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Chapter 5

Mid-Term Exam Review Project. Chapter 5. Quadrilaterals. Chapter 5: Break Down . 5-1: Properties of Parallelograms 5-2: Ways to Prove that Quadrilaterals are Parallelograms 5-3: Theorems Involving Parallelograms 5-4 : Special Quadrilaterals 5-5: Trapezoids. Chapter 5: Section 1.

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Chapter 5

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  1. Mid-Term Exam Review Project Chapter 5 Quadrilaterals

  2. Chapter 5: Break Down • 5-1: Properties of Parallelograms • 5-2: Ways to Prove that Quadrilaterals are Parallelograms • 5-3: Theorems Involving Parallelograms • 5-4 : Special Quadrilaterals • 5-5: Trapezoids

  3. Chapter 5: Section 1 Properties of Parallelograms

  4. Parallelograms A parallelogram is a quadrilateral with both pairs of opposite sides parallel. But there are more characteristics of parallelograms than just that…

  5. These additional characteristics are applied in several theorems throughout the section.

  6. Theorem 5-1 “Opposite sides of a parallelogram are congruent.”

  7. This theorem states that the sides in a parallelogram that are opposite of each other are congruent. This means that in ABCD, side AB is congruent to side CD and side AC to side BD. A B D C

  8. Theorem 5-2 “Opposite Angles of a parallelogram are congruent”

  9. This theorem states that in a parallelogram, the opposite angles of a parallelogram are congruent.

  10. Theorem 5-3 “Diagonals of a parallelogram bisect each other”

  11. This theorem states that in any parallelogram the diagonals bisect each other.

  12. Section 5-2 Ways to Prove that Quadrilaterals are Parallelograms

  13. 5 Ways to Prove that a Quadrilateral is a Parallelogram • Show that both pairs of opposite sides are parallel • Show that both pairs of opposite sides are congruent • Show that one pair of opposite sides are both congruent and parallel • Show that both pairs of opposite angles are congruent • Show that the diagonals bisect each other

  14. Theorem 5-4 “If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.”

  15. Theorem 5-4 T S R Q This theorem states that if side TS is congruent to side QR and side QT is congruent to side SR, then the quadrilateral is a parallelogram.

  16. Theorem 5-5 “If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.”

  17. Theorem 5-5 V L • Theorem 5-5 states that if side VL and side AD are both parallel and congruent, then the quadrilateral is a parallelogram. D A

  18. Theorem 5-6 “If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.”

  19. Theorem 5-6 M I • This theorem states that if angle M is congruent to angle E, and angle I is congruent to angle K, then quadrilateral MIKE is a parallelogram. K E

  20. Theorem 5-7 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

  21. Theorem 5-7 H D • This theorem states that if segment HA is congruent to segment AT, and segment MA is congruent to segment AD, that quadrilateral HDMT is a parallelogram. A M T

  22. Chapter 5: Section 3 • Theorems Involving Parallel Lines

  23. Theorem 5-8 • “If two lines are parallel, then all points on one line are equidistant from the other line.”

  24. All of the points on a line are equidistant from points on the perpendicular bisector of a parallel line.

  25. Theorem 5-9 • “If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.”

  26. Theorem 5-10 • “A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.”

  27. Theorem 5-11 • “The segment that joins the midpoints of two sides of a triangle.” 1) is parallel to the third side 2) is half the length of the third side

  28. Section 5-4 Special Parallelograms

  29. Special Parallelograms • There are four distinct special parallelograms that have unique characteristics. • These special parallelograms include: • Rectangles • Rhombi • Squares

  30. Rectangles • A rectangle is a quadrilateral with four right angles. Every rectangle, therefore, is a parallelogram.

  31. Why? • Every rectangle is a parallelogram because all angles in a rectangle are right, and all right angles are congruent. • If the both pairs of opposite angles in a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  32. Additionally: • If a rectangle is a parallelogram, then it retains all of the characteristics of a parallelogram. If this is true then name all congruencies in the following diagram. A S R D E

  33. Solution: S A R D E

  34. Rhombi • A rhombus is a quadrilateral with four congruent sides. Therefore, every rhombus is a parallelogram.

  35. Why? • Every rhombus is a parallelogram because all four sides are congruent, and therefore both pairs of opposite sides are congruent.

  36. Squares • A square is a quadrilateral with four right angles and four congruent sides. Therefore, every square is a rectangle, a rhombus, and a parallelogram.

  37. Why? • A square is a rectangle a rhombus, and a parallelogram because all four sides are congruent and all four angles are right angles.

  38. Theorems • Theorem’s 5-12 through 5-13 are very simple and self explanatory. They read as follows: • 5-12: The diagonals of a rectangle are congruent • 5-13: The diagonals of a rhombus are perpendicular. • 5-14: Each diagonal of a rhombus bisects two angles of the rhombus.

  39. Theorem 5-12 S A • According to theorem 5-12, segment SE is congruent to segment DA R D E

  40. Theorem 5-13 A • According to theorem 5-13, segment AD is congruent to segment CB. B C D

  41. Theorem 5-14 A • According to theorem 5-14, segment AD bisects angle BAC and angle BCD, and segment BC bisects angle ABD and angle ACD. B C D

  42. Theorem 5-15 “The Midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.”

  43. Theorem 5-15 A M B C According to this theorem, M is equidistant from points A, B, and C.

  44. Theorem 5-16 If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

  45. Theorem 5-16 M A • In MATH, we know that angle H is a right angle. This means that all of the angles are right angles, and that MATH is a rectangle. (According to Theorem 5-16) H T

  46. Theorem 5-17 If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

  47. Theorem 5-17 A • In MATH, if we know that segment MA is congruent to segment AT, then we know that MATH is a rhombus, because segment MA is congruent to segment TH and segment AT is congruent to segment MH. Therefore, all sides are congruent to each other. M T H

  48. Chart of Special Quadrilaterals

  49. Section 5-5 Trapezoids

  50. Trapezoid • A trapezoid is defined as a quadrilateral with exactly one pair of parallel sides.

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