6-1 Angles of Polygons The student will be able to:

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Objectives Classify polygons based on their sides and angles.

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Presentation transcript:

6-1 Angles of Polygons The student will be able to: 1. Find the sum of the measures of the interior angles of a polygon. 2. Find the sum of the measures of the exterior angles of a polygon.

A Polygon is a closed figure with sides constructed from segments A Polygon is a closed figure with sides constructed from segments. Polygons can be convex or concave and regular or irregular. Tell whether the following figures are polygons. If so, tell whether each is convex or concave and regular or irregular. yes yes yes convex convex convex regular regular irregular yes no concave irregular

Interior Angle sum Theorem – If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = 180(n – 2). Convex Polygon Number of Sides (n) Sum of Angles (S) Triangle 3 180(3 – 2) = 180 Quadrilateral 4 180(4 – 2) = 360 Pentagon 5 180(5 – 2) = 540 Hexagon 6 180(6 – 2) = 720 Heptagon 7 180(7 – 2) = 900 Octagon 8 180(8 – 2) = 1080 n sides n 180(n – 2) = 180n - 360

How many sides do we have? 4 Example 1: Find the measure of each interior angle of quadrilateral ABCD. How many sides do we have? 4 How many degrees are in the quadrilateral? 360 How do we find the sum of the interior angles? 360 = 90 + 90 + 3x + x = 135 360 = 180 + 4x -180 180 = 4x = 45 45 = x

How many sides do we have? 5 Find the measure of each interior angle of pentagon HJKLM shown. 370 = 10x How many sides do we have? 5 How many degrees are in a pentagon? 540 How do we find the sum of the interior angles? 540 = 142 + 2x + 2x + (3x + 14) + (3x + 14) 540 = 170 + 10x = 74 -170 370 = 10x = 74 37 = x = 125 = 125

sum of interior angle measures = each angle measure Example 2: The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. sum of interior angle measures = each angle measure number of congruent angles 180(n – 2) = 144 n 180n – 360 = 144n -180n -360 = -36n 10 = n

360 _ number of congruent angles = each angle measure The sum of the measures of Exterior Angles Exterior Angle Sum Theorem – If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360°. No matter how many sides there are in the convex polygon, the sum of the exterior angles of that polygon equals 360°. 360 _ number of congruent angles = each angle measure The sum of an interior angle and an exterior angle is 180°.

Example 4: Find the value of x in the diagram. 139 + 6x + 9x + 2x = 360 139 + 17x = 360 -139 17x = 221 x = 13 Example 5: Find the measure of each exterior angle of a regular dodecagon. How many sides are in a dodecagon? 12 How many degrees are in the exterior angles of a dodecagon? 360 How do we find the measure of each exterior angle? 360_______ # of congruent angles = the measure of each angle