Do Now…… 1. A triangle with a 90° angle has sides that are 3 cm, 4 cm, and 5 cm long. Classify the triangle by its sides and angles. Use the diagram for Exercises 2–6. 2. Find m  3 if m 2 = 70 and m 4 = 42. 3. Find m 5 if m  2 = 76 and m  3 = 90. 4. Find x if m  1 = 4x, m  3 = 2x + 28, and m  4 = 32. 5. Find x if m  2 = 10x, m  3 = 5x + 40, and m  4 = 3x – 4. 6. Find m  3 if m  1 = 125 and m  5 = 160.

Section 4 Polygons Objectives: To classify Polygons To find the sums of the measures of the interior & exterior s of Polygons.

Polygon: A closed plane figure. w/ at least 3 sides (segments) The sides only intersect at their endpoints Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.

Which of the following figures are polygons? No yes No

Name Polygons By Their: Classify Polygons Name Polygons By Their: Vertices Start at any vertex and list the vertices consecutively in a clockwise direction (ABCDE or CDEAB, etc) Sides Name by line segment naming convention   Angles Name by angle naming convention A, B, C, D, E

II. Also classify polygons by their Shape a) Convex Polygon – Has no diagonal w/ points outside the polygon. E A B C D b) Concave Polygon – Has at least one diagonal w/ points outside the polygon. * All polygons are convex unless stated otherwise.

Special Polygons: Equilateral Polygon – All sides are . Equiangular Polygon – All s are . Regular Polygon – Both Equilateral & Equiangular.

I. Classify Polygons by the number of sides it has. 3 4 5 6 7 8 9 10 12 n Name Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon N-gon Interior  Sum

III. Polygon Interior  sum 4 sides 2 Δs 2 • 180 = 360 4 sides 2 Δs 2 • 180 = 360 Each Interior -- Each Exterior-- Sum of exterior

5 sides 3 Δs 3 • 180 = 540 Each Interior -- Each Exterior-- Sum of exterior

6 sides 4 Δs 4 • 180 = Each Interior -- Each Exterior-- Sum of exterior

8 sides 6 Δs 6 • 180 = Each Interior -- Each Exterior-- Sum of exterior

So……What’s the pattern? Th(3-9) Polygon Angle – Sum Thm S = (n -2) 180 Sum of Interior  # of sides

So……What did you notice about the exterior angles? The sum of the measures of the exterior s of a polygon is 360°. 1 2 3 4 5 m1 + m2 + m3 + m4 + m5 = 360 For Regular Polygons = measure of one exterior  The interior  & the exterior  are Supplementary. Int + Ext = 180

Example #1 Find the interior angle sum. a. 13-gon b. decagon

Example #2 How many sides does each polygon have if its interior angle sum is: a. 2700 b. 1080

Example #3 Find x.

Example #4 Find y.

Example #5 Find x.

Example #6 Find x.

Example #7 Find x.

Example #8 How many sides does each regular polygon have if its exterior angle is: a. 120 b. 24

Example #9 How many sides does each regular polygon have if its interior angle is: a. 90 b. 144