1
Quadrilaterals
Unit 9
Name___________________
2
Classwork – Lesson 1
Features of Parallelograms
D
C
E
B
A
1. Measure the following (use mm):
Sides
AB ______ BC _______
CD _______
DA _______
Angles
mÐDAB ______
mÐABC _______
mÐ BCD_______
mÐCDA _______
Diagonals
AC _______
BD _______
AE _______
EC _______
BE ______
ED _______
Triangles Formed
Area of triangle ABC _______
Area of triangle ADC _______
Now let’s draw some conclusions. What do you know about …
Opposite sides: ___________
Consecutive angles: ___________
The diagonals: ___________
Opposite Angles: ___________
Triangles formed by the diagonals: ___________
Classwork – Lesson 1
3
Parallelogram Basics
1. Write, in words, the property of parallelograms illustrated by each figure OR illustrate the
property described:
C
D
a)
A
ABC  CDA
B
b)
3
4
3
________________________________________________
4
c)
7
5
5
________________________________________________
7
d)
45
135
________________________________________________
e)
Opposite angles are congruent
f)
Opposite sides are parallel
FLASH FORWARD: How many of these properties are necessary to PROVE that a quadrilateral is a
parallelogram? __________________
In #2-4, State the property of parallelograms that you will use, then find the missing angles and
lengths in each parallelogram.
2.
z
60
x
3.
y
y
x
Property:
Property:
4.
10
12
7
6
Property:
X= ____
X= ____
X= ____
Y= ____
Z= ____
Y= ____
Y= ____
x
y
4
Homework- Lesson 1
Identifying Parallelograms
State whether each quadrilateral is a parallelogram, is not a parallelogram, or is uncertain.
1.
45
135
2x + 4
2.
45
B
135
3.
2
4
4.
4
2
5.
D
7.
9. Quarilateral ABCD has
diagonals AC and BD
that intersect at right angles.
10. In quadrilateral ABCD,
A  D and AB  CD.
C
8.
A
6.
B
DB bisects ABC
11. In quadrilateral ABCD, AB = 2x-4,
BC = 3x, CD = x + 6, AD = 2x+4, and
the perimeter equals 54.
D
12.
C
60
110
60
A
B
Classwork- Lesson 2
5
Working With Parallelogram Properties
In # 1-4, set up an equation to solve for the missing variable(s). Justify each equation by stating the
property of parallelograms that you used. Then solve for the missing side length or angle measure.
Remember:
Properties of parallelograms:
Opposite sides are ________________ and ________________
Diagonals _________________ each other.
Consecutive angles are __________________
Opposite angles are ____________________
Diagonals divide quadrilateral into 2 __________________ triangles
1. In
ABCD, find AB, BC, CD, and AD.
4x - 12
D
4y - 6
A
ABCD, find the measure of
angles A, B, C and D.
C
B
2x + 4
.2. In
B
2x + 2
C
D
y+9
A
(5x - 7)
B
2x + 4
(2x + 5)
B
Equation:
Equation:
Justification:
Justification:
Solve:
Solve:
3. In
ABCD, find the measure of
angles A, B, C and D.
4. In
ABCD, AE is 10 inches shorter
than twice EC, and BE is 8 inches shorter
than three times ED. Find AC and BD.
C
D
(5x - 40)
2x + 4
(3x - 20)
A
D
B
C
E
B
A
B
Equation:
Equation:
Justification:
Justification:
Solve:
Solve:
Classwork- Lesson 2
6
More Practice With Parallelogram Properties
Classwork- Lesson 2
7
Homework – Lesson 2
8
Parallelogram Benchmark
I can use the properties of a parallelogram to find missing angle and segment
measures
1. In parallelogram BCDE CD || _____
2. In parallelogram EFGH EO  _____
Justification:
Justification:
E
D
H
O
B
G
O
C
E
F
3. Find AM in the parallelogram if PN  7 and AO  5.
M
N
A
P
4. Find AM in the parallelogram if PN  10 and MO  19.
O
5. Refer to the figure below.
Given: UVWX is a parallelogram, mWXV  17 , mWVX  29 , XW  41, UX  24 , UY = 15
a. Find m  WVU.
b. Find WV.
U
V
15
29°
24
Y
17°
c. Find m  XUV.
d. Find UW.
X
41
W
9
6. Use the figure below.
Given: FGHJ is a parallelogram, m  JHG = 68  , JH = 34, GH = 19
a. Find m  FJH.
b. Find JF.
F
G
K
c. Find m  GFJ.
d. Find FG.
7. For parallelogram PQLM below, if mPML  83 , then mPQL  ______ .
[A] mQLM
[B] 83
[C] 97
[D] mPQM
8. Consecutive angles in a parallelogram are always ________.
[A] supplementary angles
[B] complementary angles
[C] congruent angles
[D] vertical angles
9. Choose the statement that is NOT ALWAYS true.
For any parallelogram _______.
[A] opposite sides are congruent
[B] the diagonals are perpendicular
[C] the diagonals bisect each other
[D] opposite angles are congruen
J
68°
34
19
H
10
Classwork- Lesson 3
Classwork- Lesson 3
11
Classwork- Lesson 3
12
Rectangles, Squares & Rhombi
1. In rectangle ABCD, AB = 2x + 3y, BC = 5x – 2y, CD = 22, and AD = 17. Find x and y.
A
B
E
D
In the diagram for problems 2-7,QRST is a rectangle and QZRC is a parallelogram.
2. If QC = 2x + 1 and TC = 3x – 1,
find x.
C
3. If mTQC = 70, find mQZR.
Z
Q
R
Q
S
T
Z
R
C
C
T
4. If mRCS = 35, find mRTS.
S
5. If mQRT = mTRS, find mTCQ. Z
Z
Q
Q
R
R
C
C
T
T
S
6. If RT = x2 and QC = 4x – 6, what is the value of
x?
Z
Q
S
7. RZ = 6x, ZQ = 3x + 2y, and CS = 14 – x. Find the
values of x and y. Is QZRC a “special” parallelogram? If
so, what kind?
Z
R
C
Q
R
C
T
S
T
S
Use rectangle STUV for questions 8-11.
S
8. If m1 = 30, m2 = _______
6
7
9. If m8 = 133, m2 = _______
V
10. If m5 = 16, m3 = _______
5
4
3
U
12. ABCD is a square. If mDBC = x2 – 4x, find x.
B
A
C
11. ABCD is a rhombus. If the perimeter of
B
ABCD = 68 and BD = 16, find AC.
A
2
K
1
8
T
D
D
C
Homework- Lesson 3
13
B
A
Use rhombus ABCD for problems 14-19
13. If mBAF = 28, mACD = ______.
F
D
14. If mAFB = 16x + 6, x = _______.
16. If mBFC = 120 – 4x, x = ______.
C
15. If mACD = 34, mABC = _______.
17. If mBAC = 4x + 6 and mACD = 12x – 18, x = ______.
18. ABCD is a square. AB = 5x + 2y,
AD = 3x – y, and BC = 11. Find x and
y.
A
19. A contractor is measuring for the foundation of a building that is to
be 85 ft by 40 ft. Stakes and string are placed as shown. The outside
corners of the building will be at the points where the strings cross. He
then measures and finds WY = 93 ft and XZ = 94 ft. Is WXYZ a
B rectangle? If not, which way should stakes E and F be moved to made
WXYZ a rectangle?
85 ft
E
F
D
G
C
H
W
X
Z
Y
Given rectangle QRST
C
21. ABCD is a rectangle. Find each diagonal if AC 
and BD = 4 – c.
Q
R
X
22. If RX  QT , find mTXS.
T
23. If mRQS = 30° and QS = 13, find SR26.
40 ft
B
A
20. ABCD is a rectangle. Find the length of each
diagonal if AC = 2(x – 3) and BD = x + 5.
D
S
24. If mQST = 45° and QT = 6.2, find QR.
3c
9
Classwork- Lesson 4
14
Rectangles
1. All angles in a rectangle
measure _________.
2. If a parallelogram has congruent
diagonals, it is a _____________.
3. All rectangles are parallelograms.
TRUE FALSE
4. All parallelograms are rectangles.
TRUE
FALSE
In #5-7, State the property of rectangles that you will use, then find the missing lengths in each
rectangle.
Remember:
Rectangles have all of the properties of _________________
PLUS ______________________ and __________________________
5. ABCD is a rectangle.
Find x.
D
x+4
C
6. ABCD is a rectangle.
Find x, y, and z
C
D
z
4
A
3x - 6
B
A
7. The perimeter of rectangle ABCD
is 26. Find AB and BC.
D
C
x
x+4
y
A
B
½ x +3
B
Using only the information given in the figure, can you tell if each of the following is a rectangle?
Answer yes, no, or uncertain.
8. Parallelogram ABCD has a
right angle at A.
9. In parallelogram ABCD, A  B
10. In parallelogram ABCD,
AC = 5 and BD = 6
11. In quadrilateral ABCD, AC = 4 and BD = 4.
12.
13.
5
3
4
14.
15.
m A = (x + 20),
mC = (2x -60),
D
C
A
B
Classwork- Lesson 4
15
Square and Rhombus
1. A parallelogram with all sides
congruent must be a _________.
2. A square is a ___________ with four right angles.
3 All squares are rhombuses.
TRUE FALSE
4. All rhombuses are squares.
TRUE
FALSE
In #5-7, State the property of rhombus or square that you will use, then find the missing lengths in
each one.
Remember:
Rhombuses have all of the properties of _________________
PLUS ______________________
Squares have all of the properties of ____________________
PLUS all of the properties of ______________________
5. ABCD is a square.
Find CD.
D
6. Find the perimeter.
of rhombus ABCD
C
D
C
7. ABCD is a square. AC =
Find AD.
D
C
A
B
x + 16
A 5x - 8 B
A
4
B
8.
Classwork- Lesson 4
16
Trapezoids
Remember:
In order to conclude that a quadrilateral is a trapezoid, one and only one pair of sides must be
parallel. We call the parallel sides the bases and the other two sides the legs
In order to conclude that a trapezoid is an isosceles trapezoid, its two legs bust be congruent.
Sketch a trapezoid:
Sketch an isosceles trapezoid:
Properties of a trapezoid:
Properties of an isosceles trapezoid
Alternate interior angles are ________
Alternate interior angles are __________
Base angles are ___________
Diagonals are _____________
Midsegement = ______________________
1. AB // CD. Find the measures of
angles B, C, D.
D
2. AB // CD. Find the measures of
angles A, C, and D.
D
C
C
102
68
A
B
A
B
17
3. AB // CD. Find mC, mD,
and BC.
D
4. ABCD is an isosceles trapezoid.
AC = (4x + 7) and BD = 6x - 13.
find AC.
C
D
C
6
65
65
A
B
5. AB // CD. AC = BD = 12.
mBAD = (½x + 30) and ,
mABC = (2x - 90).
Find mBAD.
D
A
B
6. ABCD and CDEF are isosceles trapezoids.
mBAD = 50, mDEF = 30, and the ratio
of mDEA to mDAE is 3:1. Find mDAE.
F
E
C
D
A
C
B
A
B
18
Homework- Lesson 4
1. ABCD is a rhombus. AC=8
and DB = 6. Find the
perimeter.
D
2. In rhombus ABCD, BC =18.
Which cannot be the length
AC ?
C
(a) 9
E
(b) 18
(c) 35
3. In square ABCD, AE=3x
and EB = 8x - 40. Find BD.
D
C
(d) 36
E
A
A
B
B
Using only the information given in the figure, classify each of the following as a parallelogram,
rectangle, rhombus, or square. State ALL that apply.
4.
4
5.
4
4
4
4
7.
4
4
8.
Solve for the missing angle measure.
9.
4
6.
10.
Classwork- Lesson 5
19
Classifying Quadrilaterals
1. Fill in each blank with “never”, “sometimes”, or “always”
a) Parallelograms ______________ have 2 pairs of parallel sides
b) Parallelograms ______________ have 1 pair of congruent sides
c) Parallelograms ______________ have 4 right angles
d) Parallelograms ______________ have congruent diagonals
e) Parallelograms ______________ have diagonals that bisect each other
f) Parallelograms ______________ have perpendicular diagonals
g) Rectangles ______________ have 2 pairs of parallel sides
h) Rectangles ______________ have 1 pair of congruent sides
i) Rectangles ______________ have 4 right angles
j) Rectangles ______________ have congruent diagonals
k) Rectangles ______________ have diagonals that bisect each other
l) Rectangles ______________ have perpendicular diagonals
Classwork- Lesson 5
m) Rhombuses ______________ have 2 pairs of parallel sides
n) Rhombuses ______________ have 1 pair of congruent sides
o) Rhombuses ______________ have 4 right angles
p) Rhombuses ______________ have congruent diagonals
q) Rhombuses ______________ have diagonals that bisect each other
r) Rhombuses ______________ have perpendicular diagonals
s) Squares ______________ have 2 pairs of parallel sides
t) Squares ______________ have 1 pair of congruent sides
u) Squares ______________ have 4 right angles
v) Squares ______________ have congruent diagonals
w) Squares ______________ have diagonals that bisect each other
x) Squares ______________ have perpendicular diagonals
y) Trapezoids ______________ have 2 pairs of parallel sides
z) Trapezoids ______________ have 1 pair of congruent sides
aa) Trapezoids ______________ have 4 right angles
ab) Trapezoids ______________ have congruent diagonals
ac) Trapezoids ______________ have diagonals that bisect each other
ad) Trapezoids ______________ have perpendicular diagonals
20
Classwork- Lesson 6
Finish for HW
21
Directions; Work with your group to complete problems on a separate sheet. Don’t forget to
justify each step!!
22
23
Classwork- Lesson 7
Do Now
Here are some features that you might use in writing a definition of Parallelogram, Rectangle,
Rhombus, and Square.
Parallel Sides
Congruent Sides
Perpendicular Diagonals
Right Angles
Congruent Diagonals
Diagonals that Bisect Each Other
1. Write your own definitions:
a) Parallelogram __________________________________________________
__________________________________________________
b) Rectangle
__________________________________________________
__________________________________________________
c) Rhombus
__________________________________________________
__________________________________________________
d) Square
____ ______________________________________________
__________________________________________________
e) Trapezoid
__________________________________________________
__________________________________________________
2. Propose some theorems regarding the diagonals of parallelograms, rectangles, rhombuses,
and square.
Classwork- Lesson 7
24
Proving Quadrilaterals are Parallelograms
1.
Given:
ABCD is a quadrilateral,
E is the midpoint
Statement
Reason
of both AC and DB
Prove:
1) ABCD is a quadrilateral
1)
ABCD is a parallelogram
2) E is the midpoint of AC 2)
and DB
D
C
E
A
B
2.
Given:
ABCD is a quadrilateral,
ABE  CDE
Prove:
ABCD is a parallelogram
D
3)
4) BE  ED
4)
5) ABCD is a parallelogram
5)
Statement
Reason
1) ABCD is a quadrilateral
1)
2) ABE  CDE
2)
3) AE  EC
3)
4) BE  ED
4)
5) AC bisects BD
5)
6) BD bisects AC
6)
7) ABCD is a parallelogram
7)
C
E
A
3) AE  EC
B
Classwork- Lesson 7
25
3.
Since DG EF and DE GF are given, then ∠1≅ ∠4 and ∠3 ≅ ∠______ because
______________________________________________________. DF ≅ DF because
__________________. Then ΔDGF ≅ Δ _____ by ________. Therefore, DG ≅ EF by
___________________________.
4.
Given:
ABCD is a quadrilateral,
1  2,
3  4,
Prove:
ABCD is a parallelogram
D
3
1
4
A
2
B
C
Classwork- Lesson 7
26
5.
Given:
ABCD is a quadrilateral
AD @ BC,
CBD  ADB
Prove:
ABCD is a parallelogram
D
C
A
B
6.
Given:
ABF  EFB  BCE  DEC
Prove: ABCD is a parallelogram
F
A
B
E
D
C
(hint - you need to use CPCTC
to show opposite sides or angles
are congruent)
27
Homework- Lesson 7
DUE: 3/13
**Hand this page in**
You may conduct this proof however you choose.
Given:
Quadrilateral ABCD,
AD is parallel to BC,
AEF and DFE are right angles
Prove: ABCD is a parallelogram
D
A
F
E
C
B
28
Classwork- Lesson 8
Proving Special Parallelograms
1. Given : Quadrilateral ABCD,
E is the midpoint of
AC and BD,
AC  BD.
Prove: ABCD is a rectangle
D
Statement
1) ABCD is a quadrilateral
1)
2) E is the midpoint of
2)
AC and BD
3) AC and BD bisect each
other
3)
C
4) ABCD is a parallelogram
4)
B
5) AC BD
5)
6) ABCD is a rectangle
6)
E
A
2. Given: ABCD is a parallelogram,
AB ^ AD,
AB  BC
Prove: ABCD is a square
C
D
E
A
B
Reason
Classwork- Lesson 83. Given: Parallelogram ABCD,
AEB  CEB
Prove: ABCD is a rhombus
D
C
29
Since ABCD is a parallelogram and AEB  CEB are given,
We know that AEB  CEB by ___________________.
AEB and CEB are supplementary because ________________
E
_________________________________________________.
A
B
Then mAEB + mCEB = 180 since ______________________
__________________________, so mAEB + mAEB = 180 by
______________________. Therefore, m AEB = 90 because
_________________________________________________
_________________________________________________.
So we know that AC ^BD
since _______________________
____________________________. Therefore ABCD is a
rhombus because we know that all rhombuses _______________
_________________________________________________.
30
Homework- Lesson 8
DUE: 3/14
** Hand this page in **
Given:
A and D are supplementary
A and B are complementary
Prove: ABCD is a trapezoid
You may conduct this proof however you choose.
D
A
C
B
31
Classwork- Lesson 9
1.
Given :
ABCD is a quadrilateral,
BAC and ACB are
complementary,
E is the midpoint of
Proving Special Parallelograms
Statement
1) ABCD is a quadrilateral
2) E is the midpoint of
1)
2)
AC and BD
AC and BD,
Prove: ABCD is a rectangle
D
C
E
A
B
3) AC and BD bisect each
other
3)
4) ABCD is a parallelogram
4)
5) BAC and CAB
are complementary
5)
6) mBAC + mCAB = 90
6)
7) mBAC + mCAB +
mCBA = 180
7)
8) mCBA = 90
8)
9) ABCD is a rectangle
9)
Reason
Classwork- Lesson 9
32
2. Given:
ABCD is a quadrilateral,
A  C,
A and B are supplementary,
C and D are supplementary,
Prove:
ABCD is a parallelogram,
D
C
A
B
3.
Given:
ABCD is a quadrilateral,
1  2,
3  4,
Prove:
ABCD is a parallelogram
D
C
4
2
1
B3
A
B
33
Challenge Proofs
**Hand in on separate sheet for Extra Credit!!**
1.
2.
3.
4.
5.
34
Lesson 9- Homework
**Hand this page in**
You may conduct these proofs however you choose.
1.
Given:
Quadrilateral ABCD,
AB  CD,
BC  DA,
AB  BC
Prove: ABCD is a rhombus
D
C
E
A
B
2.
Given:
Parlallelogram ABCD
CBF  ADE
AE  FC
EF  DB
EF  DB
Prove:
EBFD is a quare
D
A
E
C
F
B