2. Course: Applied Geo. Aim: Polygons & Interior Angles Aim: What generalizations can be made about the measures of the angles of a quadrilateral and other polygons? Do Now: xx + 8 2x2x Find the value of x

3. Course: Applied Geo. Aim: Polygons & Interior Angles Polygons Polygon – the union of three or more coplanar segments such that 1.each endpoint is shared by exactly 2 segments. 2.the segments intersect only at their endpoint; and 3.intersecting segments are noncollinear. polygonsNOT a polygon sides vertex

4. Course: Applied Geo. Aim: Polygons & Interior Angles Polygons Polygon – the number of vertices equals the number of sides named by listing vertices in order ABCDE E A D C B adjacent sides consecutive angles

5. Course: Applied Geo. Aim: Polygons & Interior Angles Quadrilateral Quadrilateral – four-sided, four vertices polygon

6. Course: Applied Geo. Aim: Polygons & Interior Angles Aim: What generalizations can be made about the measures of the angles of a quadrilateral and other polygons? Do Now: x x + 15 2x2x Find the value of x x + 30

7. Course: Applied Geo. Aim: Polygons & Interior Angles Quadrilateral Quadrilateral – four-sided, four vertices polygon The sum of the measure of the angles of a triangle is 180 0. In quadrilateral ABCD, what is the sum of the measures of  A +  B +  C +  D? 360 o

8. Course: Applied Geo. Aim: Polygons & Interior Angles Model Problem Find the value of the variables. 65 70 110 x y 115 85 105 a b c 125 45 d 70 110 f e h g 125 i 110 55 75 105 65 110 75 125 55

9. Course: Applied Geo. Aim: Polygons & Interior Angles Model Problem Find the value of the variables.  a = 105,  b = 75,  c = 50,  d = 95,  e = 115,  f = 110,  s = 80,  t = 125,  u = 110,  x = 80,  y = 100,  z = 105 105 75 50 95 115 110 80 125 110 80 100 105 45

10. Course: Applied Geo. Aim: Polygons & Interior Angles Find the measure of each angle in the quadrilateral below: K L M N 2x2x2x – 15 x – 6x + 15 Since the sum of the angles of any quadrilateral is 360 0, then 2x + 2x – 15 + x – 6 + x + 15 = 360 6x – 6 = 360 6x = 366 x = 61  N = 2x = 122 0  M = 2x – 15 = 107 0  K = x – 6 = 55 0  L = x + 15 = 76 0 360 0 Model Problem

11. Course: Applied Geo. Aim: Polygons & Interior Angles Polygon Definitions A quadrilateral is a four-sided polygon. A pentagon is a five-sided polygon. A hexagon is a six-sided polygon. A octagon is a eight-sided polygon. A decagon is a ten-sided polygon. A concave polygon A convex polygon each interior angle is less than 180º at least one interior angle > 180º A diagonal is a line segment whose endpoints are two non-adjacent vertices of the polygon.

12. Course: Applied Geo. Aim: Polygons & Interior Angles Finding the Formula – Sketch 16 Draw a diagonal from one vertex to each of the other vertices of the polygon.

13. Course: Applied Geo. Aim: Polygons & Interior Angles Sum of Interior Angle Measures of Polygons Polygon# of sides# of triangles Sum of angle measures quadrilateral44 – 2 = 2360 pentagon55 – 2 = 3540 hexagon66 – 2 = 4720 heptagon77 – 2 = 5900 octagon88 – 2 = 61080 n-gonn sidesn – 2(n – 2)180

14. Course: Applied Geo. Aim: Polygons & Interior Angles Polygon Definitions and Theorems A regular polygon is a polygon that is both equilateral and equiangular. The sum of the degree measures of the interior angles of any polygon of n sides is 180(n - 2). hexagon: n = 6 180º 4(180º) = 720º 180(6 - 2) 180(4) = 720º 180(n - 2)

15. Course: Applied Geo. Aim: Polygons & Interior Angles a. Model Problem A regular polygon has 10 sides. a. Find the sum of the measures of all interior angles of this polygon b. Find the measure of one interior angle The sum of the degree measures of the interior angles of any polygon of n sides is 180(n - 2). n = 10 180(n - 2) 180(10 - 2) 180(8) 1440 b.

16. Course: Applied Geo. Aim: Polygons & Interior Angles a. Model Problem Find n, the number of sides of a regular polygon, if: a. Each interior angle contains 150º. b. The sum of the degree measures of all interior angles is 2,880. The sum of the degree measures of the interior angles of any polygon of n sides is 180(n - 2). n = 150 180(n – 2) = 150n 180n – 360 = 150n 30n = 360 n = 12 b. 180(n - 2) = 2,880 180n - 360 = 2,880 180n = 3240n = 18 18-sided regular polygon

17. Course: Applied Geo. Aim: Polygons & Interior Angles A B C w y x An exterior angle of a triangle is an angle formed by a side of the triangle and the extension of another side of the triangle.  w is an exterior angle of triangle ABC  x &  y are nonadjacent interior angles often called remote interior angles The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. w = x + y Exterior Angle Theorem

18. Course: Applied Geo. Aim: Polygons & Interior Angles Exterior Angles of Polygons 1 2 3 4 5 The sum of the measures of the exterior angles of any polygon, one at each vertex, is 360.  1 +  2 +  3 +  4 +  5 = 360 o pentagon

19. Course: Applied Geo. Aim: Polygons & Interior Angles Model Problem Find the measure of each exterior angle of a regular pentagon. A pentagon has 5 angles all equal in measure. 108 o The sum of the degree measures of the interior angles of any polygon of n sides is 180(n - 2). 180(5 - 2) = 540 o The sum of the measures of the exterior angles of any polygon, one at each vertex, is 360. 72 o

20. Course: Applied Geo. Aim: Polygons & Interior Angles Model Problem The measure of each exterior angle of a regular polygon is 30. Find the sum of the interior angles of the polygon. First: Find the number of sides or angles with n = number of sides 30n = 360Sum of exterior angles measures 360 Second: Use formula for sum of  measures of polygon. 180(5 - 2) = sum n = 12 180(12 - 2) = 1800 The sum of the degree measures of the interior angles of any polygon of n sides is 180(n - 2).

21. Course: Applied Geo. Aim: Polygons & Interior Angles