1. Section 3-4 Polygon Angle-Sum Theorem SPI 32A: Identify properties of plane figures from information given in a diagram Objectives: Classify Polygons Find the sums of the measures of the interior and exterior angle of polygons Polygon: closed plane figure with at least 3 sides that are segments the sides intersect only at their endpoints no adjacent sides are collinear

2. Classify Polygons Name Polygons By Their: Vertices Start at any vertex and list the vertices consecutively in a clockwise direction (ABCDE or CDEAB, etc) Sides Name by line segment naming convention Angles Name by angle naming convention

3. Classify Polygons by the Number of Sides Number of SidesName 3Triangle 4Quadrilateral 5Pentagon 6Hexagon 8Octagon 9Nonagon 10Decagon 12Dodecagon nn-gon

4. Classify Polygons as Convex or Concave Convex Polygon Has no diagonals with points outside the polygon Concave Polygon Has at least one diagonal outside the polygon

5. Starting with any side, count the number of sides clockwise around the figure. Because the polygon has 12 sides, it is a dodecagon. Classify the polygon below by its sides. Identify it as convex or concave. Think of the polygon as a star. If you draw a diagonal connecting two points of the star that are next to each other,that diagonal lies outside the polygon, so the dodecagon is concave. Classify Polygons as Convex or Concave

6. Triangle Angle-Sum Theorem 1. Draw and cut out a triangle. 2. Number the angles and tear them off. 3. Place the angles adjacent to each other. 4. Compare your results with others. What do you observe about the sum of the angles of a triangle? Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle measure 180º.

8. Polygon Angle-Sum Theorem Use the Triangle Angle-Sum Theorem to find the sum of the measures of the angles of a polygon. 1. Sketch convex polygons with 4, 5, 6, 7, and 8 sides. Construct a table to record your data in order to look for a pattern or rule to find the sum of the measures of the angles of an n-gon. 2. Divide each polygon into triangles by drawing all diagonals that are possible from one vertex. 3. Multiply the number of triangles by 180 to find the sum of the measures of the angles of each polygon. PolygonNumber sides (n) # of TrianglesSum of Interior angle measures (___∙ 180= ___) 422 ∙ 180 = 360 n (n - 2) ∙ 180

9. Polygon Angle-Sum Theorem Theorem 3-9: Polygon Angle-Sum Theorem The sum of the measures of the angles of an n-gon is (n - 2) 180. A decagon has 10 sides, so n = 10. Sum = (n – 2)(180) Polygon Angle-Sum Theorem = (10 – 2)(180) Substitute 10 for n. = 8 180 Simplify. = 1440 Find the sum of the measures of the angles of a decagon.

10. Polygon Angle-Sum Theorem The sum of the measures of the angles of a given polygon is 720. How can you use the Polygon Angle- Sum Theorem to find the number of sides in the polygon? Sum = (n – 2) 180 Write the Equation 720 = (n – 2) 180 Sub. In known values 720 = 180n – 360 Simplify 1080 = 180n Addition Prop of EQ 6 = n Hexagon (6 sides)

11. m X + m Y + m Z + m W = (4 – 2)(180) Polygon Angle-Sum Theorem m X + m Y + 90 + 100 = 360 Substitute. m X + m Y + 190 = 360 Simplify. m X + m Y = 170 Subtract 190 from each side. 2m X = 170 Simplify. m X = 85 Divide each side by 2. m X + m X = 170 Substitute m X for m Y. The figure has 4 sides, so n = 4. Find m  X in quadrilateral XYZW. Use the Polygon Angle-Sum Theorem

12. Polygon Exterior Angle-Sum Theorem Equilateral Polygon: all sides are congruent Equiangular Polygon: all angles are congruent Regular Polygon: is both equilateral and equiangular

13. Because supplements of congruent angles are congruent, all the angles marked 1 have equal measures. The hexagon is regular, so all its angles are congruent. An exterior angle is the supplement of a polygon’s angle because they are adjacent angles that form a straight angle. Below is a regular hexagon game board packaged in a rectangular box. Explain how you know that all the angles labeled 1 have equal measures. Real-world Connection