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Name: __________________________ What You’ll Learn Properties of various polygons How to calculate for the sum of interior angle in a regular polygon How to calculate for the measure of an interior angle in a regular polygon Why It’s Important Polygons are used by: Professional tilers creating designs for flooring or backsplashes. Engineers looking to keep soldiers safe. Brick layers for pathways. Key Formulas NOTE: Here n = the number of sides Sum of the interior angles: 푆 = 180(푛 − 2) 180 (푛−2) Measure of an interior angles: 푀 = 푛 360 Measure of central angle: 퐶 = 푛 The practice problems have been taken from a variety of sources including: MathWorks 12 Math at Work 12 Apprenticeship and Workplace 12 Provincial exams Grade 12 Essentials - Polygons Getting Started: Notes Measuring Angles To measure an angle: - Put the vertical marker of the protractor at the vertex (corner) of the angle you are measuring. - Make sure that the 0̊ line is along one of the legs of the angle. - Follow out to the second leg and read the measurement on the protractor. Depending on the length of the line, it may be difficult to use the outside measurements on the protractor. Marking Sides and Angles - Capital letters are used to mark vertices of a polygon. - The line segment (side) of a polygon is denoted by the two vertices (corners) it sits between. - Congruent (equal) sides are marked with dashes. - Congruent (equal) angles are marked with arcs. Examples: 1. Congruency of Angles 2. Congruency of Sides 2 Grade 12 Essentials - Polygons Getting Started: Practice NOTE: is the symbol for angles. The letter that is listed in the middle is the angle that is being measured. Ex. ABC = 60˚ means that angle B = 60˚. (We could also call this B) 1. Measure each of the following angles. a) A ABC= B C b) ACB= B BCD= A D C 2. Record the side lengths and angles for ∆ABC. AB = A= AC= B= BC= C= 3 Grade 12 Essentials - Polygons 4 Grade 12 Essentials - Polygons 5 Grade 12 Essentials - Polygons Triangles: Notes Definitions Triangle: Vertex (pl. Vertices): Classification Triangles can be classified by their… Angles 1. Right: 2. Acute: 3. Obtuse: Side Lengths 1. Equilateral: 2. Isosceles: 3. Scalene: 6 Grade 12 Essentials - Polygons Triangles: Practice 1. Circle the types of triangles that have each property. NOTE: There may be more than one right answer. a) Some sides are equal. Equilateral Isosceles Scalene b) No interior angles are equal. Equilateral Isosceles Scalene c) The sum of the interior angles is 180˚. Acute Right Equilateral The Esplanade Riel Bridge 2. Circle the types of triangles that do not have each property. a) All angles are less than 90˚. Acute Right Obtuse b) Some angles are equal. Equilateral Isosceles Scalene c) There are at least 2 equal sides. Equilateral Isosceles Scalene 3. Use the angle measures to calculate the unknown angles in each triangle. NOTE: The following are not to scale. Use calculations rather than a protractor to solve. 7 Grade 12 Essentials - Polygons 4. Use the following diagram to answer the questions below. a) What is the measure of M? b) Classify ∆MNP by angle measure and by side length. 5. Recall our activity at the start of class … a) The sum of the interior angle plus the exterior angle is the same at each vertex. What is this sum? b) Why does it make sense that each vertex has the same sum? c) Is this a property for all triangles? Explain. 8 Grade 12 Essentials - Polygons Quadrilaterals: Notes Which of the following shapes are polygons? Circle each one. Definitions Polygon: Quadrilateral: Types of Quadrilaterals Rectangle: Square: Parallelogram: Rhombus: Trapezoid: Kite: 9 Grade 12 Essentials - Polygons Irregular Quadrilateral: Concave Quadrilateral: NOTE: Some polygons are also convex, which means that there are no interior angles which are greater than 180˚. 10 Grade 12 Essentials - Polygons Activity – Properties of Quadrilaterals We can take a polygon and draw diagonals, which are line segments joining vertices that are NOT NEXT TO EACH OTHER. Example) In pairs, complete the following instructions and answer each question using the given square. A B 1. Draw the diagonal AC. Measure its length. 2. Draw the diagonal BD. Measure its length. What do you notice? 3. Label the point where the diagonals intersect as E. Measure the lengths of AE, BE, CE, and DE. What do you notice? 4. Measure DEA, AEB, BEC, and CED. What do D C you notice? 5. What is the sum of the angles where the diagonals intersect? What do we find? - The diagonals are . That is, they are the equal. - The diagonals on a square are . That is, they cross at a 90˚ angle. - The diagonals each other. That is, they cut each other in half. All regular polygons have certain properties when you draw diagonals. See the following page for an overview of the properties. 11 Grade 12 Essentials - Polygons 12 Grade 12 Essentials - Polygons Quadrilaterals: Practice 1. Determine the missing measurements using the properties of quadrilaterals. 2. State two properties that would prove that a quadrilateral is a parallelogram. 13 Grade 12 Essentials - Polygons 3. List all the quadrilaterals that could fit each description. a) Has at least one set of parallel sides b) Has four equal side lengths c) Has two equal diagonals 4. Sketch and name a quadrilateral that fits each description. a) The diagonals are equal, but the sides are not all equal. b) The diagonals are equal, and all the sides are equal. c) The diagonals are not equal, and no two sides are equal. 5. Using your knowledge of the properties of quadrilaterals, find the measures of the missing angles. What kind of quadrilateral is this? 6. Solve for the indicated length or angle, and identify the type of quadrilateral. a) b) a. 14 Grade 12 Essentials - Polygons Regular Polygons: Notes Definitions Regular Polygon: Example 1) Are the following shapes regular Common Polygons polygons? Explain. Name Number of Sides Triangle a) b) Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Activity – Measure and Sum of Interior Angles 1. Draw a square. Then draw a diagonal between two non-adjacent corners so the square is divided into triangles. 2. How many triangles were created? 3. What is the sum of the interior angles of a square? 4. What is the measure of each interior angles of a square? 5. For the regular pentagon below, repeat steps 2-5. NOTE: You will need to draw more than one diagonal to divide each shape into triangles. 15 Grade 12 Essentials - Polygons 6. For the regular hexagon below, repeat steps 2-5. Again, you will need to draw more than one diagonal to divide each shape into triangles. 7. Using your results from steps 1-7, complete the following table: Figure Number of Number of Sum of Interior Measure of Each Sides Triangles Angles Individual Angle Equilateral Triangle Square Regular Pentagon Regular Hexagon 8. Use your chart from Question 8 to answer the following questions. a. How does the number of triangles you can make in a polygon relate to the number of sides? b. How many triangles can you make in a 12-sided polygon? c. What is the sum of all the angles measures in a 12-sided polygon? d. What is the measure of each angle in a 12-sided polygon? 16 Grade 12 Essentials - Polygons Properties We can use formulas to find the measure of an interior angle, as well as the sum of the interior angles of a regular polygon. Sum of the interior angles: 푆 = 180(푛 − 2) where n is the number of sides Example 2) Find the sum of the interior angles of a hexagon. Example 3) Working backwards: The sum of the interior angles of a polygon is 900˚. Determine the number of sides of the polygon. Measure of an interior angle: If we know that all of the angles in a hexagon sum to 720˚, how can we find one angle? 180 (푛−2) 푀 = where n is the number of sides 푛 Example 4) Find the measure of an interior angle in a square. 17 Grade 12 Essentials - Polygons Measure of the central angle: We can also determine the measure of the central angles in a regular polygon. The central angle is the angle made at the center of a polygon by any two adjacent vertices of the polygon. All central angles would add up to 360º (a full circle), so the measure of the central angle is 360 divided by the number of sides 360 퐶 = where n is the number of sides. 푛 Example 5) What is the measure of the central angle in a hexagon? Example 6) A regular polygon has central angles of 45º. a) State the number of sides for this polygon. b) State the name of this polygon. 18 Grade 12 Essentials - Polygons Regular Polygons: Practice 1. Which are regular polygons? Check with a ruler and a protractor. 2. What is the measure of each interior angle in a regular octagon? 3. Given the following regular polygon: a) Calculate the sum of the interior angles in the polygon. b) State the measure of each interior angle in the polygon. 4. The sum of the interior angles is 900º. Determine the number of sides of the polygon. 19 Grade 12 Essentials - Polygons 5. A regular hexagon has a side length of 10 metres. a) State the measure of angle A, the central angle, in degrees. b) State the measure of the given diagonal in metres. 6. Draw ALL diagonals in each regular polygon.
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