5.1 Properties of Parallelograms

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Chapter 5 Quadrilaterals

Name ________________

Da
te _________________

Period _______________


5.1 Properties of Parallelograms

Vocabulary:
parallelogram





The following theorems deal with common properties of parallelograms. Proofs using these theorems will
involve what you know about parallel lines (due to the sides) and congruent triangles
(since you can split
a parallelogram into triangles).


Theorem 5
-
1

Opposites sides of a parallelogram are _______________.


Theorem 5
-
2

Opposite angles of a parallelogram are ______________.


Theorem 5
-
3

Diagonals of a parallelogram _____________________
_______.



Examples:


Chapter 5 Quadrilaterals

Name ________________

Da
te _________________

Period _______________

5.2 Ways to Prove that Quadrilaterals Are Parallelograms

There are 5 ways to prove that a quadrilateral is a parallelogram.

F
rom the def
inition of a parallelogram:


I
f both pairs of
__________________________________________________
_
, then it is a parallelogram.

The following theorems will give you the remaining 4 ways to prove a quadrilateral is a parallelogram.


Theorem 5
-
4

If both pairs of _______________________________________________________________, then the
quadrilateral is
a parallelogram.


Theorem 5
-
5

If one pair of ________________________________________________________________, then the
quadrilateral is a parallelogram.


Theorem 5
-
6

If both pairs of _______________________________________________________________, then th
e
quadrilateral is a parallelogram.


Theorem 5
-
7

If the diagonals of a quadrilateral ___________________________________, then the quadrilateral is a
parallelogram.

Open your textbooks to p173 and do Classroom Exercises 1
-
11 all. Note: #6 has two answers.

1.







6.


2.








7.


3.








8.


4.








9.


5.







10.




11.





Find x and y if the following are parallelograms.

12.





13.





14.






In summary, the
5 Ways to Prove that a Quadrilateral is a Parallelogram

1.


2.


3.


4.


5.

Chapter 5 Quadrilaterals

Name ________________

Da
te _________________

Period _______________

5.3

Th
eorems Involving Parallel Lines

Four new theorems about parallel lines. We’ll be using these with the previous theorems to find lengths of
other sides, and of course, solve for x.

Theorem 5
-
8

If two lines are parallel, then all points on one line are _____
___________________________________.





Theorem 5
-
9

If three parallel lines cut off congruent segments on one transversal, then they cut off ________________


____________________________.




Theorem 5
-
10

A line that contains the midpoint of one si
de of a triangle and is parallel to


another side passes through the ________________________________________.


Theorem 5
-
11

The segment that joins the midpoints of two sides of a triangle…


a)

Is ____________ to the third side. (converse to Theorem 5
-
10)


b)

Is _____________ as long as the third side.


Given: X is the midpoint of AB; Y is the midpoint of BC; Z is the midpoint of AC.

Note: We know that XY and AC are parallel since _____________________________.

a)

If AC = 24, then XY = _____.

b)

If AB = 10, t
hen YZ = _____.

c)

If XZ = 2x + 3, then BC = ___________.

d)

If AB = 9, BC = 8, AC = 6, then the perimeter of
Δ
XYZ = ________.

e)

If the perimeter of
Δ
XYZ = 24, then the perimeter of
Δ
ABC = ______.

f)

Name the 4 congruent triangles.



g)

If XY = 3x + 5 and AC = 8x


3, x = ______.




Given: What is noted in the diagram.

h)

If RS = 12, then ST = _____.

i)

If AB = 8, then

BC = _____.

j)

If AC = 20, then AB = _____.

k)

If AC = 10x, then BC = _____.

Chapter 5 Quadrilaterals

Name ________________

Da
te _________________

Period _______________

5.4 Special Parallelograms

rectangle



Why is every rectangle a parallelogram?



rhombus



Why is every rhombus a parallelogram?


square



Why is every square a parallelogram, r
ectangle, AND rhombus?


Note: Since all of the above are parallelograms, all of the properties from parallelograms exist. They also
have other properties.

Theorem 5
-
12

The diagonals of a rectangle are ____________.


Theorem 5
-
13

The diagonals of a rhombu
s are __________________.


Theorem 5
-
14

Each diagonal of a rhombus _________________________________.


Theorem 5
-
12 can also give you an interesting theorem about right triangles.

Theorem 5
-
15

The midpoint of the hypotenuse of a right triangle is…


_____
________________________________________.

Note: Label the median.

And the following theorems are ways to prove a parallelogram is a specific one.


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 ______________.

Given: MNOP is a rectangle.

M


N

Given: ABCD is a rhombus.


m
LMN = 29°
; LN = 12




m
B
C
E

= 62°



A



B






L

1. m
PON = _____





5. m
ACD = _____


E

2. m
PMO = _____


P


O

6. m
DEC = _____

3. PL = _______





7. m
EDC = _
____

4. MO = ______





8. m
ABC = _____

D


C





__



X

Given: T is the midpoint of XZ.




T

9. If XT = 7, then TZ = ______ and XZ = _____

and TY = _______.

10. What is the labeled median? _______

Y




Z

Chapter 5 Quadrilaterals

Name ________________

Da
te _________________

Period _______________

5.5
Trapezoids

The last of the quadrilaterals that we will cover are trapezoids.
You will need your knowledge of parallel
lines to help you.


trapezoid




base



leg



isosceles trapezoid



Theorem 5
-
18

Base angles of an isosceles trapezoid are ________
_____.



median (of a trapezoid)



Note: the median of a trapezoid is NOT the same as the median of a triangle.

Theorem 5
-
19

The median of a trapezoid …


a) is ___________ to the bases


b)

has a length equal to the _________________________________.





___



___

In trapezoid ABCD, EF is the median.

Draw EF.





D


C


1. AB = 25, DC = 13, EF = ______






E



F

2. AE = 11, FB = 8, AD = ______, BC = _____

3. AB = 29, EF = 24, DC = _____

4. AB = 7y + 6, EF = 5y


3, DC = y


5,
y = ______





A




B










In

ABC, X, Y, and Z are midpoints of their respective sides.

AB = 12, BC = 16, AC = 16

5. Name 3 trapezoids in the figure.



6. Name an isosceles trapezoid in the figure and find its perimeter.




For 7
-
8
,
given the bases lengths

of a trapezoid
, find the median.



7. base = 12, 30


8
. base = 24, 34





9. Assume an isosceles trapezoid has one angle = 62°. Find the measure of the other 3 angles.