Lesson 7.1 Angles of Polygons

Essential Question: How can I find the sum of the measures of the interior angles of a polygon?

Polygon A plane figure made of three or more segments (sides). Each side intersects exactly two other sides at their endpoints. Polygons are named by vertices in consecutive order, going CW or CCW.

Diagonals in a Polygon A segment that joins two non-consecutive vertices. The diagonals from one vertex divide a polygon into triangles.

Interior Angle Sum of a Triangle Review 4 5 3 m2 = m5 m1 = m4 180 1 2 m1 + m2 + m3 = 180 m4 + m5 + m3 = 180 1   180 7.1 Angles of Polygons September 18, 2018September 18, 2018

Polygon Interior Angle Sums Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 7.1 Angles of Polygons September 18, 2018September 18, 2018

Interiore Angle Sum in a Quadrilateral 2 3 6 1 4 m1 + m2 + m3 = 180 m4 + m5 + m6 = 180 5 m1 + m4 + m2 + m5 + m3 + m6 = 360 7.1 Angles of Polygons September 18, 2018September 18, 2018

Polygon Interor Angle Sums Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 7.1 Angles of Polygons September 18, 2018September 18, 2018

Interior Angle Sum in a Pentagon From this vertex, how many diagonals are there? 2 7.1 Angles of Polygons September 18, 2018September 18, 2018

Angle Sum in a Pentagon 3 180 180 180 180 How many triangles are there? 180 3 180 And what is the sum of the angles of each triangle? 180 180 7.1 Angles of Polygons September 18, 2018September 18, 2018

Angle Sum in a Pentagon 3  180 = 540 3 s  540 180 180 180 So what is the sum of the interior angles of a pentagon? 180 180 3  180 = 540 7.1 Angles of Polygons September 18, 2018September 18, 2018

Polygon Interior Angle Sums Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 540 7.1 Angles of Polygons September 18, 2018September 18, 2018

Interior Angle Sum in a Hexagon How many diagonals from this vertex? 3 How many triangles are formed? 4 The sum of the angles is? 4  180 = 720 4 s  720 7.1 Angles of Polygons September 18, 2018September 18, 2018

Polygon Interior Angle Sums Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 540 Hexagon 6 720 7.1 Angles of Polygons September 18, 2018September 18, 2018

Do you see a pattern? September 18, 2018September 18, 2018 7.1 Angles of Polygons September 18, 2018September 18, 2018

Polygon Interior Angle Sums Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 540 Hexagon 6 720 Octagon 8 1080 7.1 Angles of Polygons September 18, 2018September 18, 2018

What’s the pattern? A polygon with n sides can be divided into how many triangles? n – 2 The sum of the angles then is? 180(n – 2) 7.1 Angles of Polygons September 18, 2018September 18, 2018

Polygon Interior Angle Sums Sides Triangles formed by Diagonals Sum of Interior Angles Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 540 Hexagon 6 720 Octagon 8 1080 n-gon n n – 2 180(n – 2) 7.1 Angles of Polygons September 18, 2018September 18, 2018

Theorem 7.1 Polygon Interior Angles Theorem The sum of the interior angles of a convex n-gon is: memorize this! 7.1 Angles of Polygons September 18, 2018September 18, 2018

Example 1: Find the sum of the interior angles of a polygon with 14 sides. 180(14 – 2) = 180(12) = 2160 7.1 Angles of Polygons September 18, 2018September 18, 2018

Example 2: The sum of the interior angles of a polygon is 2700. How many sides does the polygon have? 17 180(n – 2) = 2700 180n – 360 = 2700 180n = 3060 n = 17 7.1 Angles of Polygons September 18, 2018September 18, 2018

Example 3: The sum of the interior angles of a polygon is 1620. How many sides does the polygon have? 11 180(n – 2) = 1620 180n – 360 = 1620 180n = 1980 n = 11 7.1 Angles of Polygons September 18, 2018September 18, 2018

Example 4: The sum of the interior angles of a polygon is 1380. How many sides does the polygon have? This is not a polygon. 180(n – 2) = 1380 180n – 360 = 1380 180n = 1740 n = 9.66666… Why must this be a whole number? 7.1 Angles of Polygons September 18, 2018September 18, 2018

7.1 Corollary to the Polygon Interior Angles Theorem The sum of the interior angles of a quadrilateral is 360.

Example 5: x + x + 55 + 55 = 360 2x + 110 = 360 2x = 250 x = 125 ° Solve for x. x + x + 55 + 55 = 360 2x + 110 = 360 2x = 250 x = 125 °

Your Turn Find the value of x in the diagram. x° + 108° + 121° + 59° = 360° x + 288 = 360 x = 72 °

Regular Polygons

Regular Polygon 𝐸 𝐼 = 180°(𝑛 −2) 𝑛 All sides congruent All angles congruent The Sum of the interior angles is 180(n – 2) Since the angles are congruent, the measure of each interior angle in a regular polygon is 𝐸 𝐼 = 180°(𝑛 −2) 𝑛 September 18, 2018 7.1 Angles of Polygons

Example 6: Find the measure of each angle of a regular pentagon. 108 7.1 Angles of Polygons September 18, 2018September 18, 2018

Example 7: Each angle of a regular polygon measures 160. How many sides does the polygon have? September 18, 2018 7.1 Angles of Polygons

Example 8: A home plate for a baseball field is shown. Is the polygon regular? Explain your reasoning. The polygon is not equilateral or equiangular. So, the polygon is not regular.

Example 8: Find the measures of ∠C and ∠E. 180° ⋅ (n − 2) x° + x° + 90° + 90° + 90° = 540° 2x + 270 = 540 2x = 270 x = 135 Therefore, ∠C= 135° and ∠E= 135° = 180° ⋅ (5 − 2) = 540°

The Sum of the Exterior Angles This always means using one exterior angle at each vertex. But not this one. or this angle. This angle… 7.1 Angles of Polygons September 18, 2018September 18, 2018

The Sum of the Exterior Angles Extend only ONE side at each vertex. Exterior Angle Exterior Angle Exterior Angle 7.1 Angles of Polygons September 18, 2018September 18, 2018

Theorem 7.2 Polygon Exterior Angles Theorem The sum of the exterior angles of any polygon, one angle at each vertex, is 360. m∠1 + m∠2 + · · · + m∠n = 360° 𝑆 𝐸 =360°

Example 9: Find the value of x in the diagram. x° + 2x° + 89° + 67° = 360° 3x + 156 = 360 3x = 204 x = 68 °

Corollary The measure of an exterior angle of a regular polygon with n sides is 𝐸 𝐸 = 360° 𝑛 7.1 Angles of Polygons September 18, 2018September 18, 2018

Example 10: Find the measure of an exterior angle of a regular 40-gon. Solution: 360/40 = 9 7.1 Angles of Polygons September 18, 2018September 18, 2018

Example 15: The trampoline shown is shaped like a regular dodecagon. Find the measure of each interior angle. = 180(12−2) 12 = 180(10) 12 = 1800 12 =150°

Example 15: The trampoline shown is shaped like a regular dodecagon. b. Find the measure of each exterior angle. = 30°

Summary The sum of the interior angles of an n-gon is 180(n – 2). The sum of the exterior angles of any polygon is 360. The measure of an interior angle of a regular polygon is 180(𝑛−2) 𝑛 . The measure of an exterior angle of a regular polygon is 360 𝑛 . 7.1 Angles of Polygons September 18, 2018September 18, 2018