G.9 Quadrilaterals Part 1

1. G.9QuadrilateralsPart 1 Parallelograms Modified by Lisa Palen

2. Definition • A parallelogram is a quadrilateral whose opposite sides are parallel. • Its symbol is a small figure:

3. Naming a Parallelogram • A parallelogram is named using all four vertices. • You can start from any one vertex, but you must continue in a clockwise or counterclockwise direction. • For example, this can be either ABCD or ADCB.

4. Basic Properties • There are four basic properties of all parallelograms. • These properties have to do with the angles, the sides and the diagonals.

5. Opposite Sides Theorem Opposite sides of a parallelogram are congruent. • That means that . • So, if AB = 7, then _____ = 7?

6. Opposite Angles • One pair of opposite angles is A and  C. The other pair is  B and  D.

7. Opposite Angles Theorem Opposite angles of a parallelogram are congruent. • Complete: If m  A = 75 and m  B = 105, then m  C = ______ and m  D = ______ .

8. Consecutive Angles • Each angle is consecutive to two other angles. A is consecutive with  B and  D.

9. Consecutive Angles in Parallelograms Theorem Consecutive angles in a parallelogram are supplementary. • Therefore, m  A + m  B = 180 and m  A + m  D = 180. • If m<C = 46, then m  B = _____? Consecutive INTERIOR Angles are Supplementary!

10. Diagonals • Diagonals are segments that join non-consecutive vertices. • For example, in this diagram, the only two diagonals are .

11. Diagonal Property When the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal. • So, P is the midpoint of . • Therefore, they bisect each other; so and . • But, the diagonals are not congruent!

12. Diagonal Property Theorem The diagonals of a parallelogram bisect each other.

13. Parallelogram Summary • By its definition, opposite sides are parallel. Other properties (theorems): • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • The diagonals bisect each other.

14. Examples • 1. Draw HKLP. • 2. Complete: HK = _______ and HP = ________ . • 3. m<K = m<______ . • 4. m<L + m<______ = 180. • 5. If m<P = 65, then m<H = ____, m<K = ______ and m<L =______ .

15. Examples (cont’d) • 6. Draw in the diagonals. They intersect at M. • 7. Complete: If HM = 5, then ML = ____ . • 8. If KM = 7, then KP = ____ . • 9. If HL = 15, then ML = ____ . • 10. If m<HPK = 36, then m<PKL = _____ .

16. Tests for Parallelograms Part 2

17. Review: Properties of Parallelograms • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • The diagonals bisect each other.

18. How can you tell if a quadrilateral is a parallelogram? • Defn: A quadrilateral is a parallelogram iff opposite sides are parallel. • PropertyIf a quadrilateral is a parallelogram, then opposite sides are parallel. • Test If opposite sides of a quadrilateral are parallel, then it is a parallelogram.

19. Proving Quadrilaterals as Parallelograms Theorem 1: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram . H G Theorem 2: E F If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .

20. Theorem: Theorem 3: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. G H then Quad. EFGH is a parallelogram. M Theorem 4: E F If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram . then Quad. EFGH is a parallelogram. EM = GM and HM = FM

21. 5 ways to prove that a quadrilateral is a parallelogram. 1. Show that both pairs of opposite sides are || . [definition] 2. Show that both pairs of opposite sides are  . 3. Show that one pair of opposite sides are both || and  . 4. Show that both pairs of opposite angles are  . 5. Show that the diagonals bisect each other .

22. Examples …… Example 1: Find the values of x and y that ensures the quadrilateral is a parallelogram. y+2 6x = 4x + 8 2x = 8 x = 4 2y = y + 2 y = 2 6x 4x+8 2y Find the value of x and y that ensure the quadrilateral is a parallelogram. Example 2: 5y + 120 = 180 5y = 60 y = 12 2x + 8 = 120 2x = 112 x = 56 5y° (2x + 8)° 120°

23. Rectangles Part 3 Lesson 6-3: Rectangles

24. Rectangles • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Definition: A rectangle is a quadrilateral with four right angles. Is a rectangle is a parallelogram? Yes, since opposite angles are congruent. Thus a rectangle has all the properties of a parallelogram. Lesson 6-3: Rectangles

25. A B E D C Properties of Rectangles Theorem: If a parallelogram is a rectangle, then its diagonals are congruent. Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles. Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle. Lesson 6-3: Rectangles

26. A B E D C Properties of Rectangles Parallelogram Properties: • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Plus: • All angles are right angles. • Diagonals are congruent. • Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles Lesson 6-3: Rectangles

27. A B 2 3 1 E 4 5 6 D C Examples……. • If AE = 3x +2 and BE = 29, find the value of x. • If AC = 21, then BE = _______. • If m<1 = 4x and m<4 = 2x, find the value of x. • If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6. x = 9 units 10.5 units x = 18 units m<1=50, m<3=40, m<4=80, m<5=100, m<6=40 Lesson 6-3: Rectangles

28. Rhombi and Squares Part 4 Lesson 6-4: Rhombus & Square

29. Rhombus • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Definition: A rhombus is a quadrilateral with four congruent sides. ≡ Is a rhombus a parallelogram? ≡ Yes, since opposite sides are congruent. Since a rhombus is a parallelogram the following are true: Lesson 6-4: Rhombus & Square

30. Rhombus Note: The four small triangles are congruent, by SSS. This means the diagonals form four angles that are congruent, and must measure 90 degrees each. ≡ ≡ So the diagonals are perpendicular. This also means the diagonals bisect each of the four angles of the rhombus So the diagonals bisect opposite angles. Lesson 6-4: Rhombus & Square

31. Properties of a Rhombus Theorem: The diagonals of a rhombus are perpendicular. Theorem: Each diagonal of a rhombus bisects a pair of opposite angles. Note: The small triangles are RIGHT and CONGRUENT! Lesson 6-4: Rhombus & Square

32. Properties of a Rhombus • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Plus: • All four sides are congruent. • Diagonals are perpendicular. • Diagonals bisect opposite angles. • Also remember: the small triangles are RIGHT and CONGRUENT! Since a rhombus is a parallelogram the following are true: . ≡ ≡ Lesson 6-4: Rhombus & Square

33. Rhombus Examples ..... Given: ABCD is a rhombus. Complete the following. • If AB = 9, then AD = ______. • If m<1 = 65, the m<2 = _____. • m<3 = ______. • If m<ADC = 80, the m<DAB = ______. • If m<1 = 3x -7 and m<2 = 2x +3, then x = _____. 9 units 65° 90° 100° 10 Lesson 6-4: Rhombus & Square

34. Square Definition: A square is a quadrilateral with four congruent angles and four congruent sides. • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. Plus: • Four right angles. • Four congruent sides. • Diagonals are congruent. • Diagonals are perpendicular. • Diagonals bisect opposite angles. Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals.

35. Squares – Examples…... Given: ABCD is a square. Complete the following. • If AB = 10, then AD = _____ and DC = _____. • If CE = 5, then DE = _____. • m<ABC = _____. • m<ACD = _____. • m<AED = _____. 10 units 10 units 5 units 90° 45° 90° Lesson 6-4: Rhombus & Square

36. Trapezoids and Kites Part 5 Lesson 6-5: Trapezoid & Kites

37. Trapezoid A quadrilateral with exactly one pair of parallel sides. Definition: The parallel sides are called bases and the non-parallel sides are called legs. Base Trapezoid Leg Leg Base Lesson 6-5: Trapezoid & Kites

38. Median of a Trapezoid The median of a trapezoid is the segment that joins the midpoints of the legs. (It is sometimes called a midsegment.) • Theorem - The median of a trapezoid is parallel to the bases. • Theorem - The length of the median is one-half the sum of the lengths of the bases. Median Lesson 6-5: Trapezoid & Kites

39. Isosceles Trapezoid A trapezoid with congruent legs. Definition: Isosceles trapezoid Lesson 6-5: Trapezoid & Kites

40. Properties of Isosceles Trapezoid 1. Both pairs of base angles of an isosceles trapezoid are congruent. 2. The diagonals of an isosceles trapezoid are congruent. B A D C Lesson 6-5: Trapezoid & Kites

41. Kite A quadrilateral with two distinct pairs of congruent adjacent sides. Definition: Diagonals of a kite are perpendicular. Theorem: Lesson 6-5: Trapezoid & Kites

42. Trapezoid Flow Chart Quadrilaterals Kite Parallelogram Isosceles Trapezoid Rhombus Rectangle Square Lesson 6-5: Trapezoid & Kites