6-1 Polygons Goal 1 Describing Polygons

A polygon is an enclosed plane figure that is made up of segments.

Polygons 3 sidedTriangle 4 sidedQuadrilateral 5 sidedPentagon 6 sidedHexagon 7 sidedHeptagon 8 sidedOctagon 9 sidedNonagon 10 sidedDecagon 12 sidedDodecagon

Identify the polygon.

Polygons Quadrilateral Kite Trapezoid Parallelogram RectangleRhombus Square Isosceles trapezoid

Quadrilaterals Quadrilaterals are four-sided polygons. <A + <B + <C + <D = 360° AB D C

Parallelogram A parallelogram is a four-sided figure with both pairs of opposite sides parallel.

Properties of a Parallelogram 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.The diagonals bisect each other. 5.Consecutive angles are supplementary.

Diagonal The diagonals of a polygon are the segments that connect any two nonconsecutive vertices.

1. AB // DC, AD // BC 2. AB =DC, AD = BC 3. <A = <C and <B = <D 4. AM = MC and MD = MB 5. <A + <B = 180 and <B + <C = 180 <C + <D = 180 and <D + <A = 180 A B CD

WXYZ is a parallegram, m<ZWX = b, and m<WXY = d. Find the values of a, b, c, and d. WX Y Z 15 18° 31° a 2c 22

Ch = GF // <DCG = DC = <DCG is supplementary to __ ∆HGC = G D C F H

In parallelogram ABCD, AB = 2x +5, m<BAC = 2y, m<B = 120, m<CAD = 21, and CD= 21. Find the values of x and y.

Quadrilateral WXYZ is a parallelogram with m<W = 47. Find the measure of angles X, Y, and Z.

Assignment Class work on page 295 problems

6-2 Tests for Parallelogram A Quadrilateral is a parallelogram if any of the following is true. Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. Both pairs of opposite angles are congruent. Diagonals bisect each other. A pair of opposite sides is both parallel and congruent.

Rectangle A rectangle is a quadrilateral with four right angles.

Properties of a Rectangle 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.The diagonals bisect each other. 5.Consecutive angles are supplementary 6.All angles are congruent 7.The diagonals are congruent

1. Explain why a rectangle is a special type of parallelogram. All rectangles are parallelograms, but not all parallelograms are rectangles.

Quadrilateral MNOP is a rectangle. Find the value of x. MO = 2x – 8; NP = 23 CN = x 2 + 1; CO = 3x + 11 MO = 4x – 13; PC = x + 7 M N O P

Use rectangle KLMN and the given information to solve each problem. M<1 = 70. Find m<2, M<5, M<6 KL M N C

6-4 Rhombus A rhombus is a quadrilateral with four congruent sides.

Properties of a Rhombus 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.The diagonals bisect each other. 5.Consecutive angles are supplementary 6.All sides are congruent 7.The diagonals are perpendicular 8.The diagonals bisect the opposite angles

Rhombus AB C D

Use rhombus BCDE and the given information to find each missing value. If m<1 = 2x + 20 and m<2 = 5x – 4, find the value of x. If BD = 15, find BF. If m<3 = y , find y. B C D E F 12 3

Assignment Page 316 Problems 4, 10-13, 21-32, 36-47

Square A square is a quadrilateral with four right angles and four congruent sides.

Properties of a Square 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.The diagonals bisect each other. 5.Consecutive angles are supplementary 6.All angles are congruent. 7.The diagonals are congruent. 8.All sides are congruent 9.The diagonals are perpendicular. 10.The diagonals bisect the opposite angles.

Polygons Quadrilateral Kite Trapezoid Parallelogram RectangleRhombus Square Isosceles trapezoid

Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. Property The angles along the legs are supplementary.

leg base

AB // DC M<A + m<D = 180 M<B + m<C = 180 AB CD

Isosceles Trapezoid Properties The legs are congruent Both pairs of base angles are congruent The diagonals are congruent

AD = BC m<A = m<B, m<D = m<C AC = BD A B C D

PQRS is an isosceles trapezoid. Find m<P, m<Q, and m<R. 50° P Q R S

Theorem 6-16 The median of a trapezoid is parallel to the bases, and its measure is one- half the sum of the measures of the bases.

XY = ½(AB + DC) A B C D XY

Find the length of the midsegment When the bases are 7 and 11 3 and 7 12 and 7 14 and 16 x

Find x x 7 4

AB = ½(EZ + IO) E Z I O AB 4x x + 8

Assignment on page 325 Problems 16-28, 46