section 5.1 - parallelograms
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Section 5.1 - Parallelograms. 11/15 Remember: a quadrilateral is any 4-sided polygon. Parallelograms are a type of quadrilateral with both pairs of opposite sides parallel. Recall: The sum of the interior angles of a parallelogram (and quadrilateral) is 360 ْ . - PowerPoint PPT PresentationTRANSCRIPT
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Section 5.1 - Parallelograms11/15
• Remember: a quadrilateral is any 4-sided polygon.
• Parallelograms are a type of quadrilateral with both pairs of opposite sides parallel.
• Recall: The sum of the interior angles of a parallelogram (and quadrilateral) is 360 ْ.• If you know a shape is a parallelogram, then it has 4 big properties (theorems).
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• Parallelograms are named by 4 vertices, start with any and go around.
Section 5.1 - Parallelograms
A B
D C
ABCD
BCDA
CDAB
CBAD DCBA
DABC
ADCB BADC
C l o c k w i s e
C o u n t e r C l o c k w i s e
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Theorem #1: The opposite sides of a parallelogram are congruent.
Section 5.1 - Parallelograms
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Theorem #1: The opposite sides of a parallelogram are congruent.
Proving Theorem #1:
Section 5.1 - Parallelograms
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Theorem #2: The opposite angles of a parallelogram are congruent
Section 5.1 - Parallelograms
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Theorem #2: The opposite angles of a parallelogram are congruent
Proving Theorem #2:
Using the proof from Theorem #1, we know (given): andby ASA
Therefore, with CPCTC, <B = <D.And because <1 = <2 and <3 = <4, by using angle addition postulate and
substitution, we can also conclude that <A = <C.
Section 5.1 - Parallelograms
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Theorem #3: The diagonals of a parallelogram bisect each other.
Section 5.1 - Parallelograms
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Theorem #3: The diagonals of a parallelogram bisect each other.
Proving Theorem #3:
Using the proof from Theorem #1 and #2,
we know (given): ΔABC = ΔCDA and ΔDBA = ΔBDC
Section 5.1 - Parallelograms
5
6 7
8
5
6 7
8
O
OO
By ASA ΔDCO = ΔBAO
By CPCTC OC = AO
By midpt theorem
AO = ½AC
therefore O is midpoint of AC
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Theorem #4: The same-side interior angles (consecutive angles) of a parallelogram are supplementary angles.
Section 5.1 - Parallelograms
A
B
C
DےA +ے B = 180 D = 180 ے+ Cےْ ْA = 180 ے+ Cے B = 180 ے+ Dےْ ْ
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The following quadrilaterals are parallelograms.
Example 1
Solve for w,x,y,z.
Example 2
Solve for j,k,l,m,n.
Parallelograms – We Do
B
D
A
C115º
10
18w x
y
z
F
H
E
G
55º 65º
jk
l40ºn
m
w = 115º x = 65ºy = 10 z = 18
j = 60º k = 55º
l = 120º m = 40º
n = 20º
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• Textbook practice: Pg 168 #2-8
Parallelograms – You Do