Lesson 6-1

1. Parallelograms Lesson 6-1 Lesson 6-1: Parallelogram

2. B C D A Parallelogram Definition: A quadrilateral whose opposite sides are parallel. • 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, the figure above can be either ABCD or ADCB. Symbol: a smaller version of a parallelogram Naming: Lesson 6-1: Parallelogram

3. A B Properties of Parallelogram P D C 1. Both pairs of opposite sides are parallel 2. Both pairs of opposite sides are congruent. 3. Both pairs of opposite angles are congruent. 4. Consecutive angles are supplementary. 5. Diagonals bisect each other but are not congruent P is the midpoint of . Lesson 6-1: Parallelogram

4. H K Examples M P L • Draw HKLP. • HK = _______ and HP = ________ . • m<K = m<______ . • m<L + m<______ = 180. • If m<P = 65, then m<H = ____,m<K = ______ and m<L =____. • Draw the diagonals with their point of intersection labeled M. • If HM = 5, then ML = ____ . • If KM = 7, then KP = ____ . • If HL = 15, then ML = ____ . • If m<HPK = 36, then m<PKL = _____ . PL KL P P or K 115° 115° 65 5 units 14 units 7.5 units 36; (Alternate interior angles are congruent.) Lesson 6-1: Parallelogram

5. 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 .

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

7. Rectangles Definition: A rectangle is a parallelogram with four right angles. • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other. A rectangle is a special type of parallelogram. Thus a rectangle has all the properties of a parallelogram. Lesson 6-3: Rectangles

8. 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

9. 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

10. Rhombus Definition: A rhombus is a parallelogram with four congruent sides. • Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • Diagonals bisect each other ≡ ≡ Since a rhombus is a parallelogram the following are true: Lesson 6-4: Rhombus & Square

11. Properties of a Rhombus Theorem: The diagonals of a rhombus are perpendicular. Theorem: Each diagonal of a rhombus bisects a pair of opposite angles. Lesson 6-4: Rhombus & Square

12. 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

13. Square Definition: A square is a parallelogram with four congruent angles and four congruent sides. • Opposite sides are parallel. • Four right angles. • Four congruent sides. • Consecutive angles are supplementary. • Diagonals are congruent. • Diagonals bisect each other. • Diagonals are perpendicular. • Each diagonal bisects a pair of opposite angles. Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals. Lesson 6-4: Rhombus & Square

14. 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

15. 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 An Isosceles trapezoid is a trapezoid with congruent legs. Isosceles trapezoid Lesson 6-5: Trapezoid & Kites

16. 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 Base Angles D C Lesson 6-5: Trapezoid & Kites

17. Median of a Trapezoid The median of a trapezoid is the segment that joins the midpoints of the legs. The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of the bases. Median Lesson 6-5: Trapezoid & Kites

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