Pearson Unit 1 Topic 6: Polygons and Quadrilaterals 6-1: The Polygon Angle-Sum Theorems Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (5)(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. (1)(C) Select tools, including real objects, manipulatives paper and pencil, and technology as appropriate, and techniques, including mental math, estimations, and number sense as appropriate, to solve problems. (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments that intersect only at their endpoints.

Each segment that forms a polygon is a side of the polygon Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal. This polygon is ABCDE or AEDCB or many other options. You may start at any letter and go in a circular motion either clockwise or counter-clockwise.

You can name a polygon by the number of its sides You can name a polygon by the number of its sides. The table shows the names of some common polygons.

Example: 1 Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon polygon, heptagon not a polygon not a polygon polygon, nonagon not a polygon

All the sides are congruent in an equilateral polygon All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is 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 no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.

Example: 2 Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave irregular, convex regular, convex irregular, concave regular, 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.

By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Remember! (1) 180°= 180° (2) 180°=360° (3) 180°=540° (4) 180°=720° (n-2) (n-2) 180°

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°.

Example: 3 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.

Example: 4 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. Substitute 16 for n and simplify. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.

Example: 5 Find the sum of the interior angle measures of a convex 15-gon. (n – 2)180° Polygon  Sum Thm. (15 – 2)180° A 15-gon has 15 sides, so substitute 15 for n. 2340° Simplify.

Example: 6 Find the measure of each interior angle of pentagon ABCDE. Polygon  Sum Thm. (5 – 2)180° = 540° mA + mB + mC + mD + mE = 540° Polygon  Sum Thm. 35c + 18c + 32c + 32c + 18c = 540 Substitute. Combine like terms. 135c = 540 Divide both sides by 135. c = 4 mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°

Example: 7 Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon  Sum Thm. Substitute 10 for n and simplify. (10 – 2)180° = 1440° Step 2 Find the measure of one interior angle. The int. s are , so divide by 10.

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.

An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Remember!

Example: 8 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. measure of one ext.  = A regular 20-gon has 20  ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°.

Example: 9 Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. Polygon  Sum Thm. A regular dodecagon has 12  ext. s, so divide the sum by 12. measure of one ext. The measure of each exterior angle of a regular dodecagon is 30°.

Example: 10 Find the value of b in polygon FGHJKL. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° Polygon Ext.  Sum Thm. 120b = 360 Combine like terms. b = 3 Divide both sides by 120.

Example: 11 Find the value of r in polygon JKLM. 4r° + 7r° + 5r° + 8r° = 360° Polygon Ext.  Sum Thm. 24r = 360 Combine like terms. r = 15 Divide both sides by 24.