1. Properties of Parallelogram
Geometry
Properties of Parallelogram
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2. Warm up
An interior angle measure of a regular polygon is given. Find the number of sides and the measure of each exterior angle:
1) 120°
2) 135°
3) 156°
hexagon, 60° ; 2) heptagon, 45° ; 3) 15 - gon, 24°
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3. Properties of Parallelograms
Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals have their own names.
A quadrilaterals with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol □. A B C D
parallelogram ABCD
□ ABCD
AB || CD; BC || DA
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4. Properties of Parallelograms
Theorem 1:
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
(□ -> opp. sides ) A B C D
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5. Proof of Theorem 1: A B C D 1 2 3 4
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6. Properties of Parallelograms
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
(□ -> opp. angles ) A B C D
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7. Properties of Parallelograms
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
(□ -> cons. ∕s supp.) A B C D
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8. Properties of Parallelograms
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
(□ -> diags. bisect each other) A B C D 1 2 3 4 Z
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9. In □ PQRS, QR = 48 cm, RT = 30 cm, and /QPS = 73°. Find each measure.
Racing application
In □ PQRS, QR = 48 cm, RT = 30 cm, and /QPS = 73°. Find each measure. Q R S P T
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10. In □KLMN, LM = 28 in., LN = 26 in. and m/LKN = 74°. Find each measure:
Now you try!
In □KLMN, LM = 28 in., LN = 26 in. and m/LKN = 74°. Find each measure:
1a) KN
1b) m/NML
1c) LO M L K N O
ANSWER: 1a) 28 in.; 1b) 74° ; 1c) 13 in.
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11. Using Properties of Parallelogram to Find Measures
ABCD is a parallelogram. Find each measure. B C D A
5x+19
(6y+5)°
(10y-1)°
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12. B C D A
5x+19
(6y+5)°
(10y-1)°
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13. ABCD is a parallelogram. Find each measure:
Now you try!
ABCD is a parallelogram. Find each measure: E F G H J
4z-9 2z
w+8 3w
2a) JG
2b) FH
2a) 12; 2b) 18
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14. Parallelograms in the coordinate plane
Three vertices of □ABCD are A(1,-2), B(-2,3) and D(5,-1). Find the coordinates of vertex C. B A D 5 6 C -3 -2 x y
Since ABCD is a parallelogram, both pairs of opposite sides must be parallel.
Step1: Graph the given points.
Step2: Find the slope of AB by counting the units from A to B.
The rise from -2 to 3 is 5.
The run from 1 to -2 is -3.
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15. Step3: Start at D and count the same number of units.5 6 C -3 -2 x y
Step3: Start at D and count the same number of units.
The rise from -1 to 4 is 5.
The run from 5 to 2 is -3.
Label (2,4) as vertex C.
Step4: Use the slope formula to verify that BC || AD.
Slope of BC = 4 – 3 = 1
2 – (-2) 4
Slope of AD = -1 – (-2) = 1
5 –
The coordinates of vertex C are (2, 4).
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16. Now you try!
3) Three vertices of □PQRS are P(-3,-2), Q(-1,4) and S(5,0). Find the coordinates of vertex R.
3) (7,6)
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17. Using properties of Parallelograms in a proofB C D A E
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18. K J H G N M L
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19. 4) Use the figure above to write a two-column proof.
Now you try! J H G N M L
4) Use the figure above to write a two-column proof.
5) 45°
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20. Now some problems for you to practice !
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21. Assessment C D A B E BD CD 3) BE 4) /ABC 5) /ADC 6) /DAB CONFIDENTIAL
1) 36; 2) 17.5; 3) 18; 4) 70°; 5) 70°; 6) 110°
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22. Find the values of x and y for which ABCD must be a parallelogram:7) 8)
7) x = 5; 8) x = 1.6; y = 1
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23. 9) Three vertices of llgm DFGH are D(-9,4), F(-1,5) and G(2,0)
9) Three vertices of llgm DFGH are D(-9,4), F(-1,5) and G(2,0). Find the coordinates of vertex H.
9) (-6,-1)
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24. P V T S R Q
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25. Properties of Parallelograms
Let’s review
Properties of Parallelograms
Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals have their own names.
A quadrilaterals with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol □. A B C D
parallelogram ABCD
□ ABCD
AB || CD; BC || DA
CONFIDENTIAL

26. Properties of Parallelograms
Theorem 1:
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
(□ -> opp. sides ) A B C D
CONFIDENTIAL

27. Proof of Theorem 1: A B C D 1 2 3 4
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28. Properties of Parallelograms
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
(□ -> opp. angles ) A B C D
CONFIDENTIAL

29. Properties of Parallelograms
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
(□ -> cons. ∕s supp.) A B C D
CONFIDENTIAL

30. Properties of Parallelograms
Theorem
Hypothesis
Conclusion
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
(□ -> diags. bisect each other) A B C D 1 2 3 4 Z
CONFIDENTIAL

31. Parallelograms in the coordinate plane
Three vertices of □ABCD are A(1,-2), B(-2,3) and D(5,-1). Find the coordinates of vertex C. B A D 5 6 C -3 -2 x y
Since ABCD is a parallelogram, both pairs of opposite sides must be parallel.
Step1: Graph the given points.
Step2: Find the slope of AB by counting the units from A to B.
The rise from -2 to 3 is 5.
The run from 1 to -2 is -3.
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32. Step3: Start at D and count the same number of units.5 6 C -3 -2 x y
Step3: Start at D and count the same number of units.
The rise from -1 to 4 is 5.
The run from 5 to 2 is -3.
Label (2,4) as vertex C.
Step4: Use the slope formula to verify that BC || AD.
Slope of BC = 4 – 3 = 1
2 – (-2) 4
Slope of AD = -1 – (-2) = 1
5 –
The coordinates of vertex C are (2, 4).
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33. Using properties of Parallelograms in a proofB C D A E
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34. You did a great job today!
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