1. Drill 1)If two angles of a triangle have a sum of 85 degrees find the third angle. 2) The three angles of a triangle are 2x, 3x, and 2x + 40 find each angle.

2. Polygons 2.2 Polygons

3. Polygon Is a closed figure with at least three sides, so that each segment intersects exactly two segments at their endpoints.

4. Polygon Terminology SidesVertex Interior A B CD E F Diagonal Consecutive Vertices

5. A polygon can also be classified as convex or concave. If all of the diagonals lie in the interior of the figure, then the polygon is ______. convex If any part of a diagonal lies outside of the figure, then the polygon is _______. concave Naming Polygons

6. Types of Polygons # of Sides Name/Draw 3 4 5 6 7 8 9 10 12 QUADRILATERAL PENTAGON HEXAGON OCTAGON HEPTAGON NONAGON DECAGON DODECAGON TRIANGLE

7. Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 1) Draw a convex quadrilateral. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Make a table like the one below.

8. Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 1) Draw a convex pentagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? pentagon 5 2 3 3(180) = 540

9. Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 1) Draw a convex hexagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720

10. Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 1) Draw a convex heptagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900

11. Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 1) Any convex polygon. 2) All possible diagonals from one vertex. 3) How many triangles? pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900 n -gon n n - 3 n - 2 ( n – 2)180 Theorem 10-1 If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

12. 57° 48° 74° 55° 54° 72° In §7.2 we identified exterior angles of triangles. Likewise, you can extend the sides of any convex polygon to form exterior angles. The figure suggests a method for finding the sum of the measures of the exterior angles of a convex polygon. When you extend n sides of a polygon, n linear pairs of angles are formed. The sum of the angle measures in each linear pair is 180. sum of measure of exterior angles sum of measures of linear pairs sum of measures of interior angles = = – –n180180(n – 2) =–180n180n + 360 =360 sum of measure of exterior angles

13. Polygon Interior Angle-Sum Theorem The sum of the measures of the interior angles of an n-gon is (n-2)180. Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

17. Homework Pages 79 – 80 #’s 1 – 4, 10 - 26