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Quadrilaterals

Quadrilaterals . Chapter 6. Parallelogram. A quadrilateral with both pairs of opposite sides parallel. Rhombus. A parallelogram with four congruent sides. Rectangle. A parallelogram with four right angles. Square . A parallelogram with four congruent side and four right angles.

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Quadrilaterals

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  1. Quadrilaterals Chapter 6

  2. Parallelogram • A quadrilateral with both pairs of opposite sides parallel

  3. Rhombus • A parallelogram with four congruent sides.

  4. Rectangle • A parallelogram with four right angles

  5. Square • A parallelogram with four congruent side and four right angles.

  6. Kite • A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent

  7. Trapezoid • A quadrilateral with exactly one pair of parallel sides

  8. Isosceles Trapezoid • A trapezoid whose nonparallel opposite sides are congruent.

  9. Seven important types of quadrilaterals … • Parallelogram-has both pairs of opposite sides parallel • Rhombus-has four congruent sides • Rectangle-has four right angles • Square-has four congruent sides and four right angles • Kite-has two pairs of adjacent sides congruent and no opposite opposite sides congruent.

  10. Continued…. • Trapezoid-has exactly one pair of parallel sides. (you have same side interior angles) • Isosceles trapezoid-is a trapezoid whose non-parallel opposite sides are congruent

  11. Quadrilaterals Parallelograms Kites Rhombuses Rectangles Squares Trapezoids

  12. Classifying by Coordinate Method • Do you remember the slope formula? • Do you remember the distance formula that finds the distance between two points?

  13. Do you remember how to tell if two lines are parallel? • Do you remember how to tell if two lines are perpendicular?

  14. Graph quadrilateral QBHA. First, find the slope of each side. slope of QB = slope of BH = slope of HA = slope of QA = 4 – 4 –4 – 10 9 – 9 8 – (–2) 4 – 9 10 – 8 9 – 4 –2 – (–4) 5 2 5 2 = = = = – 0 0 BH is parallel to QA because their slopes are equal. QB is not parallel to HA because their slopes are not equal. Classifying Quadrilaterals Determine the most precise name for the quadrilateral with vertices Q(–4, 4), B(–2, 9), H(8, 9), and A(10, 4).

  15. Next, use the distance formula to see whether any pairs of sides are congruent. QB = ( –2 – ( –4))2 + (9 – 4)2 = 4 + 25 = 29 HA = (10 – 8)2 + (4 – 9)2 = 4 + 25 = 29 BH = (8 – (–2))2 + (9 – 9)2 = 100 + 0 =10 QA = (– 4 – 10)2 + (4 – 4)2 = 196 + 0 = 14 Classifying Quadrilaterals (continued) One pair of opposite sides are parallel, so QBHA is a trapezoid. Because QB = HA, QBHA is an isosceles trapezoid.

  16. In parallelogram RSTU, mR = 2x – 10 and m S = 3x + 50. Find x. If lines are parallel, then interior angles on the same side of a transversal are supplementary. m R + m S = 180 Draw quadrilateral RSTU. Label R and S. RSTU is a parallelogram. Given ST || RU Definition of parallelogram Classifying Quadrilaterals

  17. (continued) (2x – 10) + (3x + 50) = 180 Substitute 2x – 10 for m R and 3x + 50 for m S. 5x + 40 = 180 Simplify. 5x = 140 Subtract 40 from each side. x = 28 Divide each side by 5. Classifying Quadrilaterals

  18. Properties of Parallelograms 6.2

  19. Theorem 6-1: • Opposite sides of a parallelogram are congruent.

  20. Theorem 6.2 • Opposite angles of a parallelogram are congruent.

  21. Find the measure of the numbered angles for each parallelogram.

  22. Theorem 6.3 • The diagonals of a parallelogram bisect each other.

  23. Find the value of x in each parallelogram.

  24. Theorem 6.4 • If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Then If

  25. 6.3 Proving That a Quadrilateral is a Parallelogram

  26. Warm Up: • Given: • Prove:

  27. Theorem 6-5: • If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

  28. Given: • Prove:

  29. Theorem 6.6: • If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  30. Given: • Prove:

  31. Theorem 6.7: • If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

  32. Given: • Prove:

  33. Theorem 6.8: • If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

  34. Given: • Prove:

  35. Warm Up: • Given: • Prove:

  36. Warm Up: Show algebraically that ABCD is a parallelogram A (-1, 4) B (3, 2) C (2, -2) D (-2, 0) Hint: Think of the choices you have that prove a quadrilateral is a parallelogram…

  37. 6.4 Special Parallelograms

  38. Theorem 6.9: • Each diagonal of a rhombus bisects two angles of the rhombus.

  39. Theorem 6.10: • Diagonals of a rhombus are perpendicular.

  40. Theorem 6.11: • Diagonals of a rectangle are congruent.

  41. Example 3 Use Diagonals of a Rhombus ABCDis a rhombus. Find the value ofeach variable.

  42. Finding Angle Measure • MNPQ is a rhombus and m<N =120°. Find the measures of the numbered angles. N P 3 120° 4 1 2 M Q

  43. Finding Angles Measures • Find the measures of the numbered angles in the rhombus. m<1 : m<2 and m<3: 50° 1 m<4: 3 4 2

  44. Checkpoint 90 ANSWER 12 ANSWER 45 ANSWER Use Diagonals Find the value of x. 4. rhombusABCD 5. rectangleEFGH 6. squareJKLM

  45. Given Rectangle EFGH, find the value of x, y, FG, GH, and EH.

  46. Is the Parallelogram a Rhombus or a Rectangle?

  47. Theorem 6.12: • If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.

  48. Theorem 6.13: • If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

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