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Geometry Properties of Parallelogram

Geometry Properties of Parallelogram. 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°. Properties of Parallelograms.

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Geometry Properties of Parallelogram

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  1. Geometry Properties of Parallelogram CONFIDENTIAL

  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° CONFIDENTIAL

  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 parallelogram ABCD □ ABCD AB || CD; BC || DA D C CONFIDENTIAL

  4. Properties of Parallelograms Theorem 1: A B D C CONFIDENTIAL

  5. Proof of Theorem 1: 1 3 2 4 A B D C CONFIDENTIAL

  6. Properties of Parallelograms A B D C CONFIDENTIAL

  7. Properties of Parallelograms A B D C CONFIDENTIAL

  8. Properties of Parallelograms 1 A B 3 Z 2 4 D C CONFIDENTIAL

  9. Racing application In □ PQRS, QR = 48 cm, RT = 30 cm, and /QPS = 73°. Find each measure. T Q R P S CONFIDENTIAL

  10. 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 O M L N K CONFIDENTIAL

  11. Using Properties of Parallelogram to Find Measures ABCD is a parallelogram. Find each measure. 5x+19 (6y+5)° (10y-1)° B C A D CONFIDENTIAL

  12. 5x+19 (6y+5)° (10y-1)° B C A D CONFIDENTIAL

  13. Now you try! ABCD is a parallelogram. Find each measure: F G 4z-9 J w+8 2a) JG 2b) FH 3w 2z H E CONFIDENTIAL

  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. y C Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. B -3 5 5 x 6 -2 0 D 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. -3 A CONFIDENTIAL

  15. 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. y C B -3 5 5 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 – 1 4 The coordinates of vertex C are (2, 4). x 6 -2 0 D -3 A CONFIDENTIAL

  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. CONFIDENTIAL

  17. Using properties of Parallelograms in a proof A) E B C A D CONFIDENTIAL

  18. K H J G L N M CONFIDENTIAL

  19. Now you try! H J G L N M 4) Use the figure above to write a two-column proof. CONFIDENTIAL

  20. Now some problems for you to practice ! CONFIDENTIAL

  21. Assessment • BD • CD • 3) BE 4) /ABC 5) /ADC 6) /DAB E C D B A CONFIDENTIAL

  22. Find the values of x and y for which ABCD must be a parallelogram: 8) 7) CONFIDENTIAL

  23. 9) Three vertices of llgm DFGH are D(-9,4), F(-1,5) and G(2,0). Find the coordinates of vertex H. CONFIDENTIAL

  24. Q S T P R V CONFIDENTIAL

  25. 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 parallelogram ABCD □ ABCD AB || CD; BC || DA D C CONFIDENTIAL

  26. Properties of Parallelograms Theorem 1: A B D C CONFIDENTIAL

  27. Proof of Theorem 1: 1 3 2 4 A B D C CONFIDENTIAL

  28. Properties of Parallelograms A B D C CONFIDENTIAL

  29. Properties of Parallelograms A B D C CONFIDENTIAL

  30. Properties of Parallelograms 1 A B 3 Z 2 4 D C 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. y C Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. B -3 5 5 x 6 -2 0 D 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. -3 A CONFIDENTIAL

  32. 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. y C B -3 5 5 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 – 1 4 The coordinates of vertex C are (2, 4). x 6 -2 0 D -3 A CONFIDENTIAL

  33. Using properties of Parallelograms in a proof A) E B C A D CONFIDENTIAL

  34. You did a great job today! CONFIDENTIAL

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