1. GEOMETRY HELP Name the polygon. Then identify its vertices, sides, and angles. The polygon can be named clockwise or counterclockwise, starting at any vertex. Possible names are ABCDE and EDCBA. Its vertices are A, B, C, D, and E. Its angles are named by the vertices, A (or EAB or BAE), B (or ABC or CBA), C (or BCD or DCB), D (or CDE or EDC), and E (or DEA or AED). Its sides are AB or BA, BC or CB, CD or DC, DE or ED, and EA or AE. Quick Check The Polygon Angle-Sum Theorems LESSON 3-5 Additional Examples

2. GEOMETRY HELP 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. The Polygon Angle-Sum Theorems LESSON 3-5 Additional Examples Quick Check

3. GEOMETRY HELP 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. The Polygon Angle-Sum Theorems LESSON 3-5 Additional Examples Quick Check

4. GEOMETRY HELP 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. Quick Check The Polygon Angle-Sum Theorems LESSON 3-5 Additional Examples

5. GEOMETRY HELP Because supplements of congruent angles are congruent, all the angles marked 1 have equal measures. Sample: 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. A regular hexagon is inscribed in a rectangle. Explain how you know that all the angles labeled 1 have equal measures. The Polygon Angle-Sum Theorems LESSON 3-5 Additional Examples Quick Check