(over Lesson 10-1) Slide 1 of 1 1-1a

Diagonals and Angle Measure What You'll Learn You will learn to find measures of interior and exterior angles of polygons. Vocabulary Nothing New!

Diagonals and Angle Measure Number of Diagonals from One Vertex Make a table like the one below. 1) Draw a convex quadrilateral. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? 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

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Draw a convex pentagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? 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 pentagon 5 2 3 3(180) = 540

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Draw a convex hexagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? 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 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Draw a convex heptagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? 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 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900

Diagonals and Angle Measure Number of Diagonals from One Vertex 1) Any convex polygon. 2) All possible diagonals from one vertex. 3) How many triangles? 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 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.

Diagonals and Angle Measure In §7.2 we identified exterior angles of triangles. Likewise, you can extend the sides of any convex polygon to form exterior angles. 48° 57° 74° The figure suggests a method for finding the sum of the measures of the exterior angles of a convex polygon. 72° 55° 54° 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 = – = n•180 – 180(n – 2) = 180n – 180n + 360 sum of measure of exterior angles = 360

Diagonals and Angle Measure Theorem 10-2 In any convex polygon, the sum of the measures of the exterior angles, (one at each vertex), is 360. Java Applet

Diagonals and Angle Measure End of Section 10.2