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Unit 5 - Quadrilaterals

Unit 5 - Quadrilaterals. MM1G3 d. Essential Quesitons. Parallelograms Examples. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. K L. J H.

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Unit 5 - Quadrilaterals

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  1. Unit 5 - Quadrilaterals MM1G3 d

  2. Essential Quesitons

  3. Parallelograms Examples

  4. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. K L J H

  5. There are 5 properties associated with parallelograms. Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other.

  6. Given: JKLH is a parallelogram Prove: K L 1 4 3 2 J H Since diagonal KH is also a transversal, angle 1 and angle 2 are congruent as they are alternate interior angles. Likewise, angle 3 and angle 4 are congruent. Since KH is congruent to itself, triangle JKH and triangle LHK are congruent by ASA. (By drawing diagonal JL, it can similarly be proven that triangle JKL and triangle LHJ are congruent and, thus, .)

  7. Example 1: Find the perimeter of parallelogram . Solution: Opposite sides of a parallelogram are congruent. So, ZY = 12 cm and WZ = 8 cm. Therefore, P = 12cm + 8cm + 12cm + 8cm. P = 40 cm Z Y W 8 cm 12 cm X

  8. Given: JKLH is a parallelogram Prove: K L 1 4 3 2 J H

  9. Given: JKLH is a parallelogram Prove: Diagonals KH and JL bisect each other Statements Reasons K L 1 5 4 M 3 6 2 J H

  10. Example 2: is a parallelogram. What is the length, in units, of ? A B 3x 2y 4x y+2 D C

  11. Summary A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Properties of parallelograms: Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other.

  12. Investigate Parallelograms

  13. Try This: B

  14. Try This D

  15. Try This: D

  16. Try This: C Solve for w:

  17. Try This: B

  18. RectanglesExamples

  19. A rectangleis a parallelogram with four right angles. Since ABCD is also a parallelogram, it has the following properties: A D B C

  20. A D B C A rectangle has an additional property – the diagonals are congruent. In rectangle ABCD above, .

  21. To investigate these properties, click below. http://www.geogebra.org/en/upload/files/english/dfreeston/rectangle_properties.html As you drag point A or point B, you should notice the values in the left margin changing. By clicking on various pieces of the rectangle, the measurement will be highlighted. You can verify the properties of the rectangle. Notice that as you drag point A or point B, the properties of the rectangle remain. For example, the diagonals are always congruent and bisect each other.

  22. Example 1:

  23. Example 2: Solution:

  24. Summary A rectangleis a parallelogram with four right angles. Properties of a rectangle: Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other. The diagonals are congruent

  25. Investigate Rectangles

  26. D

  27. Try This: Check this! B

  28. Try This: D

  29. Try This: D

  30. Essential Questions

  31. RhombusExamples

  32. A rhombus is a parallelogram with all sides congruent. A B D C Since ABCD is also a parallelogram, the following statements are true.

  33. A rhombus has two additional properties. • the diagonals are perpendicular • the diagonals bisect opposite angles A B D C

  34. To investigate these properties, click below. http://www.geogebra.org/en/upload/files/english/dfreeston/rhomprop.html As you drag point B or point C, you should notice the measurements changing. However, you should see that the diagonals are always perpendicular. You should also see each of the angles are bisected, no matter the size of the rhombus. Notice that as you drag point B or point C, the rhombus will have the 5 properties for a parallelogram as well. For example, the diagonals bisect each other and the opposite angles are congruent.

  35. Example 1: W

  36. Example 2:

  37. Summary A rhombusis a parallelogram with all sides congruent. Properties of a rhombus: Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. Consecutive angles are supplementary. The diagonals bisect each other. The diagonals are perpendicular. The diagonals bisect opposite angles.

  38. Investigate Rhombi

  39. Try This: B

  40. Try This: B

  41. Try This: D

  42. Try This: C

  43. SquaresExamples

  44. A square is a parallelogram with four right angles and all sides congruent. Notice that a square meets the requirements for both a rectangle (four right angles) and a rhombus (all sides congruent). A B D C

  45. To investigate the properties of a square, click below. http://www.geogebra.org/en/upload/files/english/dfreeston/square_properties.html As you drag point B, you should notice the measurements changing. However, you should see that the diagonals are always perpendicular and congruent. You should also see that the interior angles of the square are always 900 . Notice that as you drag point B, the square will have the 5 properties for a parallelogram as well. For example, the diagonals bisect each other and the consecutive angles are supplementary.

  46. Example 1:

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