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## Sum of Interior and Exterior Angles in Polygons

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**Sum of Interior and Exterior Angles in Polygons**Essential Question – How can I find angle measures in polygons without using a protractor? Key Standard – MM1G3a**Warm-Up**(Triangles)**Polygons**• A polygon is a closed figure formed by a finite number of segments such that: 1. the sides that have a common endpoint are noncollinear, and 2. each side intersects exactly two other sides, but only at their endpoints.**Polygons**• Can be concave or convex. Concave Convex**Polygons are named by number of sides**Triangle 3 4 Quadrilateral Pentagon 5 Hexagon 6 Heptagon 7 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon**Regular Polygon**• A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular polygon.**Draw a:** Quadrilateral Pentagon Hexagon Heptagon Octogon • Then draw diagonals to create triangles. • A diagonal is a segment connecting two nonadjacent vertices (don’t let segments cross) • Add up the angles in all of the triangles in the figure to determine the sum of the angles in the polygon. • Complete this table**3**1 180° 4 2 2 · 180 = 360° 5 3 3 · 180 = 540° 4 4 · 180 = 720° 6 7 5 5 · 180 = 900° 8 6 6 · 180 = 1080° n n - 2 (n – 2) · 180°**Polygon Interior Angles Theorem**The sum of the measures of the interior angles of a convex n-gon is (n – 2) • 180. Examples – • Find the sum of the measures of the interior angles of a 16–gon. • If the sum of the measures of the interior angles of a convex polygon is 3600°, how many sides does the polygon have. • Solve for x. (16 – 2) 180 = 2520° (n – 2) 180 = 3600 180n = 3960 180 180 n = 22 sides 180n – 360 = 3600 + 360 + 360 (4 – 2)*180 = 360 4x - 2 108 108 + 82 + 4x – 2 + 2x + 10 = 360 6x = 162 6 6 6x + 198 = 360 2x + 10 82 x = 27**There are two sets of angles formed when the sides of a**polygon are extended. • The original angles are called interior angles. • The angles that are adjacent to the • interior angles are called exterior angles. • These exterior angles can be formed when any side is extended. • What do you notice about the interior angle and the exterior angle? • What is the measure of a line? • What is the sum of an interior angle with the exterior angle? Draw a quadrilateral and extend the sides. They form a line. 180° 180°**If you started at Point A, and followed along the sides of**the quadrilateral making the exterior turns that are marked, what would happen? You end up back where you started or you would make a circle. What is the measure of the degrees in a circle? A D B C 360°**Polygon Exterior Angles Theorem**• The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360°. • Each exterior angle of a regular polygon is 360 n where n is the number of sides in the polygon**Example**Find the value for x. Sum of exterior angles is 360° (4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360 7x + 248 = 360 – 248 – 248 7x = 112 7 7 (4x – 12)⁰ 68⁰ 60⁰ x = 16 54⁰ (3x + 13)⁰ 65⁰ What is the sum of the exterior angles in an octagon? What is the measure of each exterior angle in a regular octagon? 360° 360°/8 = 45°**Objectives:**• recall the primary parts of a triangle • show that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side • solve for the length of an unknown side of a triangle given the lengths of the other two sides. • solve for the range of the possible length of an unknown side of a triangle given the lengths of the other two sides • determine whether the following triples are possible lengths of the sides of a triangle**Triangle Inequality Theorem**B • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC AB + AC > BC AC + BC > AB C A**Is it possible for a triangle to have sides with the given**lengths? Explain. a. 3 ft, 6 ft and 9 ft • 3 + 6 > 9 b. 5 cm, 7 cm and 10 cm • 5 + 7 > 10 • 7 + 10 > 5 • 5 + 10 > 7 c. 4 in, 4 in and 4 in • Equilateral: 4 + 4 > 4 (NO) (YES) (YES)**Solve for the length of an unknown side (X) of a triangle**given the lengths of the other two sides. a. 6 ft and 9 ft • 9 + 6 > x, x < 15 • x + 6 > 9, x > 3 • x + 9 > 6, x > – 3 • 15 > x > 3 b. 5 cm and 10 cm c. 14 in and 4 in The value of x: |a - b|<x< a + b 5 < x < 15 10< x < 18**Solve for the range of the possible value/s of x, if the**triples represent the lengths of the three sides of a triangle. • Examples: a. x, x + 3 and 2x b. 3x – 7, 4x and 5x – 6 c. x + 4, 2x – 3 and 3x d. 2x + 5, 4x – 7 and 3x + 1**OBJECTIVES:**• recall the Triangle Inequality Theorem • state and identify the inequalities relating sides and angles • differentiate ASIT (Angle – Side Inequality Theorem) from SAIT (Side – Angle Inequality Theorem) and vice-versa • identify the longest and the shortest sides of a triangle given the measures of its interior angles • identify the largest and smallest angle measures of a triangle given the lengths of its sides**INEQUALITIES RELATING SIDES AND ANGLES:**ANGLE-SIDE INEQUALITY THEOREM: • If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. If AC > AB, then mB > mC. SIDE-ANGLE INEQUALITY THEOREM: • If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If mB > mC, then AC > AB. C B A**EXAMPLES:**O E • List the sides of each triangle in ascending order. a. e. c. R 70 61 J 73 59 P N M L 31 JR, RE, JE ME & EL, ML PO, ON, PN I d. b. E A P 42 46 U E 79 AT, PT, PA UE, IE, UI T**Objectives:**• recall the parts of a triangle • define exterior angle of a triangle • differentiate an exterior angle of a triangle from an interior angle of a triangle • state the Exterior Angle theorem (EAT) and its Corollary • apply EAT in solving exercises • prove statements on exterior angle of a triangle**Exterior Angle of a Polygon:**• an angle formed by a side of a and an extension of an adjacent side. • an exterior angle and its adjacent interior angle are linear pair 3 2 1 4**Exterior Angle Theorem:**• The measure of each exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. • m1 = m3 + m4 3 2 1 4**Exterior Angle Corollary:**• The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles. • m1 > m3 and m1 > m4 3 2 1 4**Examples: Use the figure on the right to answer nos. 1- 4.**• The m2 = 34.6 and m4 = 51.3, solve for the m1. • The m2 = 26.4 and m1 = 131.1, solve for the m3 and m4. • The m1 = 4x – 11, m2 = 2x + 1 and m4 = x + 18. Solve for the value of x, m3, m1 and m2. • If the ratio of the measures of 2 and 4 is 2:5 respectively. Solve for the measures of the three interior angles if the m1 = 133. 1 3 2 4