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Kuta Software - Infinite Geometry Name___________________________________ Angles and Their Measures Date________________ Period____ Find the measure of each angle to the nearest degree. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ©H g2B0u1I1Z sKRuytNax QSpoEfWtOwbaMrZeo ELJLECM.P q SAUlKlr Zroiog4hMtds9 9rje7sze5ruvtepdM.w J CM2aDd2eu uwniVtwh5 OIqncf0iSnAiptNef BGFe7oWmVektDrIyI.n -1- Worksheet by Kuta Software LLC 1 Draw an angle with the given measurement. 11) 90° 12) 70° 13) 120° 14) 105° 15) 31° 16) 166° 17) 144° 18) 53° ©8 42m0m1r1K 0KjuotGam vS3oaf8t8wnaNroeQ nLTLHCk.o x QA7lald ErbiNgthdtxsC 6rzeUsheJr6vRevdB.F P cMUardseC qwwi9t5hQ NIZndfOion6iXtdey xG5eMo3mweytQrny0.V -2- Worksheet by Kuta Software LLC 2 Defining & Drawing Shapes Find the definitions of each of the shapes below: SquareRectangleTriangleRight triangleEquilateral triangle- Draw a triangle below that has one angle that is 35° and one angle that is 45°. Draw a square below that has 2.5 inch sides (use a ruler and a protractor). 3 Draw a triangle below with 60° angles and 3 inch sides Draw a rectangle below with 2 sides that are 3.75 inches long and two sides that are 5.25 inches long (use a ruler and a protractor) 4 Naming Polygons Find the names of the following Polygons: Polygon’s Name Definition A polygon with 3 sides A polygon with 4 sides A polygon with 5 sides A polygon with 6 sides A polygon with 7 sides A polygon with 8 sides A polygon with 9 sides A polygon with 10 sides A polygon with 12 sides A polygon with 20 sides Find the definition of a Regular Polygon: Find the definition of an Interior Angle: Shape #1 Triangle Measure each of the angles on the triangle to the right. What is the total (sum) of their measurements? Number of Sides Total Degrees Is this the same for all triangles? 5 Interior Angles of Polygons Now using what you learned about the interior angle total (sum) of a triangle, prove the total angles of these other polygons by dividing the shapes into the fewest number of triangles as possible. Use those triangles to calculate the sums of the new shapes. 1. Lexi was solving the problems above when she noticed a pattern. “Aha”, she exclaimed, “the number of triangles you can draw in each polygon is related to the number of sides of that polygon!”. Explain what she means. How can this be used to create a shortcut for finding the sum of the interior angles of any polygon? 2. Use Lexi’s discovery to find the sum of the interior angles of a regular polygon that has 8 sides. 3. How would you find the angle measurement of one of the angles in #2? 6 Interior Angle Theorem The sum of the interior angles of any polygon is equal to the number of sides of the polygon (n) minus 2 (this is the number of triangles in each polygon) then multiplied by 180 (the sum of the degrees inside a triangle). 𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚 = 𝑛 − 2 ∗ 180° If the polygon is regular, you can find the measurement of each of the interior angles (if it is regular all the angles will be the same) by dividing the interior angle sum by the number of sides (n). 𝑅𝑒𝑔𝑢𝑙𝑎𝑟 𝑝𝑜𝑙𝑦𝑔𝑜𝑛 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 = 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚 ÷ 𝑛 4. Find the sum of the interior angles of a decagon. 5. Find the sum of the interior angles of a regular polygon that has 16 sides. 6. What is the measure of each angle of the regular polygon in #5? 7. Find the number of sides in a polygon whose sum of the interior angles is 1440. 8. Find the number of sides in a polygon whose sum of the interior angles is 1800. 9. What is the measure of each angle of the regular polygon in #8? 7 10. Find the sum of the degree of the measures of the interior angles of a regular polygon that has 13 sides. 11. What is the measure of one interior angle in the regular polygon in #10? 12. Find the number of sides in a polygon whose sum of the interior angles is 2700. 13. Find the sum of the degree measures of the interior angles of a regular polygon that has 15 sides. 14. What is the measure of one interior angle in the regular polygon in #13? More Polygon Measurements Complete the following problem with your group: Dog Pen Problem: Ms. Mary got a new dog and wants to make the largest possible pen for him. She has 60 feet of fencing. What is the largest area the pen can have? 8 Before you move on to the next part of the packet, you must have your 6 recommended Khan Skills mastered (out of this list…please follow the order below): Nets of 3D figures Area of parallelogram Area of triangle Surface area using nets Surface area Area of composite shapes Area of trapezoids Area challenge Volume 1 Volume with fractions Volume with Unit Cubes Volume Word Problems Review… 9 Putting it all together…The Lantern Project Time to choose a polyhedron for your Lantern Project – Use the websites below to search through many different polyhedron shapes. Once you have decided on the one you want, print out the net and fill in the name of the polyhedron below. http://www.senteacher.org/worksheet/12/NetsPolyhedra.html http://www.korthalsaltes.com/ Name of the polyhedron you chose ___________________________________________________ Now that you have chosen your Polyhedron net for your lantern, you need to make some calculations to determine how to enlarge your figure. What is the total number of faces of your polyhedron? _________________ Your polyhedron is made up of one or more 2-dimensional polygons (such as a triangle, square, hexagon, etc.) You will need to find the angle measurements and side lengths of each of these polygons. Use the space below to record these measurements: Name of Polygon 1 ________________________ Sum of interior angles of polygon 1 ___________________ Measurement of each angle ________________ (If polygon is regular, show calculation below. If not, carefully measure each angle) Length of each side (list all lengths if they are not the same) ______________________________ Name of Polygon 2 ________________________ Sum of interior angles of polygon __________________ Measurement of each angle ________________ (If polygon is regular, show calculation below. If not, carefully measure each angle) Length of each side (list all lengths if they are not the same) ______________________________ 10 Name of Polygon 3 ________________________ (if needed) Sum of interior angles of polygon _________________ Measurement of each angle ________________ (If polygon is regular, show calculation below. If not, carefully measure each angle) Length of each side (list all lengths if they are not the same) ______________________________ ******************************************************************************************* Now it is time to see what your polyhedron will look like once it is completely constructed. Go ahead and fold your net. Diameter of your polyhedron ___________ (measure best you can from one side to the other) Based on this diameter and the shape of your polyhedron, determine the scale factor for your enlarged polyhedron. Remember that there will need to be enough space for a light bulb to fit inside. Scale factor for lantern _____________ STOP Before moving on you must get your scale factor approved by a teacher. Teacher initials ________ ENLARGEMENT: With your new measurements and your scale factor, you can begin to enlarge your polygons to provide you with a template to use in construction. Use blank paper to precisely construct each polygon to the enlarged size. You will turn these constructions in before you begin building. 11 Surface Area Calculations One more step before we begin our lamp construction. We need to calculate the total surface area of our Polyhedron in order to estimate the tissue paper and ModPodge we will need for the construction. Use this worksheet to calculate the surface area of each enlarged polygon in your polyhedron. Total up your values on the bottom of the back page and give your teacher these results. Name of Polygon #1 ________________________ Formula for calculating the surface area of this polygon __________________________________ Calculations: How many faces of this polygon do you have in the polyhedron? __________ Total surface area of polygon #1: _____________________ x __________________ = ___________________ surface area of polygon # of faces total surface area of polygon #1 -------------------------------------------------------------------------------------------------------------------------------------------------Name of Polygon #2 ________________________ Formula for calculating the surface area of this shape __________________________________ Calculations: How many faces of this polygon do you have in the polyhedron? __________ Total surface area of polygon #2: _____________________ x __________________ = ___________________ surface area of polygon # of faces total surface area of polygon #2 -------------------------------------------------------------------------------------------------------------------------------------------------12 Name of Polygon #3 ________________________ (if needed) Formula for calculating the surface area of this polygon __________________________________ Calculations: How many faces of this polygon do you have in the polyhedron? __________ Total surface area of polygon #3 (if needed): _____________________ x __________________ = ___________________ surface area of polygon # of faces total surface area of polygon #3 -------------------------------------------------------------------------------------------------------------------------------------------------TOTAL Total surface area __________________+ ________________ + ________________ = total surface area of #1 total surface area of #2 total surface area of #3 Congratulations! You are ready to begin building your lantern! Please turn in this packet and all extra drawings/calculations to receive your Supplies Tracker and supplies. 13