Rhombuses Rectangles and Squares Lesson 6 4 Rhombuses

Rhombuses, Rectangles, and Squares Lesson 6. 4: Rhombuses, Rectangles, and Squares 1

Theorems About Parallelograms B A D Opposite sides are congruent. Opposite angles are congruent. C Consecutive angles are supplementary. Diagonals bisect each other.

Special Parallelograms Rhombus Def: A parallelogram with four congruent sides. Properties of a rhombus: Ø Diagonals are perpendicular Since it is a parallelogram: üOpposite sides are congruent. üOpposite angles are congruent. üConsecutive angles are supplementary. Ø Each diagonal bisects a pair of opposite angles üDiagonals bisect each other.

Rhombus Examples. . . Given: ABCD is a rhombus. Complete the following. 1. 9 units If AB = 9, then AD = ______. 2. 65° If m<1 = 65, the m<2 = _____. 3. 90° m<3 = ______. 4. If m∠D = 50, the m∠A = 130° ______. Lesson 6. 4: Rhombuses, Rectangles, and Squares 4

Special Parallelograms Rectangle Def: A parallelogram with four right angles. Properties of a rectangle: Ø Diagonals are congruent. Since it is a parallelogram: üOpposite sides are congruent. üOpposite angles are congruent. üConsecutive angles are supplementary. üDiagonals bisect each other.

Rectangle Examples……. 1. If AE = 3 x +2 and BE = 29, find the value of x. x = 7 units 11 units 2. If AC = 22, then BE = _______. 3. If m∠ 2 = 40, find m∠ 1, m∠ 3, m∠ 4, m∠ 5 and m∠ 6. m∠ 1=50, m∠ 3=40, m∠ 4=100, m∠ 5=80, m∠ 6=40 A B 1 2 3 5 D 4 E 6 Lesson 6. 4: Rhombuses, Rectangles, and Squares C 6

Special Parallelograms SQUARE Def: A parallelogram with four congruent sides and four right angles. Since it is a parallelogram: Since it is also a rhombus and rectangle: üOpposite sides are congruent. üOpposite angles are congruent. üConsecutive angles are supplementary. üDiagonals bisect each other. Diagonals are perpendicular Ø Each diagonal bisects a pair of opposite angles Ø Ø Diagonals are congruent.

Squares – Examples…. . . Given: ABCD is a square. Complete the following. 1. units and DC = 10 units If AB = 10, then AD =10 _____. 2. units If CE = 5, then DE = 5_____. 3. m∠ABC =90° _____. 4. m∠ACD = 45° _____. 5. 90° m∠AED = _____. Lesson 6. 4: Rhombuses, Rectangles, and Squares 8

1) Given the rhombus below find all labeled variables. b x 6 in w 40º a y z The figure is a rhombus, so by definition all side are congruent a = 6 in b = 6 in Each diagonal bisects a pair of opposite angles w = 40° Diagonals are perpendicular x = 90° Triangle Sum Theorem ( = 180°) 40° + 90° + y = 180° y = 50° Each diagonal bisects a pair of opposite angles z = 50°

Lesson 6. 5 Trapezoids and Kites Lesson 6. 5: Trapezoid & Kites 10

Lesson Objective: I will learn properties of trapezoids and kites so I can find missing angles. Trapezoids and Kites Lesson 6. 5: Trapezoid & Kites 11

Trapezoid Definition: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases and the non-parallel sides are called legs. Leg Trapezoid Base An Isosceles trapezoid is a trapezoid with congruent legs. Isosceles trapezoid Lesson 6. 5: Trapezoid & Kites 12

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 Lesson 6. 5: Trapezoid & Kites C 13

Kites Definition: A quadrilateral that has two pairs of consecutive congruent sides. Opposite sides of a kite are not congruent. Lesson 6. 5: Trapezoid & Kites 14

Properties of Kites 1. The diagonals of a kite are perpendicular. 2. A kite has exactly one pair of opposite angles that are congruent. C D B A Lesson 6. 5: Trapezoid & Kites 15

Flow Chart Quadrilaterals Parallelogram Kite Trapezoid Rhombus Rectangle Isosceles Trapezoid Square Lesson 6. 5: Trapezoid & Kites 16

Practice Problems 100° y x Solve for x and y. Same side interior angles are supplementary 79° 100 + x = 180 x = 80° 79 + y = 180 y = 101° Lesson 6 -5: Trapezoid & Kites 17

Practice Problems Solve for x, y and z. 95° y Base angles are congruent in isosceles trapezoids. x = 95° x z Same side interior angles are supplementary. 95 + y = 180 y = 85° Base angles are congruent in isosceles trapezoids. z = 85° Lesson 6 -5: Trapezoid & Kites 18

Practice Problems Solve for x. 40° x x 35° Angles opposite the consecutive sides are congruent. Angles in a quadrilateral sum to 360° 40 + 35 + x = 360° 75 + 2 x = 360° 2 x =285° x =142. 5° Lesson 6 -5: Trapezoid & Kites 19