8.2 Angles in Polygons

1 In Polygons417 GoalFind the measures ofinterior and exteriorangles of Words interior angle p. 181 exterior angle p. 181 The definitions for interior Angles and exterior Angles can be extendedto include Angles formed in any polygon. In the diagrams shownbelow, interior Angles are red, and exterior Angles are anglesinterior in PolygonsFind the sum of the measures of theinterior Angles of a convex heptagon has 7 sides. Use the Polygon Interior Angles Theorem and substitute 7 for n.(n22)p1808 5 (722)p1808 Substitute 7 for The sum of the measures of the interior Angles of a convex heptagon is the Polygon Interior Angles TheoremPolygon Interior Angles TheoremWordsThe sum of the measures of theinterior Angles of a convex polygonwith nsides is (n22) 1ma2 1.

2 1man5(n22)p1808 THEOREM suggests the Polygon Interior Angles Theoremshown 1 of 7In Exercises 1 3, find the measure of the measure of an interior angle of a regular polygon withtwelve 8 Polygons and AreaFind Measures of Interior AnglesFind the measure of aAin the polygon has 6 sides, so the sum of themeasures of the interior Angles is:(n22)p1808 5(622)p1808 54p1808 the measures of the interior Angles and set the sum equal to 11368 1888 11428 11058 1maA57208 The sum is The measure of aAis the Measure of an Interior AngleFind the measure of an interior angle ofa regular sum of the measures of the interiorangles of any octagon is:(n22)p1808 5(822)p1808 56p1808 the octagon is regular, each angle has the same , divide 10808by 8 to find the measure of one interior angle.

3 }108808}51358 ANSWER The measure of an interior angle of a regular octagon is Angles of a Regular Polygon8881428136813681058 DEFABCSKILLSREVIEWTo review solving anequation, see p. 671. Student HelpPage 2 of 7 exterior AnglesThe diagrams below show that the sum of themeasures of the exterior Angles of the convex polygon is 3608. 1 Shade one exterior 2 Cut out the 3 Arrange the exteriorangle at each to form sum of the measures of the exterior Angles of a convex polygon doesnot depend on the number of sides that the polygon in Polygons419 Find the value of the Polygon exterior AnglesTheorem, set the sum of the measures of the exterior Angles equal to 3608.

4 958 1858 12x8 1x8 53608 Polygon exterior Angles Theorem180 13x5360 Combine like 180 from each each side by The value of xis the Measure of an exterior AnglePolygon exterior Angles TheoremWordsThe sum of the measures of theexterior Angles of a convex polygon,one angle at each vertex, is 1ma2 1..1man53608 THEOREM circle contains twostraight Angles . So,there are 1808 1 1808,or 3608, in a It!180818083608orn55 Page 3 of 7420 Chapter 8 Polygons and aninterior angleand an exterior angleof polygon ABCDE shown at the the formula that is used to find the sum of the measures of the interior Angles of any convex polygon with your answer from Exercise 2 to find the sum of the measuresof the interior Angles of a convex polygon with 9 an equation to find the measure of a1.

5 Do not solve the CheckVocabulary CheckGuided the value of exterior Angle MeasuresPage 4 of in Polygons421 Extra PracticeSee p. 1:Exs. 8 16 Example 2:Exs. 17 20 Example 3:Exs. 21 23 Example 4:Exs. 24, 25 Homework HelpInterior Angles of Regular PolygonsFind the measure of an interiorangle of the regular AlgebraFind the value of of Interior Angle MeasuresFind the sum of the measures ofthe interior Angles of the convex with nSidesFind the sum of the measures of the interiorangles of the convex polygon with Angle MeasuresFind the measure of and ApplicationsHOMEWORKHELPE xtra help with problemsolving in Exs.

6 21 23 is at HelpIC L A S S Z O N E . C O MPage 5 of 7422 Chapter 8 Polygons and FurnitureYou are constructing a regular hexagonalwooden bench like the one shown. On the bench, a1 ca2. Find themeasures of a1 and a2 so that you know what angle to use to cut the pieces of the number of sides of the regular polygon with thegiven exterior angle home plate for a baseball field is a pentagon as shown. the polygon regular? Explain why or why is the sum of the interior Angles ? the measures of aCand a strip of lined paper and tie it into an overhand knotas shown.

7 Gently flatten the knot to form a pentagon. The pentagonshould be the measure of an interior angle of a regular the interior Angles of your knot with a protractor. Useyour answer to Exercise 29 to determine whether your pentagon is ReasoningSelect the word that makes the statement sum of the measures of the exterior Angles of a convexpolygon, one at each vertex, is (always, sometimes, never) sum of the measures of the interior Angles of a convexoctagon is (always, sometimes, never) measures of the exterior Angles of a convex polygon are(always, sometimes, never) It!

8 DAEBCBASEBALLHome plate isused to find the placement ofthe foul lines on the 6 of in Polygons423 Multiple ChoiceIn Exercises 39 and 40, use the diagram is the value of x? is the measure of aP? XF398XG788XH1568XJ2348 Using a CentroidEis the centroid of TABC. Find BEand ED. (Lesson ) ValueEvaluate. (Skills Review, p. 662)44. 25 45. 11 46. 248 47. 0 48. 49. 22 50. 51. Decide whether the polygon is regular. Explain your answer.(Lesson ) the measure of a1.(Lesson ) 1 Algebra SkillsCDABEABCDEADCBEM ixed ReviewStandardized TestPracticeRSTUPP10282x8130812081348x8 Page 7 of 7