Chapter 6 Quadrilaterals

Chapter Objectives Define a polygon and its characteristics Identify a regular polygon Interior Angles of a Quadrilateral Theorem Properties of Parallelograms Using coordinate geometry to prove parallelograms Compare rhombuses, rectangles, and squares Identify trapezoids and kites Midsegment Theorem for Trapezoids Calculate area of trapezoids, kites, rhombuses, rectangles, and squares

Lesson 6.1 Polygons

Lesson 6.1 Objectives Identify a figure to be a polygon. Recognize the different types of polygons based on the number of sides. Identify the components of a polygon. Use the sum of the interior angles of a quadrilateral.

Definition of a Polygon A polygon is plane figure (two-dimensional) that meets the following conditions. It is formed by three or more segments called sides. The sides must be straight lines. Each side intersects exactly two other sides, one at each endpoint. The polygon is closed in all the way around with no gaps. Each side must end when the next side begins. No tails. Polygons Not Polygons

Polygon Parts Each segment that is used to close a polygon in is called a side. Where each side ends is called a vertex. A vertex is simply a corner of the polygon. vertices sides

Types of Polygons Number of Sides Type of Polygon 3 4 5 6 7 8 9 10 12 Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon

Concave v Convex A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that never leave the interior of the polygon, then it is convex. A polygon is concave if a line that contains a side of the polygon contains a point in the interior of the polygon. Take any two points in the interior of the polygon. If you can draw a line between the two points that does leave the interior of the polygon, then it is concave. Concave polygons have dents in the sides, or you could say it caves in.

Example 1 No! Yes Yes Concave Convex Determine if the following are polygons or not. If it is a polygon, classify it as concave or convex. No! Yes Yes Concave Convex

Regular Polygons A polygon is equilateral if all of its sides are congruent. A polygon is equiangular if all of its interior angles are congruent. A polygon is regular if it is both equilateral and equiangular. The best way to draw these is to label each sides and angle with the proper congruent marks.

Diagonals of a Polygon A diagonal of a polygon is a segment that joins two nonconsecutive vertices. A diagonal does not go to the point next to it. That would make it a side! Diagonals cut across the polygon to all points on the other side. There is typically more than one diagonal.

Theorem 6.1: Interior Angles of a Quadrilateral Theorem The sum of the measures of the interior angles of a quadrilateral is 360o. 1 2 3 4 m 1 +m 2 + m 3 + m 4 = 360o

Homework 6.1 In Class 1-11 p325-328 HW 12-46, 54-59 Due Tomorrow

Properties of Parallelograms Lesson 6.2 Properties of Parallelograms

Lesson 6.2 Objectives Define a parallelogram Identify properties of parallelograms Use properties of parallelograms to determine unknown quantities of the parallelogram

Definition of a Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Theorem 6.2: Congruent Sides of a Parallelogram If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Theorem 6.3: Opposite Angles of a Parallelogram If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Example 2 d = 53 d + 15 = 68 m = 101 x = 11 y = 8 c – 5 = 20 c = 25 Find the missing variables in the parallelograms.   d = 53  d + 15 = 68 m = 101 x = 11 y = 8   c – 5 = 20 c = 25

Theorem 6.4: Consecutive Angles of a Parallelogram If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Q P R S m P + m S = 180o m Q + m R = 180o m P + m Q = 180o m R + m S = 180o

Theorem 6.5: Diagonals of a Parallelogram If a quadrilateral is a parallelogram, then its diagonals bisect each other. Remember that means to cut into two congruent segments.

Example 3 Find the indicated measure in  HIJK HI GH KH HJ m KIH 16 Theorem 6.2 GH 8 Theorem 6.6 KH 10 HJ Theorem 6.6 & Seg Add Post m KIH 28o AIA Theorem m JIH 96o Theorem 6.4 m KJI 84o Theorem 6.3

Homework 6.2 HW Due Tomorrow Quiz Wednesday 20-37, 47-54, 60, 61 p333-336 20-37, 47-54, 60, 61 Due Tomorrow Quiz Wednesday Lessons 6.1-6.3

Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving Quadrilaterals are Parallelograms

Lesson 6.3 Objectives Verify that a quadrilateral is a parallelogram. Utilize coordinate geometry with parallelograms

Theorem 6.6: Congruent Sides of a Parallelogram Converse If both pairs of opposite sides are congruent, then it is a parallelogram.

Theorem 6.7: Opposite Angles of a Parallelogram Converse If both pairs of opposite angles are congruent, then it is a parallelogram.

Theorem 6.8: Consecutive Angles of a Parallelogram Converse If an angle of a quadrilateral is supplementary to its consecutive angles, then it is a parallelogram. Q P R S m P + m S = 180o m P + m Q = 180o m Q + m R = 180o m R + m S = 180o

Theorem 6.9: Diagonals of a Parallelogram Converse If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Theorem 6.10: Opposite Sides of a Parallelogram If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

Example 4 Theorem 6.10 Theorem 6.6 Theorem 6.7 Theorem 6.9 Theorem 6.8 Which theorem would you use to show the following are parallelograms? Theorem 6.10 Theorem 6.6 Theorem 6.7 Theorem 6.9 Theorem 6.8 or Theorem 6.7 Theorem 6.6 or Theorem 6.10

Homework 6.3 In Class HW Due Tomorrow Quiz Friday 1-7 9-29, 45-47 p342-345 HW 9-29, 45-47 skip 15-16 Due Tomorrow Quiz Friday Lessons 6.1-6.3

Rhombuses, Rectangles, and Squares Lesson 6.4 Rhombuses, Rectangles, and Squares

Lesson 6.4 Objectives Identify characteristics of a rhombus. Identify characteristics of a rectangle. Identify characteristics of a square.

Rhombus A rhombus is a parallelogram with four congruent sides. The rhombus corollary states that a quadrilateral is a rhombus if and only if it has four congruent sides.

Theorem 6.11: Perpendicular Diagonals A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Theorem 6.12: Opposite Angle Bisector A parallelogram is a rhombus iff each diagonal bisects a pair of opposite angles.

Rectangle A rectangle is a parallelogram with four congruent angles. The rectangle corollary states that a quadrilateral is a rectangle iff it has four right angles.

Theorem 6.13: Four Congruent Diagonals A parallelogram is a rectangle iff all four segments of the diagonals are congruent.

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

Square Corollary A quadrilateral is a square iff it s a rhombus and a rectangle. So that means that all the properties of rhombuses and rectangles work for a square at the same time.

Example 5 Classify the parallelogram. Explain your reasoning. Rhombus Must be supplementary    Rhombus Square Rectangle Diagonals are perpendicular. Theorem 6.11 Square Corollary Diagonals are congruent. Theorem 6.13

Homework 6.4 In Class HW Due Tomorrow 1, 3-11 p351-354 HW 12-46 evens, 55-58, 66, 67 Due Tomorrow

Lesson 6.5 Trapezoids and Kites

Lesson 6.5 Objectives Identify properties of a trapezoid. Recognize an isosceles trapezoid. Utilize the midsegment of a trapezoid to calculate other quantities from the trapezoid. Identify a kite.

Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called the bases. The nonparallel sides are called legs. The angles formed by the bases are called the base angles.

Isosceles Trapezoid If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

Theorem 6.14: Bases Angles of a Trapezoid If a trapezoid is isosceles, then each pair of base angles is congruent. That means the top base angles are congruent. The bottom base angles are congruent. But they are not all congruent to each other!

Theorem 6.15: Base Angles of a Trapezoid Converse If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

Theorem 6.16: Congruent Diagonals of a Trapezoid A trapezoid is isosceles if and only if its diagonals are congruent. Notice this is the entire diagonal itself. Don’t worry about it being bisected cause it’s not!!

Example 6 Find the measures of the other three angles. Supplementary  because of CIA  83o 127o 127o   97o 83o Supplementary because of CIA

Midsegment The midsegment of a trapezoid is the segment that connects the midpoints of the legs of a trapezoid.

Theorem 6.17: Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. It is the average of the base lengths! A B C D M N MN = 1/2(AB + CD)

Example 7 Find the length of the midsegment. RT = 1/2(WX + ZY)

Kite A kite is a quadrilateral that has two pairs of consecutive sides that are congruent, but opposite sides are not congruent. It looks like the kite you got for your birthday when you were 5! There are no sides that are parallel.

Theorem 6.18: Diagonals of a Kite If a quadrilateral is a kite, then its diagonals are perpendicular.

Theorem 6.19: Opposite Angles of a Kite If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. The angles that are congruent are between the two different congruent sides. You could call those the shoulder angles. NOT

Example 8 Find the missing angle measures. 60 + K + 50 + M = 360 64o 125o  125o 88o 60 + K + 50 + M = 360 But K  M 88 + 120 + 88 + J = 360 60 + M + 50 + M = 360 296 + J = 360 110 + 2M = 360 J = 64 2M = 250 M = 125 K = 125

Example 9 Find the lengths of all the sides of the kite. Use Pythagorean Theorem! Cause the diagonals are perpendicular!! a2 + b2 = c2 Find the lengths of all the sides of the kite. Round your answer to the nearest hundredth. a2 + b2 = c2 52 + 52 = c2 a2 + b2 = c2 25 + 25 = c2 7.07 7.07 52 + 122 = c2 50 = c2 25 + 144 = c2 c = 7.07 169 = c2 13 13 c = 13

Homework 6.5 In Class HW Due Tomorrow Test Monday 3-9 p359-362 HW 10-39, 51, 52, 57-64 Due Tomorrow Test Monday November 12

Special Quadrilaterals Lesson 6.6 Special Quadrilaterals

Lesson 6.6 Objectives Create a hierarchy of polygons Identify special quadrilaterals based on limited information

Polygon Hierarchy Polygons Pentagons Triangles Quadrilaterals Parallelogram Trapezoid Kite Rhombus Rectangle Isosceles Trapezoid Square

How to Read the Hierarchy NEVER How to Read the Hierarchy SOMETIMES ALWAYS Polygons Triangles Quadrilaterals Pentagons Rhombus Rectangle Trapezoid Parallelogram Kite Square Isosceles Trapezoid But a parallelogram is sometimes a rhombus and sometimes a square. So that means that a square is always a rhombus, a parallelogram, a quadrilateral, and a polygon. However, a parallelogram is never a trapezoid or a kite.

Using the Hierarchy Remember that a square must fit all the properties of its “ancestors.” That means the properties of a rhombus, rectangle, parallelogram, quadrilateral, and polygon must all be true! So when asked to identify a figure as specific as possible, test the properties working your way down the hierarchy. As soon as you find a figure that doesn’t work any more you should be able to identify the specific name of that figure.

Homework 6.6 In Class HW Due Tomorrow Test Friday 2-7 8-35, 55-65 p367-370 HW 8-35, 55-65 Due Tomorrow Test Friday November 7

Areas of Triangles and Quadrilaterals Lesson 6.7 Areas of Triangles and Quadrilaterals

Lesson 6.7 Objectives Find the area of any type of triangle. Find the area of any type of quadrilateral.

Postulate 22: Area of a Square Postulate The area of a square is the square of the length of its side. A = s2 s

Area Postulates Postulate 23: Area Congruence Postulate If two polygons are congruent, then they have the same area. Postulate 24: Area Addition Postulate The area of a region is the sum of the areas of its nonoverlapping parts.

Theorem 6.20: Area of a Rectangle The area of a rectangle is the product of a base and its corresponding height. Corresponding height indicates a segment perpendicular to the base to the opposite side. A = bh h b

Theorem 6.21: Area of a Parallelogram The area of a parallelogram is the product of a base and its corresponding height. Remember the height must be perpendicular to one of the bases. The height will be given to you or you will need to find it. To find it, use Pythagorean Theorem a2 + b2 = c2 A = bh h b

Theorem 6.22: Area of a Triangle The area of a triangle is one half the product of the base and its corresponding height. The base for this formula is the segment that is perpendicular to the height. It may be a side of the triangle, it may not! h h h b b b

Theorem 6.23: Area of a Trapezoid The area of a trapezoid is one half the product of the height and the sum of the bases. The height is the perpendicular segment between the bases of the trapezoid. A = ½ h (b1+b2) b1 h b2

Theorem 6.24: Area of a Kite d1 d2 The area of a kite is one half the product of the lengths of the diagonals. A = ½ d1d2 d1 d2

Theorem 6.25: Area of a Rhombus The area of a rhombus is equal to one half the product of the lengths of the diagonals. A = ½ d1d2 d1 d2

Homework 6.7 In Class HW Due Tomorrow Test Monday 3-13 p376-380 HW 14-38 evens, 50-52, 60, 61 Due Tomorrow Test Monday November 12