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 360ْ. • If you know a shape is a parallelogram, then it has 4 big properties (theorems).
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 Presentation
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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).
• 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
Theorem #1: The opposite sides of a parallelogram are congruent.
Section 5.1 - Parallelograms
Theorem #1: The opposite sides of a parallelogram are congruent.
Proving Theorem #1:
Section 5.1 - Parallelograms
Theorem #2: The opposite angles of a parallelogram are congruent
Section 5.1 - Parallelograms
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
~~ ~
~
Theorem #3: The diagonals of a parallelogram bisect each other.
Section 5.1 - Parallelograms
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
~ ~
~~~
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 ْ