answer.
1. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
2. The diagonals of a quadrilateral bisect each other.
3. If one pair of opposite sides of a quadrilateral are parallel and congruent, the quadrilateral is a
parallelogram.
23
4. A square
is a rhombus.
Applied
Geometry
Unit 10 Quadrilaterals CW 9.6
Proofs with Rhombi, Rectangles, and Squares
HW: Worksheet C
Name:____________________________
Warm-up:
24
Today we will work on proofs involving
rectangles, rhombi, and squares.
Example 1:
The vertices of quadrilateral PQRS are P  0, 2  , Q (4,8) , R  7,6  , and S  3,0 .
Use slopes to prove that PQRS is a rectangle.
25
Example 2:
Quadrilateral MATH has vertices M  1, 4  , A(4, 7) , T  7, 2  , and H  2, 1 .
Prove that MATH is a square.
26
Example 3:
Statements
Reasons
27
Applied Geometry Unit 10 Quadrilaterals CW 9.7
Proofs with Rhombi, Rectangles, and Squares
HW: Worksheet D
Name:___________________________
Today is a proof lab day…
Work together with a partner to complete the proofs!
Proof #1:
Quadrilateral ABCD has vertices A  3,6  , B (6, 0) , C  9, 9  , and D  0, 3 .
Prove that ABCD is a parallelogram but NOT a rhombus.
28
Proof #2:
Quadrilateral ABCD has vertices A  2, 1 , B (2,3) , C  4,1 , and D  0, 3 .
Prove that ABCD is a rectangle.
29
Proof #3:
Given: Rectangle ABCD,
DE  FC
Prove: ADE  BCF
(Remember, you can use all Parallelogram reasons
for a Rectangle!)
Statements
Reasons
30
Proof #4:
Given: Square ABCD
Prove: Diagonals are congruent
(Must first prove ADC  BCD )
*Remember, you can use all Parallelogram reasons
for a Square! You can also use Rectangle and Rhombus
reasons for a Square, but must first state that this figure is a Rectangle and a Rhombus!
Statements
Reasons
31
Applied Geometry Unit 10 Quadrilaterals CW 9.8 Name: ___________________________
Properties of a Trapezoid, Isosceles Trapezoid, and Kite
HW: Worksheet E
***Review the Quadrilateral Family Tree on Page 1***
Isos. Trap has congruent base angles
Isos. Trap has supplementary same side consecutive angles
Midsegment of trap = average of bases its || to
Midsegment of trap meets midpts of non-|| legs
Hint: Make congruent triangles to see which angles are
congruent!
32
A kite has perpendicular diagonals
33
Coordinate Proof Example:
Quadrilateral MATH has vertices M 1,1 , A(2,5) , T  5, 7  , and H  7,5 .
a.)
Prove that MATH is a trapezoid.
b.)
Determine if MATH is an isosceles trapezoid.
Additional Practice:
34
Applied Geometry Unit 10 Quadrilaterals CW 9.9
Name:___________________________
Proofs with Trapezoids, Isosceles Trapezoids, and Kites
HW: Worksheet F
Example 1:
Quadrilateral ABCD has vertices A 1, 2 , B (13, 4) , C  6,8 , and D  2, 4  .
Prove that ABCD is a trapezoid but NOT an isosceles trapezoid.
35
Example 2:
Given:
Prove:
TRAP is a trapezoid with TA  RP
RPA  TAP
36
Example 3
Quadrilateral ABCD has vertices A  0, 4  , B(0, 8) , C  3,1 , and D  3, 4 .
Prove that ABCD is a trapezoid but NOT an isosceles trapezoid.
37
Example 4
Quadrilateral ABCD has vertices A  6,3 , B(3, 6) , C  9,6  , and D  5, 8 .
Prove that ABCD is a trapezoid but NOT an isosceles trapezoid.
38
Applied Geometry Unit 10 Quadrilaterals CW 9.10 Name:__________________________
Special Quadrilaterals
CW/HW: Finish 9.10 Notes
Activity: Use your Family Tree to help you!
Practice:
39
40
41
Applied Geometry Unit 10 Quadrilaterals CW 9.11
Unit 9 Quadrilaterals Review Day #1
HW: CW 9.12 (Review Day #2)
Name: ______________________
Directions: In #1-6, use the information in and below each diagram and the properties of various
quadrilaterals to find x, y, and z, as required. Label answers with appropriate units.
D
1.
C
x
120°
A
70°
U
2.
60°
B
C
A
x
S
O
4.
z
50°
T
Parallelogram RSTU
D
3.
y
z
70°
R
Quadrilateral ABCD
x
N
130°
z
x
y
y
L
B
Rectangle ABCD
M
Isosceles Trapezoid LMNO
5.
6.
T
S
H
y
G
120°
30°
x
x
z
Q
E
R
Rhombus QRST
y
20°
F
Trapezoid EFGH
7.
a.)
Heptagon
b.) 30-gon
42
8.
a.)
9.
1080°
b.)
2520°
Solve for x.
10.
11.
Find the length of the midsegment or find the value of x.
12.
13.
43
14.
Which statement is true?
(1)
(2)
(3)
(4)
15.
Which quadrilateral does not necessarily have congruent diagonals?
(1)
16.
True
(2)
rhombus
(4)
rectangle
False
diagonals are congruent
(2)
opposite sides are congruent (4)
diagonals are perpendicular
adjacent sides are congruent
congruent and bisect the angles to which they are drawn
congruent and do not bisect the angles to which they are drawn
not congruent and bisect the angles to which they are drawn
not congruent and do not bisect the angles to which they are drawn
diagonals are congruent
sides and angles are congruent
opposite sides and opposite angles are congruent
diagonals bisect each other and are perpendicular to each other
A parallelogram must be a rectangle if the opposite angles are
(1) congruent
21.
(3)
A quadrilateral must be a square if
(1)
(2)
(3)
(4)
20.
square
A parallelogram must be a square if the diagonals are
(1)
(2)
(3)
(4)
19.
(2)
To prove that a parallelogram is a rectangle, it is
sufficient to show that the
(1)
(3)
18.
isosceles trapezoid
True or False: All squares are similar to each other.
(1)
17.
All parallelograms are quadrilaterals.
All parallelograms are rectangles.
All quadrilaterals are trapezoids.
All trapezoids are parallelograms.
(2) equal in measure
(3) supplementary
(4) complementary
In quadrilateral ABCD, mA  x 10 , mB  2x  10 , mC  2x  70 , and mD  3x  50 .
What kind of quadrilateral is ABCD and why?
Type of Quadrilateral: ____________________
Why? ____________________________________________________________________________
44
22.
In parallelogram ABCD, AB  3x  2 , DC  10 x 12 , and AD  5x  2 .
What kind of parallelogram is ABCD? Justify your answer.
Type of Parallelogram: ____________________
Justification: _____________________________________________________________________
23.
24.
In rectangle ABCD, diagonals AC and BD intersect at P. If CP  40 and BD  2 x 12 .
What is the value of BP ?
25.
WXYZ is a parallelogram, YA is an altitude to WX , and YA  AX . Find mZ .
W
Z
26.
A
X
Y
List two ways to prove a quadrilateral is a parallelogram in a coordinate geometry proof:
1._______________________________________________________________
2._______________________________________________________________
45
Applied Geometry Unit 10 Quadrilaterals CW 9.12
Quadrilaterals Review Day #2
1.
Name: ______________________
Quadrilateral QUAD has vertices Q  1,1 , U (3, 4) , A 1,5 , and D  3, 2 .
Prove that QUAD is a parallelogram.
46
2.
Quadrilateral ABCD has vertices A  5, 0  , B (2,9) , C  4,7  , and D  1, 2 .
Use slopes to prove that ABCD is a rectangle.
47
3. Given: Rectangle ABCD
E is the midpoint of DC
Prove: DAE  CBE
Statements
Reasons
48
4.
Quadrilateral ABCD has vertices A  3, 2  , B(2, 6) , C  2,7  , and D 1,3 .
Prove that ABCD is a rhombus.
49
5.
Quadrilateral PQRS has vertices P  0, 0  , Q (4,3) , R  7, 1 , and S  3, 4  .
Show that PQRS is a square.
50
6.
Quadrilateral DEFG has vertices D  4,0 , E (0,1) , F  4, 1 , and G  4, 3 .
Prove that ABCD is a trapezoid but NOT an isosceles trapezoid
51