Chapter 6 Quadrilaterals

Types of Polygons Triangle – 3 sides Quadrilateral – 4 sides Pentagon – 5 sides Hexagon – 6 sides Heptagon – 7 sides Octagon – 8 sides Nonagon – 9 sides Decagon – 10 sides Dodecagon – 12 sides All other polygons = n-gon

Lesson 6.1 : Angles of Polygons Interior Angle Sum Theorem  The sum of the measures of the interior angles of a polygon is found by S=180(n-2) Ex: Hexagon Exterior Angle Sum Theorem  The sum of the measures of the exterior angles of a polygon is 360 no matter how many sides.

Lesson 6.1 : Angles of Polygons Find the measure of an interior and an exterior angle for each polygon.  24-gon  3x-gon Find the measure of an exterior angle given the number of sides of a polygon  260 sides

Lesson 6.1: Angles of Polygons The measure of an interior angle of a polygon is given. Find the number of sides.  175  A pentagon has angles (4x+5), (5x-5), (6x+10), (4x+10), and 7x. Find x =5 360/5= 72

A. Find the value of x in the diagram.

Lesson 6.2: Parallelograms Properties of Parallelograms Opposite sides of a parallelogram are congruent Opposite angles in a parallelogram are congruent Consecutive angles in a parallelogram are supplementary If a parallelogram has 1 right angle, it has 4 right angles. The diagonals of a parallelogram split it into 2 congruent triangles The diagonals of a parallelogram bisect each other A parallelogram is a quadrilateral with both pairs of opposite sides parallel

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A. ABCD is a parallelogram. Find AB. B. ABCD is a parallelogram. Find m  C. C. ABCD is a parallelogram. Find m  D.

A. If WXYZ is a parallelogram, find the value of r, s and t.

What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)?

Lesson 6.3 : Tests for Parallelograms If…  Both pairs of opposite sides are parallel  Both pairs of opposite sides are congruent  Both pairs of opposite angles are congruent  The diagonals bisect each other  One pair of opposite sides is congruent and parallel Then the quadrilateral is a parallelogram

Determine whether the quadrilateral is a parallelogram. Justify your answer.

Which method would prove the quadrilateral is a parallelogram?

Determine whether the quadrilateral is a parallelogram.

Find x and y so that the quadrilateral is a parallelogram.

COORDINATE GEOMETRY Graph quadrilateral QRST with vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.

Given quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, –5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram. (The graph does not determine for you)

Foldable Fold the construction paper in half both length and width wise Unfold the paper and hold width wise Fold the edges in to meet at the center crease Cut the creases on the tabs to make 4 flaps

Lesson 6.4 : Rectangles Characteristics of a rectangle:  Both sets of opp. Sides are congruent and parallel  Both sets opp. angles are congruent  Diagonals bisect each other  Diagonals split it into 2 congruent triangles  Consecutive angles are supplementary  If one angle is a right angle then all 4 are right angles In a rectangle the diagonals are congruent. If diagonals of a parallelogram are congruent, then it is a rectangle.

Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

Quadrilateral RSTU is a rectangle. If m  RTU = 8x + 4 and m  SUR = 3x – 2, find x.

Quadrilateral EFGH is a rectangle. If m  FGE = 6x – 5 and m  HFE = 4x – 5, find x.

Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula.

6.5: Squares (special type of parallelogram) A quadrilateral with 4 congruent sides Characteristics of a square:  Both sets of opp. sides are congruent and parallel  Both sets of opp. angles are congruent  Diagonals bisect each other  Diagonals split it into 2 congruent triangles  Consecutive angles are supplementary  If an angle is a right angle then all 4 angles are right angles  Diagonals bisect the pairs of opposite angles  Diagonals are perpendicular A square is a rhombus and a rectangle.

Lesson 6.5 : Rhombi (special type of parallelogram) A quadrilateral with 4 congruent sides Characteristics of a rhombus:  Both sets of opp. sides are congruent and parallel  Both sets of opp. angles are congruent  Diagonals bisect each other  Diagonals split it into 2 congruent triangles  Consecutive angles are supplementary  If an angle is a right angle then all 4 angles are right angles In a rhombus:  Diagonals are perpendicular  Diagonals bisect the pairs of opposite angles

A. The diagonals of rhombus WXYZ intersect at V. If m  WZX = 39.5, find m  ZYX. B. The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

A. ABCD is a rhombus. Find m  CDB if m  ABC = 126. B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

QRST is a square. Find n if m  TQR = 8n + 8.

QRST is a square. Find QU if QS = 16t – 14 and QU = 6t + 11.

Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain.

Kite Two sets of consecutive sides are congruent Diagonals are perpendicular

6.6: Trapezoids A quadrilateral with exactly 1 pair of opposite parallel sides (bases), 2 pairs of base angles, and 1 pair of non- parallel sides (legs) Isosceles Trapezoid:  A trapezoid with congruent legs and congruent base angles Diagonals of an isosceles trapezoid are congruent Median (of a trapezoid):  The segment that connects the midpoints of the legs The median is parallel to the bases base leg Base angle Median = ½ (base + base) AB CD AC = BD

A. Each side of the basket shown is an isosceles trapezoid. If m  JML = 130, KN = 6.7 feet, and LN = 3.6 feet, find m  MJK. B. Each side of the basket shown is an isosceles trapezoid. If m  JML = 130, KN = 6.7 feet, and JL is 10.3 feet, find MN.

In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x.

WXYZ is an isosceles trapezoid with median Find XY if JK = 18 and WZ = 25.

A. If WXYZ is a kite, find m  XYZ.