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Polygons

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.

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

Lesson 6-4: Rhombus & Square 5 Rhombus Definition:A rhombus is a parallelogram with four congruent sides. Since a rhombus is a parallelogram the following are true: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other ≡ ≡

Lesson 6-4: Rhombus & Square 6 Rhombus Examples..... Given: ABCD is a rhombus. Complete the following. 1.If AB = 9, then AD = ______. 2.If m<1 = 65, the m<2 = _____. 3.m<3 = ______. 4.If m<ADC = 80, the m<DAB = ______. 5.If m<1 = 3x -7 and m<2 = 2x +3, then x = _____. 9 units 65° 90° 100° 10

Lesson 6-4: Rhombus & Square 7 Square 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. Definition:A square is a parallelogram with four congruent angles and four congruent sides. 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 8 Squares – Examples…... Given: ABCD is a square. Complete the following. 1.If AB = 10, then AD = _____ and DC = _____. 2.If CE = 5, then DE = _____. 3.m<ABC = _____. 4.m<ACD = _____. 5.m<AED = _____. 10 units 5 units 90° 45° 90°

Lesson 6-5: Trapezoid & Kites 9 Properties of Isosceles Trapezoid 2. The diagonals of an isosceles trapezoid are congruent. 1. Both pairs of base angles of an isosceles trapezoid are congruent. A B C D Base Angles

KITE 1. Two sides of adjacent sides congruent 2. Diagonals are perpendicular Note: opposite sides are not congruent Note: diagonals do not bisect each other