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## Triangle

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**Each segment that forms a polygon is a side of the polygon.**The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.**You can name a polygon by the number of its sides. The table**shows the names of some common polygons. 3 Triangle 8 Octagon 4 9 Nonagon Quadrilateral 5 Pentagon 10 Decagon 6 12 Hexagon Dodecagon 7 n Heptagon N-gon**Remember!**A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Example 1 Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon polygon, heptagon not a polygon polygon, nonagon not a polygon**All the sides are congruent in an equilateral polygon. All**the angles are congruent in an equiangular polygon. A regular polygonis one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.**Example 2:**Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex regular, convex regular, convex**In each convex polygon, the number of triangles formed is**two less than the number of sides n. So the sum of the angle measures of all these triangles is (n — 2)180°. Example 3A: Find the sum of the interior angle measures of a convex heptagon. (n – 2)180° Polygon Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify.**Find the measure of each interior angle of a regular 16-gon.**Example 3B: Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon Sum Thm. Substitute 16 for n and simplify. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.**Find the measure of each interior angle of pentagon ABCDE.**Example 3C: (5 – 2)180° = 540° Polygon Sum Thm. Polygon Sum Thm. mA + mB + mC + mD + mE = 540° 35c + 18c+ 32c+ 32c+ 18c= 540 Substitute. 135c= 540 Combine like terms. c= 4 Divide both sides by 135. mA = 35(4°)= 140° mB = mE = 18(4°)= 72° mC = mD = 32(4°)= 128°**Remember!**An exterior angle is formed by one side of a polygon and the extension of a consecutive side. In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.**measure of one ext. =**Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20 sides and 20 vertices. sum of ext. s = 360°. Polygon Sum Thm. A regular 20-gon has 20 ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°.**Example 4B: Finding Interior Angle Measures and Sums in**Polygons Find the value of b in polygon FGHJKL. Polygon Ext. Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b°= 360° 120b= 360 Combine like terms. b= 3 Divide both sides by 120.**Example 5: Art Application**Ann is making paper stars for party decorations. What is the measure of 1? 1 is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°. A regular pentagon has 5 ext. , so divide the sum by 5.