# Unit 8. Quadrilaterals. Academic Geometry Spring Name Teacher Period

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1 Unit 8 Quadrilaterals Academic Geometry Spring 2014 Name Teacher Period

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5 Unit 8 at a glance Quadrilaterals This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test, and apply conjectures about quadrilaterals. Students will be able to identify quadrilaterals by the given properties and apply the properties to solve both purely mathematical and real world situations. Essential Questions How are polygons related? What properties do they share? What are the differences and similarities between the types of quadrilaterals (square, rectangle, rhombus, parallelogram, trapezoid, and kite)? And how does knowing this help me? In Unit 8, students will Apply the terms convex, concave, n-gon, equilateral, equiangular, and regular to describe polygons when solving problems involving area, perimeter and circumference in both real-world and purely mathematical situations; Formulate, test, and apply conjectures (based on explorations and concrete models) involving the sum of the measures of interior and exterior angles of convex polygons and the measures of each interior and exterior angle of a regular polygon to solve problems in both real-world and purely mathematical situations; Formulate, test, and apply conjectures (based on explorations and concrete models) involving the properties of parallelogram (including angle and side measure relationships) to solve problems in both real-world and purely mathematical situations; Formulate, test, and apply conjectures (based on explorations and concrete models) involving the conditions that ensure a quadrilateral is a parallelogram including opposite side, opposite angle and diagonal relationships and solve problems in both real-world and purely mathematical situations requiring axiomatic and coordinate approaches; Formulate, test, and apply conjectures (based on explorations and concrete models) involving the properties of rhombuses, rectangles and squares to solve problems in both real-world and purely mathematical situations; 4

6 Formulate, test, and apply conjectures (based on explorations and concrete models) involving the properties of trapezoids and kites to solve problems in both real-world and purely mathematical situations requiring axiomatic or coordinate geometry approaches; Compare and contrast quadrilaterals (parallelograms, rhombuses, rectangles, squares, trapezoids, kites) and their properties to identify them and to solve problems in both real-world and purely mathematical situations. 5

7 Vocabulary concave polygons convex polygons diagonal equiangular exterior angles interior angles isosceles trapezoid kite midsegment n-gons opposite angle parallelogram quadrilateral rectangle regular polygons rhombus square trapezoid vertices 6

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9 Meet the Quadrilateral Family *Parallelogram I have: - 2 sets of parallel sides - 2 sets of congruent sides - opposite angles congruent - consecutive angles supplementary - diagonals bisect each other *Rhombus I have all of the properties of the parallelogram PLUS - 4 congruent sides - diagonals bisect angles - diagonals perpendicular *Rectangle I have all of the properties of the parallelogram PLUS - 4 right angles - diagonals congruent *Square Hey, look at me! I have all of the properties of the parallelogram AND the rectangle AND the rhombus. I have it all! *Quadrilateral I have exactly four sides. 0 *Trapezoid I have only one set of parallel sides. *Isosceles Trapezoid I have: - only one set of parallel sides - base angles congruent - legs congruent - diagonals congruent - opposite angles supplementary *Kite I have: - 2 sets of consecutive congruent sides - only one pair of opposite angles congruent - diagonals perpendicular 8

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11 8.1 NOTES Essential Vocabulary Polygon Not Polygons Concave Convex Equilateral Equiangular Regular Triangle Octagon Quadrilateral Nonagon Pentagon Decagon Hexagon Dodecagon Heptagon n-gon (an n-sided shape) ANY shape can be called n -gon based on the number of sides. Polygons with more than 10 sides are usually referred to as n -gons Ex. 14-gon, 32-gon, 100-gon 10

12 Interior and Exterior Angles of a Polygon In a polygon, two vertices that are endpoints of the same side are called consecutive vertices. A diagonal of a polygon joins two non-consecutive vertices of a polygon. Consecutive vertices Choose any vertex and draw every diagonal possible from that vertex. Notice that when you draw all the diagonals of a polygon from one vertex, you divide the polygon into. Recall that the triangle sum theorem states. For each polygon in the table, draw all the diagonals from one vertex. Complete the table. Polygon # of Sides # of triangles formed Interior Angle Sum of Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon n-gon POLYGON INTERIOR ANGLES THEOREM: For an n-sided convex polygon, the sum of all the interior angles is. 11

13 The exterior angle sum of a polygon does not depend on the number of sides on the polygon. To prove this, use the diagrams below to calculate each exterior angle of the polygons. Remember that an interior angle and its adjacent exterior angle form a linear pair (so their sum is ). triangle quadrilateral pentagon 50⁰ 85⁰ 110⁰ 130⁰ 125⁰ 115⁰ 85⁰ 55⁰ 90⁰ 40⁰ 95⁰ 100⁰ Interior angle sum 180⁰ 360⁰ 540⁰ Exterior angle sum POLYGON EXTERIOR ANGLES THEOREM: The sum of all the exterior angles of a convex polygon (one exterior angle at each vertex is always. Example 1: Find the sum of the measures of the interior angles of a convex octagon. Example 2: The sum of the measures of the interior angles of a convex polygon is 1440⁰. How many sides does the polygon have? 12

14 Example 3: Find the value of x. Example 4: A trampoline is shaped like a regular dodecagon (12 sides). Find the measure of each interior and exterior angle. Example 5: Find the value of x. 13

15 8.1 HOMEWORK Find the sum of the measures of the interior angles of the indicated convex polygon gon gon The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides Find the value of

16 8. The measures of the interior angles of a convex octagon are and What is the measure of the smallest interior angle? Find the measures of an interior angle and an exterior angle of the indicated polygon. 9. Regular Octagon 10. Regular 100-gon 11. The side view of a storage shed is shown below. Find the value of. Then determine the measure of each angle. 15

17 8.2 NOTES Properties of Parallelograms DEFINITION: A parallelogram is a quadrilateral with. You can call this figure: Parallelogram PQRS or. In, and, by definition. Other properties: If a quadrilateral is a parallelogram, then. THM 8.3 THM 8.4 THM 8.5 THM 8.6 Ex1: Find the values of x and y. Your thoughts: 1) Is it a parallelogram? YES, b/c the opp. sides are parallel (def.) 2) Opposite sides are congruent, so x + 4 = 12. 3) Opposite angles are congruent, so y = 65. Ex 2: 16

18 Ex 3-6: Ex 7: The measure of one interior angle of a parallelogram is 50 degrees more than 4 times the measure of another angle. Find the measure of each angle. (Make a sketch and label it.) Ex 8: In LMNO, the ratio of LM to MN is 4:3. Find LM if the perimeter of LMNO is 28. Ex 9: The diagonals of LMNO intersect at point P. What are the coordinates of P? Hint: What do you need to know? 17

19 Ex 10: Is the quadrilateral formed by the lines on the graph a parallelogram? Hint: What do you need to know? 18

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21 8.2 HOMEWORK Find the measure of the indicated angle in the parallelogram. 1. Find 2. Find Find the value of each variable in parallelogram Find the indicated measure in

22 Use the diagram of Points and are midpoints of and Find the indicated measure In parallelogram the ratio of to is. Find if the perimeter of is. 21

23 8.3 NOTES Proving a Quadrilateral is a Parallelogram Examples: 22

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25 8.3 HOMEWORK What theorem can you use to show that the quadrilateral is a parallelogram? For what value of is the quadrilateral a parallelogram?

26 The vertices of quadrilateral are given. Draw in a coordinate plane and show that it is a parallelogram. 11. ( ) ( ) ( ) ( ) 12. ( ) ( ) ( ) ( ) 25

27 8.4NOTES Rhombuses, Squares and Rectangles In this lesson, you will learn about three special types of parallelograms: Rhombus Rectangle Square A parallelogram with four right angles (equiangular). A parallelogram with four congruent sides (equilateral). SPECIAL NOTES ABOUT SQUARES: A parallelogram with four congruent sides and four right angles (regular). Since a square has four congruent sides, it is also a. Since a square has four right angles, it is also a. Diagonals of Rhombuses and Rectangles 26

28 Examples: Name each quadrilateral parallelogram, rectangle, rhombus, and square for which the statement is true. 1. It is equiangular. 2. It is equiangular and equilateral. 3. It is diagonals are perpendicular. 4. Opposite sides are congruent. 5. The diagonals bisect each other. 6. The diagonals bisect opposite angles. Classify the special quadrilateral. Explain your reasoning. Then find the values of and In the rhombus to the right, given. Find all the other angles. 27

29 10. Given rectangle CERT and, find each measure. 11. Given rectangle CERT, (1) If and, solve for. (2) If and, solve for. 28

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33 8.4 HOMEWORK For any rhombus decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning For any rectangle decide whether the statement is always or sometimes true. Draw a diagram and explain your reasoning Classify the quadrilateral. Explain your reasoning

34 Classify the special quadrilateral. Explain your reasoning. Then find the values of and The diagonals of rhombus intersect at Given that and find the indicated measure In preparation for a storm, a window is protected by nailing boards along its diagonals. The lengths of the boards are the same. Can you conclude that the window is a square? Explain. 33

35 8.5/8.6 NOTES Trapezoids & Kites DEFINITION: A trapezoid is a quadrilateral with. The parallel sides are called the. The non-parallel sides are called the. Since a trapezoid has two bases, it has two pairs of. DEFINITION: An isosceles trapezoid is one in which the. Think of it as an isosceles triangle with the top cut off by a segment parallel to a base. *An isosceles trapezoid has: (THM 8.14) (THM 8.16) 34

36 DEFINITION: The midsegment of a trapezoid is a segment that connects the of its. (THM 8.17) The midsegment of a trapezoid is parallel to each base and measures on half the sum of the base lengths. If MN is the midsegment of trapezoid ABCD, 1 then MN AB, MN DC, and MN (AB CD) 2 **In other words, Examples: 1. Find and. 2. Find the length of the midsegment. 3. Solve for. 4. In trapezoid, and. The midsegment is. Sketch and its midsegment and find. 35

37 DEFINITION: A kite is a quadrilateral with. THREE RULES FOR KITES 36

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39 UNIT 8 PERFORMANCE TASK Performance of a Lifetime You are a dance choreographer and have been asked by the Houston Rockets to come up with a 45 second dance that will be showcased during a playoff game. You have been allowed to use the entire court for the performance and have been asked to make sure that all sides of the viewing audience will be able to see the performance without using the Jumbotron. Your goal is to fill the 94 x 50 foot court by placing the dancers in the shape of a polygon. You want to make sure that you use as much of the court as possible while making sure that there is an equal amount of unused space at each corner of the court. 1. Determine where the vertices of your polygon must fall to develop an equal amount of unused space so all fans can see the performance and the owner of the Houston Rockets will be happy. How would you find these points and how do you know the unused space is equal at each corner of the court? 2. What type of polygon is created? How can you justify your answer? Justify all of the polygon s properties. 38

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