Understanding Polygon Angles and Theorems
Explore the angles of polygons, including interior and exterior angle sums, theorems 3-13 and 3-14, properties of regular polygons, and measurements of angles in various polygon types. Discover the relationships between sides, vertices, and angles to deepen your geometric knowledge.
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Section 3-5 Angles of a Polygon
many angles two endpoint collinear Yes Yes No No No Yes
a point of in the polygon interior the Yes No Yes No
Number of Sides 3 4 5 6 7 8 9 10 n Name of Polygon triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon n-gon
consecutive vertices C D E A E D E A B A B C B C D D C B
two nonconsecutive vertices
Theorem 3-13 The sum of the measures of the angles of a convex polygon with n sides is _____________________. (n 2) 180
1440 1. What is the interior angle sum of a decagon? _____ (n 2)180 (10 2)180 (8)180 1080 2. What is the interior angle sum of an octagon? _____ (n 2)180 (8 2)180 (6)180 3600 3. What is the interior angle sum of 22-gon? ________ (n 2)180 (22 2)180 (20)180
Theorem 3-14 The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is __________. 360
4. What is the exterior angle sum of a pentagon? _______________________ 360 5. What is the exterior angle sum of a decagon? _______________________ 360
Regular Polygon a polygon that is both ____________ and ________________. equilateral equiangular 6. What is the measure of one interior angle of a regular hexagon? __________ 120 ? 2 180 ? 6 2 180 6 angle = angle = 7. What is the measure of one interior angle of a regular pentagon? _____________________ ? 2 180 ? angle = 108 5 2 180 5 angle =
8. What is the measure of one interior angle of a regular decagon? ____________________ 144 ? 2 180 ? 10 2 180 10 angle = angle =
9. What is the measure of one exterior angle of a regular hexagon? _______________________ angle = 360 ? 60 angle = 360 6 10. What is the measure of one exterior angle of a regular octagon? _____________________ angle = 360 ? 45 angle = 360 8
11. What polygon has an exterior angle measuring 24 ? ______________________ 15-gon 24 1 = 360 24n = 360 n = 15 angle = 360 24 = 360 ? ? ? 12. What polygon has an interior angle measuring 135 ? ______________________ ? 2 180 ? 135 = (n 2)180 = 135n octagon ? 2 180 ? angle = 135 1 = ? 2 180 ? 360 = 45n 8 = n
Complete with always, sometimes, or never. 13. The sum of the measures of the exterior angles of any polygon, one angle at each vertex, is __________ 360 . always 14. The sum of the measures of the angles of a convex polygon is ______________ 360 . sometimes 15. A segment joining two vertices of a polygon is ________________ a diagonal. sometimes
16. The sum of the measures of the exterior angles of a polygon _____________ depends on the number of sides of the polygon. never always 17. A regular polygon is __________ equilateral. sometimes 18. An equiangular polygon is _________ regular.
19. Three of the angles of a quadrilateral have measures 90, 60, and 115. The fourth angle has measure________. 95 (4 2) 180 = 360 90 + 60 + 115 + x = 360
HOMEWORK: page 104 #2-22 even
