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## Geometry Unit 7: Polygons

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**Geometry Unit 7:Polygons**Mr. Andrew Volk Spring 2014**Unit 7 Preview in Numbers**• Unit Lessons: 6 • Instructional Days: 6 • Review Days: 1 • Test Days: 1 • Total Homework Problems: ? • Total Objectives: 17**Unit 7 Preview of Topics**• Two-Dimensional Figures • Parallelograms • Tests of Parallelograms • Rectangles, Rhombus, and Square • Trapezoid and Kites • Area of Parallelograms and Triangles • Area of Trapezoids, Rhombi, and Kites**Two-Dimensional Figures**Lesson 7.1**Lesson 7.1 Objectives**The student will be able to… • Identify and name polygons. • Find perimeter, circumference and area of two dimensional figures. • Find and use the sum of measures of the interior angles. • Find and use the sum of the measures of the exterior angles.**Virginia SOL Standard G.9 and G10**• G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • G.10 The student will solve real-world problems involving angles of polygons.**Naming Polygons by sides**• 3 Triangle • 4 Quadrilateral • 5 Pentagon • 6 Hexagon • 7 Heptagon • 8 Octagon • 9 Nonagon • 10 Decagon • 11 Hendecagon • 12 Dodecagon**Finding the sum of the interior angles of a polygon**• 3 Triangle: 180 • 4 Quadrilateral: 360 • 5 Pentagon: 180*3=540 • 6 Hexagon: 180*4=720 • 7 Heptagon • 8 Octagon • 9 Nonagon • 10 Decagon????? (10-2)*180 = 8*180=1440 • 11 Hendecagon • 12 Dodecagon**In a regular polygon, all the angles and sides are the same**size. To find the measure of a single angle: Divide the Sum by the Number of Sides. For example: In a regular pentagon, the sum of the interior angles is 540 and the number of sides is 5. °**In your notebooks:Find the measure of a single angle in each**regular polygon. 180 720 360 To find the measure of a single angle: Divide the Sum of the Interior Angles by the Number of Sides/Angles.**To Find the Missing Angle, with an irregular polygon:**Step 1: Find the sum of the interior angles for the polygon (count number of sides, and use formula) Step 2: Add up all the given angles. Step 3: Set two sums equal to each other. Step 4: Solve for x**To Find the Missing Angle:**2 135 1 x Step 1: Find the sum of the interior angles for the polygon (ignore given angles, count number of sides, and use formula) 180(n – 2) = 180(7 – 2) = 180(5) = 900° 3 90 100 7 145 4 95 170 6 5**To Find the Missing Angle:**135 x Step 2: Add up all the given angles. 100 + 95 + 170 + 145 + 90 + 135 + x = 735 + x 90 100 145 95 170**To Find the Missing Angle:**135 x Step 3: Set two sums equal to each other. Step 4: Solve for x (you will need to subtract) 735 + x = 900 -735 -735 x = 165° 90 100 145 95 170**Find the sum of the exterior angles of a polygon**• ALWAYS SUMS TO 360 DEGREES!**Exterior Angle Example**70+60+65+40+y = 360 235+y = 360 y = 125**Finding Perimeter and Area**• Perimeter is the sum of all the side lengths • Area can be found using the appropriate formula.**Guided Practice**Practice this now by completing the guided practice problems: Worksheet 7.1**Independent Practice**To practice this and solidify your learning please complete Homework Set 7.1**Parallelograms**Lesson 7.2**Lesson 7.2 Objectives**The student will be able to… • Recognize and apply the properties of the sides and angles of parallelograms. • Recognize and apply the properties of the diagonals of parallelograms.**Virginia SOL Standard G.9 and G10**• G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • G.10 The student will solve real-world problems involving angles of polygons.**Consecutive Angles**• Two angles whose vertices are the endpoints of the same side are called consecutive angles.**Opposite Angles**• Two angles that are not consecutive are called opposite angles.**Consecutive Sides**• Two sides of a quadrilateral that meet are called consecutive sides.**Opposite Sides**• Two sides that are not consecutive are called opposite sides.**Properties of Parallelograms: Sides and Angles**• A parallelogram is a type of quadrilateral whose pairs of opposite sides are parallel.**Properties of Parallelograms: Sides and Angles**• If a quadrilateral is a parallelogram, then… • (1) its opposite sides are congruent, • (2) its opposite angles are congruent, • (3) its consecutive angles are supplementary. • (4) if one angle is a right angle then all angles are right angles.**Example 1**• Find x and y.**Properties of Parallelograms: Diagonals**• When we refer to the diagonals of a parallelogram, we are talking about lines that can be drawn from vertices that are not connected by line segments.**Properties of Parallelograms: Diagonals**• If a quadrilateral is a parallelogram, then… • (1) its diagonals bisect each other, and • (2) each diagonal splits the parallelogram into two congruent triangles.**Example 4**• Find the value of each variable**Guided Practice**Practice this now by completing the guided practice problems: Worksheet 7.2**Independent Practice**To practice this and solidify your learning please complete Homework Set 7.2**Tests of Parallelograms**Lesson 7.3**Lesson 7.3 Objectives**The student will be able to… • Recognize the conditions that ensure a quadrilateral is a parallelogram. • Prove that a set of points form a parallelogram in the coordinate plane.**Virginia SOL Standard G.9 and G10**• G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. • G.10 The student will solve real-world problems involving angles of polygons.**Tests for Parallelograms**• 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. • A pair of opposite sides is both parallel and congruent.**Example 4a:**Find x so that the quadrilateral is a parallelogram. 3(x-2) 4x-1 Opposite sides of a parallelogram are congruent.**Example 4a:**Substitution Distributive Property Subtract 3x from each side. Add 1 to each side. Answer: When x is 7, ABCD is a parallelogram.**D**E G F Example 4b: Find y so that the quadrilateral is a parallelogram. Opposite angles of a parallelogram are congruent.**Example 4b:**Substitution Subtract 6y from each side. Subtract 28 from each side. Divide each side by –1. Answer: DEFG is a parallelogram when y is 14.**a.**b. Answer: Answer: Your Turn: Find m and n so that each quadrilateral is a parallelogram.**Parallelograms on the Coordinate Plane**• We can use the Distance Formula and the Slope Formula to determine if a quadrilateral is a parallelogram on the coordinate plane. • Just pick one of the tests… and apply either or both of the formulas.**Example 5a:**COORDINATE GEOMETRYDetermine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and D(1, –1) is a parallelogram. Use the Slope Formula.**Answer: Since opposite sides have the same slope,**Therefore, ABCD is a parallelogram by definition. Example 5a: If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.**Example 5b:**COORDINATE GEOMETRYDetermine whether the figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and S(1, –3)is a parallelogram. Use the Distance and Slope Formulas.**Example 5b:**First use the Distance Formula to determine whether the opposite sides are congruent.