Quadrilaterals

1. Quadrilaterals Chapter 5 Pre-AP Geometry

2. Objectives • Apply the definition of a parallelogram and the theorems about properties of a parallelogram. • Prove that certain quadrilaterals are parallelograms. • Apply theorems about parallel lines. • Apply the midpoint theorems for triangles. • Apply the definitions and identify the special properties of a rectangle, a rhombus, and a square. • Determine when a parallelogram is a rectangle, rhombus, or square. • Apply the definition and identify the properties of a trapezoid and an isosceles trapezoid.

3. Parallelograms Lesson 5.1 Pre-AP Geometry

4. Objective Apply the definition of a parallelogram and the theorems about properties of a parallelogram.

5. Definition Parallelogram A quadrilateral with two sets of parallel sides.

6. Properties of Parallelograms • The diagonals of a parallelogram bisect each other. • Opposite sides of a parallelogram are congruent. • Opposite angles of a parallelogram are congruent. • Each diagonal bisects the parallelogram into two congruent triangles. Parallelogram

7. Theorems 5-1 Opposite sides of a parallelogram are congruent. 5-2 Opposite angles of a parallelogram are congruent. 5-3 Diagonals of a parallelogram bisect each other.

8. Practice • Name all pairs of parallel lines. • Name all pairs of congruent angles. • Name all pairs of congruent segments. • What is the sum of the measures of the interior angles of a parallelogram? • What is the sum of the measures of the exterior angles of a parallelogram?

9. Review – True or False • Every parallelogram is a quadrilateral. • Every quadrilateral is a parallelogram. • All angles of a parallelogram are congruent. • All sides of a parallelogram are congruent.

10. Written Exercises Problem Set 5.1, p. 169: # 2 – 32 (even) skip # 16

11. Proving Quadrilaterals are Parallelograms Lesson 5.2 Pre-AP Geometry

12. Objective Prove that certain quadrilaterals are parallelograms.

13. Quadrilaterals and Parallelograms • A quadrilateral is a polygon with 4 sides. • A parallelogram is a quadrilateral whose opposite sides are parallel (the top and bottom are parallel and the left and right are parallel).

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

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

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

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

18. Five 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.

19. Review Answer with always, sometimes, or never. • The diagonals of a quadrilateral bisect each other. • If the measures of two angles of a quadrilateral are equal, then he quadrilateral is a parallelogram. • If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram. • If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. • To prove a quadrilateral is a parallelogram, it is enough to show that one pair of opposite sides is parallel.

20. B C E A D Practice State the definition of theorem that enables you to deduce, from the information provided, that quadrilateral ABCD is a parallelogram. • BE = EX; CE = EA • BAD  DCB; ADC  CBA • BC || AD; AB || DC • BC  AD; AB  DC

21. Written Exercises Problem Set 5.2 p. 174 # 2, 4, 8, 10, 14, 20, 22

22. Theorems Involving Parallel Lines Lesson 5.3 Pre-AP Geometry

23. Objective Apply theorems about parallel lines and the segment that joins the midpoints of two sides of a triangle.

24. l m Theorem 5-8 If two lines are parallel, then all points on one line are equidistant from the other line.

25. l m n Theorem 5-9 If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

26. A D M N B C 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. A M N B C 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 as long as the third side. BM = AM, CN = AN BC = 2(MN) MN = ½(BC)

28. Written Exercises Problem Set 5.3 p. 180 # 2-20 even, p. 182 # 1-6

29. Special Parallelograms Lesson 5.4 Pre-AP Geometry

30. Objectives Apply the definitions and identify the special properties of a rectangle, a rhombus, and a square. Determine when a parallelogram is a rectangle, rhombus, or square.

31. Rectangle A parallelogram with four right angles. Both pairs of opposite angles are congruent. Every rectangle is a parallelogram.

32. Rhombus A quadrilateral with four congruent sides. Both pairs of opposite sides are congruent. Every rhombus is a parallelogram.

33. Square A quadrilateral with four right angles and four congruent sides. Both pairs of opposite angles and opposite sides are congruent. A square is also a rectangle, a rhombus, and a parallelogram.

34. Theorem 5-12 The diagonals of a rectangle are congruent.

35. Theorem 5-13 The diagonals of a rhombus are perpendicular.

36. Theorem 5-14 Each diagonal of a rhombus bisects two angles of the rhombus.

37. Theorem 5-15 The midpoint of the hypotenuse of a right angle is equidistant from the three vertices.

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

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

41. Practice Reply with always, sometimes, or never. • A square it a rhombus. • The diagonals of a parallelogram bisect the angles of the parallelogram. • A quadrilateral with one pair of sides congruent is a parallelogram. • The diagonals of a rhombus are congruent. • A rectangle has consecutive sides congruent. • A rectangle has perpendicular diagonals. • The diagonals of a rhombus bisect each other. • The diagonals of a parallelogram are perpendicular bisectors of each other.

42. Written Exercises Problem Set 5.4 p. 187 # 1-10, 12-30 evens

43. Trapezoids Lesson 5.5 Pre-AP Geometry

44. Objective Apply the definitions and identify the properties of a trapezoid and an isosceles trapezoid.

45. base leg leg base Definition Trapezoid A quadrilateral with exactly one pair of parallel sides.

46. D C A B Definition Isosceles Trapezoid In an isosceles trapezoid, the base angles are equal, and so are the other pair of opposite sides AD and BC.

47. Theorem 5-18 Base angles of an isosceles trapezoid are congruent.

48. Median of a Trapezoid The segment that joins the midpoints of the legs of a trapezoid. Median of a Trapezoid

49. Theorem 5-19 The median of a trapezoid: (1) is parallel to the base; (2) has a length equal to the average of the base lengths.

50. Practice • In trapezoid ABCD, EF is a median. • If AB = 25 and DC = 13, then EF = _____. • If AE = 11 and FB = 8, then AD = _____ and BC = _____. • If AB = 29 and EF = 24, then DC = _____. • If AB = 7y + 6 and EF = 5y – 3 and DC = y – 5, • then y = _____.