Transcript 6-1
6-1 Angles of Polygons You named and classified polygons. • Find and use the sum of the measures of the interior angles of a polygon. • Find and use the sum of the measures of the exterior angles of a polygon. Definitions A polygon is a plane figure whose sides are three or more coplanar segments that intersect only at their endpoints (the vertices). Consecutive sides cannot be collinear, and no more than two sides can meet at any one vertex. Definition A diagonal of a polygon is a line segment whose endpoints are any two nonconsecutive vertices of the polygon. diagonal Classification of Polygons 3 sides Triangle 4 sides Quadrilateral 5 sides Pentagon 6 sides Hexagon 7 sides Heptagon 8 sides Octagon 9 sides Nonagon 10 sides Decagon 12 sides Dodecagon 20 sides Icosagon Polygon interior angles What happens to the sum of the degrees of the inside angles as the number of sides increases? 180° 360° 540° 720° Is there an easier way to figure out the sum of the degrees of the inside angles for any polygon? Angle-Sum Theorem for Polygons The sum of the measures of the interior angles of a convex polygon with n sides is given by S = (n − 2)180° Page 393 A. Find the sum of the measures of the interior angles of a convex nonagon. A nonagon has nine sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures. (n – 2) ● 180= (9 – 2) ● 180 n=9 = 7 ● 180 or 1260 Simplify. Answer: The sum of the measures is 1260. B. Find the measure of each interior angle of parallelogram RSTU. Step 1 Find x. Since n=4, the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon. Sum of measures of interior angles Substitution Combine like terms. Subtract 8 from each side. Divide each side by 32. Step 2 Use the value of x to find the measure of each angle. mR = 5x mS = 11x + 4 = 5(11) or 55 = 11(11) + 4 or 125 mT = 5x mU = 11x + 4 = 5(11) or 55 = 11(11) + 4 or 125 Answer: mR = 55, mS = 125, mT = 55, mU = 125 A. Find the sum of the measures of the interior angles of a convex octagon. A. 900 B. 1080 C. 1260 D. 1440 A pottery mold makes bowls that are in the shape of a regular heptagon. Find the measure of one of the interior angles of the bowl. A. 130° B. 128.57° C. 140° D. 125.5° The sum of the measures of the interior angles of a convex polygon is 900°. Find the number of sides of the polygon. S = (n− 2)180° 900 = (n − 2)180° 900÷180 = (n − 2)180° ÷180 5=n−2 5+2=n−2+2 7=n The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon. Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides. S = 180(n – 2) Interior Angle Sum Theorem (150)n = 180(n – 2) 150n = 180n – 360 0 = 30n – 360 360 = 30n 12 = n S = 150n Distributive Property Subtract 150n from each side. Add 360 to each side. Divide each side by 30. Answer: The polygon has 12 sides. The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. A. 12 B. 9 C. 11 D. 10 Polygon exterior angles What happens to the sum of the degrees of the outside angles as the number of sides increases? 360° 360° 360° 360° Is there an easier way to figure out the sum of the degrees of the outside angles for any polygon? Exterior Angle Theorem for Polygons The sum of the measures of the exterior angles of a convex polygon (one at each vertex) is 360° Page 396 Find the sum of the interior angles AND exterior angles for the polygon 7 sides Heptagon Interior angle sum = (n − 2)180° = (7 − 2)180° = (5)180° = 900° Exterior angle sum = 360° Find the sum of the interior angles AND exterior angles for the polygon 22-gon Interior angle sum = (n − 2)180° = (22 − 2)180° = (20)180° = 3600° Exterior angle sum = 360° A. Find the value of x in the diagram. Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x. 5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) +(5x + 5) = 360 (5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 + (–12) + 3 + 5] = 360 31x – 12 = 360 31x = 372 x = 12 Answer: x = 12 B. Find the measure of each exterior angle of a regular pentagon. A. 72 B. 60 C. 45 D. 90 How do you find the sum of the measures of the interior angles of a convex polygon? S = (n −2)180° How do you find the measure of one of the interior angles of a convex polygon? S = (n −2)180°/n What is the sum of the measure of the exterior angles of a convex polygon? 360° How do you find the measure of one of the exterior angles of a convex polygon? 360°/n 6-1 Assignment Page 398, 13-37 odd