Polygon - a closed plane figure formed by three or more line segments. * Closed, no openings! * Line segments, no curved sides! * Segments cannot intersect except at corners! No overlapping or continuing past the vertex! You can name a polygon by the number of its sides. The table shows the names of some common polygons. Identifying Polygons
Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, hexagon not a polygon Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.
polygon, heptagon not a polygon Your Turn! Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. Side of the polygon – Each segment that forms a polygon
Vertex of the polygon – The common endpoint of two sides
Diagonal – A segment that connects any two nonconsecutive vertices
B. C. A Your Turn! A Name one of the following:
E B A diagonal ______
A side ______D C A vertex ______Equilateral polygon - all the sides are congruent – look for dash marks!
Equiangular polygon – All the angles are congruent – look for arc marks!
Regular polygon - both equilateral and equiangular. If a polygon is not regular, it is called irregular. A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If any interior angle is greater than 180, the polygon is concave. If no diagonal contains points in the exterior, then the polygon is convex. Note: A regular polygon is always convex. Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, convex regular, convex Your Turn! Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon. In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n — 2)180°. 1. Find the sum of the interior angle measures of a convex heptagon.
(n – 2)180° Polygon Sum Thm.
(7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify. 2. Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures.
(n – 2)180° Polygon Sum Thm.
(16 – 2)180° = 2520° Substitute 16 for n and simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 16. Find the measure of each interior angle of pentagon ABCDE.
(5 – 2)180° = 540° Polygon Sum Thm.
mA + mB + mC + mD + mE = 540° Polygon Sum Thm.
35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135.
mA = 35(4°) = 140° mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128° Your Turn! Find the sum of the interior angle measures of a convex 15-gon.
Find the measure of each interior angle of a regular decagon. Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20 sides and 20 vertices.
sum of ext. s = 360°. Polygon Sum Thm.
A regular 20-gon has 20 ext. s, so divide the sum by 20. measure of one ext. = Find the value of b in polygon FGHJKL.
Polygon Ext. Sum Thm.
15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°
120b = 360 Combine like terms.
b = 3 Divide both sides by 120. Your Turn! Find the measure of each exterior angle of a regular dodecagon.
Find the value of r in polygon JKLM. Write about it… Use the terms learned from the lesson to describe the figure as specifically as possible. Your answer should be a paragraph with at least four complete sentences.