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Angle Relationships in Polygons Definitions: • in a Regular Polygon, All Are Equal and All

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Angle Relationships in Polygons Definitions: • in a Regular Polygon, All Are Equal and All Name: Date: MPM 1D Spiral 4 U4-L7 – Angle Relationships in Polygons Definitions: • in a regular polygon, all are equal AND all are equal • examples of regular polygons: equilateral triangle, square A polygon with every interior angle less than • Note the difference Convex 180°; any straight line through it crosses, at between an Polygon most two sides. convex and a concave polygon A polygon with at least one interior angle Concave greater than 180°; a straight line through it may Polygon cross more than two sides. Investigation 1 a) b) c) d) e) f) Use your observations to complete the following table and corresponding graph: Number Sum of Number of Interior of Sides triangles Angles a) 3 b) 4 c) 5 d) 6 e) 7 f) 8 g) 120 h) 143 i) 155 j) 1440 In general, the relation between number of sides a sum of the interior angles of a polygon can be written as Use your formula to calculate the values for g-j. 2 Investigation 2 For each regular polygon below, use a ruler to extend one of the sides as shown in the pentagon below. Calculate the measure of each interior and exterior angle. Determine the sum of one exterior angle and its corresponding interior angle. Determine the sum of all exterior angles. • In a polygon, at each vertex, the sum of the and angles is . • Consider the equilateral triangle to the right: o the triangle has equal sides, therefore there are equal angles o since the sum of the angles in a triangle is 180 ° , each interior angle S can be calculated as follows: each interior angle= , where S n represents the sum of the interior angles in a polygon and n represents the number of sides o extend each side similar to the diagram above o each interior angle is : o each exterior angle is: o state the sum of the exterior angles: • complete the steps above for each regular polygon below and complete the table each each number sum of interior exterior sum of the number type of polygon of interior angle of a angle of exterior of sides triangles angles regular a regular angles polygon polygon equilateral a) 3 1 180 60 120 360 triangle b) 4 c) 5 d) 6 e) 7 f) 122 g) 12 h) 170 i) 12 n 3 Conclusions: • The sum of the exterior angles of a regular polygon is • The exterior angle of a regular polygon can be calculated in the following ways: (i) each exterior angle = the sum of the exterior angles divided by the number of sides 360 each exterior angle = n or 180 (n − 2) (ii) calculate each interior angle using , then subtract the answer from 180 n For example, given a regular hexagon, determine the measure of each exterior angle. Method (i) the sum of the exterior angles divided by the number of sides: 360 360 = = 60 Each exterior angle is 60° n 6 Method (ii) calculate each interior angle and subtract it from 180 0 180 (n − 2) each interior angle = each exterior angle = 180 − 120 n 180 (6 − 2) = = 60 ° 6 = 120 ° Angle Properties – Grade 8 Review Scalene Triangle: Isosceles Triangle: Equilateral Triangle: • no equal sides • two equal sides • all sides equal • no equal angles • the two base angles are • all angles equal equal Acute Triangle: all three angles are less than 90 degrees Right Triangle: one right angle (90 degrees) Obtuse Triangle: one angle greater than 90 but less than 180 degrees 4 Complementary Angles: add to 90° Supplementary Angles: add to 180° x = __________ y = __________ Fill in the blanks: • the sum of the interior angles in a triangle is a b c • the measure of the exterior angle in a triangle is the sum of c b a Examples 1. Find the sum of the interior angles in a polygon that has 12 sides. 2. Find the measure of each interior angle of a regular polygon with 17 sides. 3. If the interior angle of a regular polygon is 174°. How many sides does the polygon have? 5 In your binder. 4. The measure of each exterior angle of a regular polygon is 10°. How many sides does the polygon have? 5. Find the value of x for the polygon shown to the right. 6. Find the measure of the indicated angle. a) b) a b x 55 ° c d e 44 ° f 165 ° c) d) 95 ° 150 ° y z x 134° 70 ° y Note: Diagrams are not drawn to scale. Practice 1. Find the sum of the interior angles of a polygon with a) 10 sides b) 15 sides c) 20 sides 2. Find the measure of each interior angle of a regular polygon with a) 12 sides b) 32 sides 3. How many sides does a polygon have if the sum of its interior angles is a) 540° b) 1800° c) 3060° 4. Complete the table below by filling in the missing values. number of number of sum of the each number of diagonals Polygon triangles in interior exterior sides from one the polygon angles angle vertex a) decagon b) icosagon c) pentacontakaipentagon d) hectogon 5. Determine the measure of each exterior angle in a regular 11- sided polygon as shown to the right. 6 6. Determine the measure of each interior angle in a regular 15-sided polygon. 7. Determine the number sides in a regular polygon if each interior angle is a) 140° b) 168° 8. Determine the value of x for the polygon shown to the right. 9. How many sides does a Canadian dollar coin have? a) What is the measure between the adjacent sides of the coin? b) Suggest reasons why the Royal Canadian Mint chose this shape. 10. Determine the unknown values. a) b) c) d) e) f) g) h) i) x 42 ° y x y j) y k) x x y -------------------------------------------------------------------------------------------------------------------------------------------- Solutions: 1. 1440 o, 2340 o, 3240 o 2. 150 o, 168.75 o 3. 5,12,19 4a) 10, 7,8,1440 o,144 o,36 o b) 20, 17, 18, 3240 o, 162 o, 18 o c) 55, 52,53,9540 o,173 o,7 o d) 100, 97,98,17640 o,176.4 o,4.6 o 5. 32.7 o 6. 156° 7a) 9 b) 30 8. 48 9a) 11 b) 147.27° 10a) x=60°,y=30°, z=120° b) x=50°, y=z=25° c) a=35°, b=145°, c=60°, d=85°, e=95° d) x=35° e) x=140° f) a=100°, b=128°, y=30° g) x=98° h) a=75°, b=105° .
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