QUADRILATERALS 4-SIDED POLYGONS

Properties of Parallelograms Review By its definition, opposite sides are parallel. Other properties : Opposite sides are congruent. The diagonals bisect each other. Opposite angles are congruent. Consecutive angles are supplementary.

Rectangles

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

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

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 6 5 4 3 2 1 E D C B A m<1=50O, m<3=40O, m<4=80O, m<5=100O, m<6=40O

Rhombuses and Squares

Rhombus ≡ ≡ Definition: A rhombus is a quadrilateral with four congruent sides. Is a rhombus a parallelogram? ≡ Yes, since opposite sides are congruent. ≡ Therefore… Opposite sides are parallel and congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.

Properties of a Rhombus Theorem: The diagonals of a rhombus are perpendicular. Theorem: Each diagonal of a rhombus bisects opposite angles. Note: The small triangles are RIGHT and CONGRUENT!

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

Square Definition: A square is a quadrilateral with four congruent angles and four congruent sides. Is a square a parallelogram? Yes, since both opposite sides and opposite angles are congruent. It is also a rectangle and a rhombus. Therefore… Opposite sides are parallel and congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Plus: Diagonals are congruent and and perpendicular. Diagonals bisect opposite angles.

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°

Trapezoids and Kites

Trapezoid Definition: A quadrilateral with exactly one pair of opposite sides parallel. The parallel sides are called bases and the non-parallel sides are called legs. Consecutive angles between the bases are supplementary. Base Trapezoid Leg Leg Base

Midsegment of a Trapezoid The midsegment of a trapezoid is the segment that joins the midpoints of the legs. Theorem - The midsegment of a trapezoid is parallel to the bases. Theorem - The length of the midsegment is one-half the sum of the lengths of the bases. Midsegment

Trapezoids – Examples… Label trapezoid RSTV with bases RS and TV. RS_____ TV If mTSR = 75o, then mSTV = ________ Find the midpoint of ST and label it B. Find the midpoint of RV and label it C. Connect B and C to create the ____________. BC is parallel to ____________. If RS = 8 and TV = 12, then BC = ___. R S C B 105o V T midsegment RS and TV 10

6. Find the value of x and m 7. Solve for y 8 (15y – 9)° m 63° x° 14 x° 63° m (90 – 4y)° (15y – 9)°

Isosceles Trapezoid Definition: A trapezoid with congruent legs.

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

IscocelesTrapezoids – Examples… Solve for the missing values. 4x -5 6x - 9 J M K L 3y - 5 7 + y JL = 4x - 2 MK = 3x + 8 57 y x x = 57o 4x – 5 = 6x - 9 – 5 = 2x - 9 57 + y = 180 4 = 2x y = 123o 2 = x

Kite Definition: A quadrilateral with two distinct pairs of consecutive congruent sides. Note: opposite sides are NOT congruent! Theorem: Diagonals of a kite are perpendicular. Theorem: Exactly one pair of opposite angles is congruent. Note: the congruent angles are created by the noncongruent adjacent sides. The pair of opposite angles not congruent is bisected by the diagonal.

Kites – Examples… Solve for the missing values. x 21 15 y g + 35 = 90 28˚ g˚ h˚ 35˚ x 21 15 y g + 35 = 90 x = 21 h + 28 = 90 g = 65 h = 62 y = 15

Flow Chart Kite Quadrilaterals Parallelogram Rectangle Rhombus Square Trapezoid Rhombus Rectangle Isosceles Trapezoid Square