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.

Nonexamples

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.

Then draw diagonals to create triangles. 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 Polygon # of sides # of triangles Sum of interior angles

Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon n-gon 3 1 Polygon # of sides # of triangles Sum of interior angles Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon n-gon 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 2x + 10 6x + 198 = 360 82 x = 27

Draw a quadrilateral and extend the sides. 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? They form a line. 180° 180°

You end up back where you started or you would make a circle. 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°

Triangle Inequality (Triangle Inequality Theorem)

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 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)

The value of x: |a - b|< x < a + b 5 < x < 15 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

Examples: a. x, x + 3 and 2x b. 3x – 7, 4x and 5x – 6 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

TRIANGLE INEQUALITY (ASIT and SAIT)

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 mB > mC. SIDE-ANGLE INEQUALITY THEOREM: If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If mB > mC, then AC > AB. C A B

EXAMPLES: List the sides of each triangle in ascending order. a. c. e. 61 70 J 73 59 P N M L 31 JR, RE, JE PO, ON, PN ME & EL, ML b. I d. E A P 42 46 U E 79 AT, PT, PA UE, IE, UI T

Triangle Inequality (Exterior Angle Theorem)

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 1 2 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. m1 = m3 + m4 3 1 2 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. m1 > m3 and m1 > m4 3 1 2 4

Examples: Use the figure on the right to answer nos. 1- 4. The m2 = 34.6 and m4 = 51.3, solve for the m1. The m2 = 26.4 and m1 = 131.1, solve for the m3 and m4. The m1 = 4x – 11, m2 = 2x + 1 and m4 = x + 18. Solve for the value of x, m3, m1 and m2. 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 m1 = 133. 1 3 2 4