Lesson 3-5 Angles of a Polygon (page 101) Essential Question How can you apply parallel lines (planes) to make deductions?

POLYGON: A plane figure formed by coplanar segments (sides) such that: (1)each segment intersects exactly two other segments, one at each endpoint; and … (2)no two segments with a common endpoint are collinear. Polygon means “many angles.”

Examples of Polygons B CD E A Written: polygon ABCDE or ABCDE (write vertices in consecutive order)

Example of a Polygon

Example of a Figure that is not a Polygon

CONVEX POLYGON: A polygon such that no line containing a side of the polygon contains a point in the interior of the polygon.

Nonconvex Polygons - examples

DIAGONAL: A segment joining two non - consecutive vertices of a polygon.

Nonconvex Polygons will have a diagonal in the exterior.

Activity: Draw all the diagonals from one vertex on each polygon, then complete the chart. 180º

# of sides of polygon Name of Polygon# of diagonals from 1 vertex # of triangles formed sum of angle measures n triangle º quadrilateral º pentagon º hexagon º septagon º octagon º nonagon º decagon º undecagon º dodecagon º n-gon n - 3n - 2 (n-2)  180º

The sum of the measures of the angles of a convex polygon with n sides is (n - 2)  180º. Theorem 3-13 Theorem 3-14 The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360º. Refer to page 97, #10 and page 104, #7 to verify this th m.

Paper Polygon Proof We will do this proof another day

Example # 1. If a convex polygon has 24 sides (24-gon), then … (a)… the interior angle sum is ___________. (b)… the exterior angle sum is ___________. 3960º 360º (n - 2)  180º= (24 - 2)  180º = (22)  180º = 3960º

Example # 2. (a) Find the value of “x” º x º x = _____ 150º x = 540 n = 5 angle sum = 540º x = 540 x = 50

Example # 2. (b) Find the value of “x” º x º x = _____ 150º 60º x = 360 x = 360 x = 100 n = 4 angle sum = 360º

Example # 2. (c) Find the value of “x” º x º x = _____ 160º 60º 120º 140º n = 6 angle sum = 720º x = 720 x = 720 x = 90

Example # 2. (d) Find the value of “x”. 100 x º x = _____ 60º x º x + x + x + 60 = x = 300 x = 100 n = 4 angle sum = 360º

REGULAR POLYGON: A polygon that is both equiangular and equilateral. Look at some examples using your template. template #4 template #3 template #2

REGULAR POLYGON The angle measure of a regular polygon with n sides … … has every interior angle = … has every exterior angle = n = the number of sides, which is also the number of angles.

Example # 3. Find the measure of each interior angle and each exterior angle of a regular pentagon. Each interior angle has measure _____. Each exterior angle has measure _____. 108º 72º These angles will always be supplementary!

Example # 4. How many sides does a regular polygon have if the measure of each exterior angle is 45º? The polygon has ___ sides. 8

Example # 5. How many sides does a regular polygon have if the measure of each interior angle is 150º? The polygon has ___ sides. 12

Example # 5. A better way! How many sides does a regular polygon have if the measure of each interior angle is 150º? The polygon has ___ sides. 12 Remember: interior angle + exterior angle = 180º 150º + exterior angle = 180º exterior angle = 30º

Assignment Written Exercises on pages 104 & 105 RECOMMENDED: 11, 13, 15, 19 REQUIRED: 8, 10, 16, 17, 20, 21, 22, 23 Prepare for a quiz on Lessons 3-4 & 3-5 How can you apply parallel lines (planes) to make deductions?