GEOMETRY
CHAPTER 5:
QUADRILATERALS
NAME
Page 1 of 36
Section 5-1: Properties of Parallelograms
Definition of a ____________________ (
) – a quadrilateral with both pairs of opposite sides parallel
I. Record the length of each side (in cm) and the measure of each angle of this parallelogram. Use the
measurements to make three conclusions.
A
AB = ________
BC = ________
DC = ________
AD = ________
m A = _______
m B = _______
m C = _______
m D = _______
D
B
C
1. ____________________________________________________________________
2. ____________________________________________________________________
3. ____________________________________________________________________
II. Using a ruler, draw in ̅̅̅̅ and ̅̅̅̅ . Label the intersection of these segments X. Find the following
measures. Make a fourth conclusion.
AC = ________
DB = ________
AX = ________
DX = ________
CX = ________
BX = ________
4. ____________________________________________________________________
III. Given: AB = 8, BC = 6, AC = 6.1, BD = 11.8, m A = 130°
Mark the given information in the parallelogram below. Draw in the diagonals and label their intersection “X”.
Find the following without using a ruler. (Base your answers on the 4 conclusions you made above).
DC = _______
AD = ________
m B = ______
m C = _______
m D = ______
AX = ________
DX = ________
BX = ________
CX = ________
Page 2 of 36
THEOREM
Opposite sides of a parallelogram are __________________.
THEOREM
Opposite __________ of a parallelogram are congruent.
THEOREM
Diagonals of a parallelogram ___________ each other.
Examples:
Classify each statement as true or false.
1. Every parallelogram is a quadrilateral.
1. ______________________
2. Every quadrilateral is a parallelogram.
2. ______________________
3. All angles of a parallelogram are congruent.
3. ______________________
4. All sides of a parallelogram are congruent.
4. ______________________
5. In
RSTU, ̅̅̅̅ ̅̅̅̅.
5. ______________________
6. In
ABCD, if m A = 50°, then m C = 130°.
6. ______________________
7. In
XWYZ, ̅̅̅̅
7. ______________________
8. In
ABCD, ̅̅̅̅ and ̅̅̅̅ bisect each other.
̅̅̅̅̅ .
8. ______________________
In Exercises 9 and 10, quad RSTU is a parallelogram. Find the values of x, y, a, and b.
6
R
9.
y°
S
x°
10.
R
S
x°
9
y°
9
12
b
b
45°
80°
U
T
a
35°
U
T
Each figure in Exs. 11 and 12 is a parallelogram with its diagonals drawn. Find the values of x and y.
3y + 4
11.
12.
18
4x
8
13
4y – 2
22
2x + 8
Page 3 of 36
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Page 5 of 36
Section 5-2: Ways to Prove that Quadrilaterals Are Parallelograms
1. Show that both pairs of opposite sides are ______________.
(definition)
2. Show that both pairs of opposite sides are _______________.
(theorem)
3. Show that one pair of opposite sides are ____________
and _______________. (theorem)
4. Show that both pairs of opposite angles are ________________.
(theorem)
5. Show that the diagonals __________ each other.
(theorem)
Examples:
Complete with always, sometimes, or never.
1. The diagonals of a quadrilateral ___________________ bisect each other.
2. If the measures of two angles of a quadrilateral are equal, then the quadrilateral is ____________________ a
parallelogram.
3. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is
_________________ a parallelogram.
4. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is _________________
a parallelogram.
5. To prove a quadrilateral is a parallelogram, it is ____________ enough to show that one pair of opposite
sides is parallel.
Page 6 of 36
Decide if each quadrilateral must be a parallelogram. If the answer is yes, state the definition or theorem that
applies.
6.
17
7.
4
4
5
5
17
__________________________________
__________________________________________
__________________________________
__________________________________________
8.
9.
100°
80°
6
6
100°
80°
__________________________________
__________________________________________
__________________________________
__________________________________________
10.
5
8
8
_______________________________
5
_______________________________
State the principal definition or theorem that enables you to deduce, from the information given, that quad.
ABCD is a parallelogram.
11. BE = ED; CE = EA
12.
BAD  DCB; ADC  CBA
_____
_____
_____
B
C
_____
13. BC || AD ; AB || DC
E
_____
_____
_____
_____
14. BC  AD , AB  DC
_____
_____
_____
A
D
_____
15. BC || AD ; BC  AD
Page 7 of 36
A
Opposite sides are parallel.
D
Opposite sides are congruent.
Opposite angles are congruent.
B
C
U
L
Consecutive angles are
supplementary.
A diagonal of a
divides it into 2  triangles.
The diagonals of a
bisect each other.
(Note: Bisect means to cut in half.)
A
P
PAUL
Page 8 of 36
Page 9 of 36
Section 5-3: Theorems Involving Parallel Lines
Theorem 1: If 2 lines are parallel, then all points on one line are _________________ from
the other line.
1.
What do you think equidistant means?
_________________________________________________
4
x
y
2.
According to the theorem above, x = ____ and y = _____.
3.
Why do you think the 3 lines were drawn perpendicularly to the 2 parallel lines?
________________________________________________
Theorem 2: If 3 parallel lines cut off congruent segments on one transversal, then they cut
off _________________ segments on every transversal.
7
x
4.
y
6
8
z
According to Theorem 5-9 on the previous page, x = _____, y = _____,
and z = _____.
Page 10 of 36
5.
Solve for a.
5a
10
10
15
Theorem 3: A line that contains the midpoint of one side of a triangle and is parallel to another
side passes through the ___________________ of the third side.
A
B
D
C
E
6. According to Theorem 3, if AB = BC and ̅̅̅̅
̅̅̅̅ , then _____________.
7. If AC = 14, then AB = _____ and BC = _____.
8. If AD = 14, then DE = _____ and AE = _____.
Theorem 4: The segment that joins the midpoints of 2 sides of a triangle:
1._________________________________________
2._________________________________________
9. Using the diagram above, if B is the midpoint of ̅̅̅̅ and D is the midpoint
of ̅̅̅̅ , then _____ =
(_____) and ̅̅̅̅ || _____.
10. If AB = BC = 6 and AD = DE = 8 and BD = 12, then CE = _____.
Page 11 of 36
11. Given: R, S, and T are midpoints of the sides of  ABC.
Complete the table.
B
a.
c.
A
BC
AC
12
14
18
15
22
b.
S
R
AB
T
ST
RT
RS
9
7.8
10
10
C
12. Given: ⃡ || ⃡ || ⃡ ;
̅̅̅̅
̅̅̅̅
Complete.
R
S
T
A
a. If RS = 12, then ST = ______.
b. If AB = 8, then BC = ______.
B
c. If AC = 10x, then BC = ______.
C
13. Given: Points X, Y, and Z are midpoints of ̅̅̅̅ , ̅̅̅̅ , and ̅̅̅̅ .
Complete.
B
a. If AC = 24, then XY = ______.
b. If AB = k, then YZ = ______.
X
Y
c. If XZ = 2k + 3, then BC = _______.
d. If AB = 9, BC = 8, AC = 6, then the perimeter
of XYZ = ______.
A
Z
C
e. If the perimeter of XYZ = 24, then the
perimeter of ABC = ______.
f. Name all congruent triangles.
Page 12 of 36
14. Name all points that must be midpoints of the sides of  ABC.
C
11
L
M
11
B
A
N
15. Given: AB = BC = CD
U
•
Complete.
T
•
a. If RS = 6, then SU = ______.
S
•
•D
•C
b. If RT = 6x + 2 and TU = 10, then x = ____.
RR
•B
•
•A
Page 13 of 36
Page 14 of 36
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Page 16 of 36
Review for quiz on 5-1 to 5-3
I.
Quadrilateral KLMN is a parallelogram. Complete each statement.
1.
̅̅̅̅̅ 
2.
̅̅̅̅̅ 
_____
4.
1
_____
M
4
5
6
NML  _____
3.
5.
N
_____
3
X
7
K
KNM is supplementary to
8
_____ or
1
2
L
_____.
_________________________________________________
II.
Decide if it is possible to prove that the quadrilateral below is a
parallelogram. Answer yes if it is possible. Answer no if it is not possible.
6.
̅̅̅̅  ̅̅̅̅; ̅̅̅̅
7.
̅̅̅̅
̅̅̅̅; ̅̅̅̅
8.
̅̅̅̅
̅̅̅̅; ̅̅̅̅
̅̅̅̅
9.
1
3; 2
4
10.
̅̅̅̅
̅̅̅̅; ̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
_____
_____
F
E
4
_____
_____
3
G
1
A
2
B
_____
_________________________________________________
Page 17 of 36
III. Complete each statement with the number that makes the statement true.
11.
DE = _____
A
12.
m ADE = _____
54°
10
13.
m ABC = __
12
14.
m C = ____
15.
EC = _____
D
B
16.
AC = _____
17.
DF = _____
18.
m DFC = _____
E
41°
C
F
18
T
19.
If PQ = 5, then PS = _____.
P
20.
W
If YZ = 7, then XY = _____.
Q
21.
If QX = 12, then PW = _____.
22.
If WZ = 24, then WX = _____.
R
S
X
Y
Z
Page 18 of 36
IV.
Solve for x and y in each parallelogram.
23.
x = _____ y = _____
24.
x = _____ y = _____
(3x + 4)°
20
(5x + 16)°
26
(6x + 2)
(2y)
(4y)°
Page 19 of 36
Section 5-4: Special Parallelograms
I. Write the briefest definition that you can for each of the following
figures. Imagine that someone is using your definition to draw the
figure.
Parallelogram - __________________________________________
______________________________________________________
Rectangle - _____________________________________________
______________________________________________________
Rhombus - ______________________________________________
______________________________________________________
Square - _______________________________________________
______________________________________________________
PARALLELOGRAM
RECTANGLE
RHOMBUS
SQUARE
Page 20 of 36
II. THEOREMS
THEOREM 5-12
The ______________ of a rectangle are congruent.
THEOREM 5-13
The diagonals of a rhombus are ___________________.
THEOREM 5-14
Each diagonal of a rhombus ___________ two angles of the rhombus.
THEOREM 5-15
The midpoint of the hypotenuse of a right triangle is __________________ from the
three vertices.
THEOREM 5-16
If an angle of a parallelogram is a right angle, then the parallelogram is a
________________________.
THEOREM 5-17
If two consecutive sides of a parallelogram are congruent, then the parallelogram is a
______________________.
III. Answer the following questions about the 4 figures in section I.
1.
Which ones are parallelograms? __________________________
_________________________________________________
2.
Is a rectangle a square? If not, why? _____________________
_________________________________________________
3.
Is a square a rectangle? If not, why? _____________________
_________________________________________________
4.
Is a rhombus a square? If not, why? ______________________
_________________________________________________
5.
Is a square a rhombus? If not, why? ______________________
_________________________________________________
6.
Is a rhombus a rectangle? If not, why? ____________________
_________________________________________________
Page 21 of 36
Examples:
I.
Quadrilateral ABDC is a rectangle.
AD = 12
BC = _____
A
B
BE = _____
EC = _____
E
mEAC = 60
mBDA = _____
mBAE = _____
C
D
mACE = _____
II.
A
Quadrilateral ABCD is a rhombus.
B
mBAE = 50; AD = 4
mAEB = _____
E
mDAE = _____
mACD = _____
mABE = _____
DC = _____
D
C
Page 22 of 36
III. Quadrilateral ABCD is a square.
A
B
AC = 12
EB = _____
E
ED = _____
mBEC = _____
mEBC = _____
D
IV.
ABC is a right
C
; M is the midpoint of ̅̅̅̅.
A
If AM = 7, then MB = _____,
AB = _____, and CM = _______.
M
If AB = x, then AM = _____,
MB = ______, and MC = ______.
C
B
Page 23 of 36
Page 24 of 36
Rectangles, Rhombuses and Squares
Rectangle
with 4 right  s
Diagonals are 
Rhombus
with 4  sides
Diagonals are 
Square
with 4 right  s &
4  sides
Diagonals are  & 
Page 25 of 36
Page 26 of 36
Section 5-5: Trapezoids
1.
Trapezoid - ___________________________________
____________________________________________
2.
The parallel sides of a trapezoid are the ______________.
3.
The other sides are the _____________ .
4.
A trapezoid with congruent legs is called an ___________
________________.
Theorem : _________________________________________
________________________________________________.
Diagonals of an isosceles trapezoid are ______________.
Example: Given trapezoid ABCD is isosceles, find mA, mB, and mC.
B
A
D
110
C
Page 27 of 36
The median of a trapezoid is the segment that ______________
________________________________________________.
Theorem: The median of a trapezoid
1) __________________________________
2) _________________________________
__________________________________
Given:
EF
is a median of trapezoid ABCD.
a. If AB = 5 and CD = 15, find EF.
A
B
E
F
D
C
b. If AB = x – 3, CD = 2x - 4, and EF = 10, find x.
A
E
D
B
F
C
Page 28 of 36
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Page 31 of 36
Trapezoids
1
2
3
4
Exactly one pair of parallel sides
Exactly two pairs of consecutive angles are
supplementary.
 1 &  2,
 3 &  4 are supplementary.
base 1
l eg
l eg
base 2
Parallel sides are bases; non-parallel sides are legs.
Isosceles trapezoids have congruent legs.
Page 32 of 36
Page 33 of 36
Page 34 of 36
Review for Chapter 5 Test
I.
Write the letter of every special quadrilateral that has the given property.
A. Parallelogram
B. Rectangle
C. Rhombus
D. Square
1.
All angles are right angles.
__________________
2.
All sides are congruent.
__________________
3.
Diagonals are congruent.
__________________
4.
Diagonals bisect each other.
__________________
5.
Diagonals are perpendicular.
__________________
6.
The quadrilateral is equiangular & equilateral.
__________________
7.
Both pairs of opposite sides are parallel.
__________________
8.
Exactly one pair of opposite sides is parallel.
__________________
9.
Both pairs of opposite sides are congruent.
__________________
10.
Each diagonal bisects two angles.
__________________
II.
Quadrilateral DKRT is a parallelogram.
11.
If DK = 12 and KR = 8, then TR = _____.
12.
13.
If DR = 28 and KT = 18, then HR = _____.
If ̅̅̅̅ ̅̅̅̅, then the parallelogram must be a
______________. (rectangle, rhombus, square)
14.
If m 1 = 30 and m 8 = 40, then m RTD = _____.
15.
If m 2 = 45 and m 3 = 55, then m 6 = _____.
16.
If TH = 2x + 1 and KH = 4x, then x = _____.
E. Trapezoid
D
4
5
T
3
2
H
6
7
K
1
8
R
Page 35 of 36
III.
State whether the given information is sufficient to prove that quad. MNOP is a
parallelogram.(yes or no)
17.
̅̅̅̅
̅̅̅̅; ̅̅̅̅
̅̅̅̅
____________
P
18.
19.
1  5; 4  8
̅̅̅̅
̅̅̅̅̅; ̅̅̅̅
O
____________
7
̅̅̅̅̅
5
____________
20.
S
1
____________
21.
7  3; 4  8
____________
22.
1  2; 3  4
____________
̅̅̅̅
6
8
̅̅̅̅
̅̅̅̅; ̅̅̅̅̅
̅̅̅̅
Given: ̅̅̅̅̅
23.
If RS = 8, then TV = _____.
24.
If RV = 21, then RT = _____.
25.
If RT = 3x and SV = x + 8, then x = _____.
V.
Given: ̅̅̅̅̅ is the median of trapezoid HIJK.
26.
If KJ = 7 and HI = 15, then LM = _____.
N
̅̅̅̅
R
W
S
X
T
28.
If trapezoid HIJK is isosceles
and m I = 85, then m K = _____.
29.
If HI = 4x, LM = 2x + 3,
and KJ = x – 2, then x = _____.
Y
V
K
If HI = 22 and LM = 17, then KJ = _____.
3
M
IV.
27.
4
2
L
H
Z
J
M
I
Page 36 of 36