Section 8.2

Find the measures of the interior angles of a polygon. Find the measures of the exterior angles of a polygon.

Interior angle Exterior angle

8.1 Polygon Interior Angles Sum 8.2 Polygon Exterior Angles Sum

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!

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral Pentagon Hexagon Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral 4 Pentagon Hexagon Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral 42 Pentagon Hexagon Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon Hexagon Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon 5 Hexagon Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon 53 Hexagon Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon 6 Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon 64 Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon 9 n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon 97 n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon ° =1260 ° n - gon

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon ° =1260 ° n - gon n

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon ° =1260 ° n - gon nn-2

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon ° =1260 ° n - gon nn-2 (n-2) 180 °

Polygon# of sides # of triangles Sum of measures of interior  ’s Triangle 31 1 ● 180  =180  Quadrilateral ° =360 ° Pentagon ° =540 ° Hexagon ° =720 ° Nonagon ° =1260 ° n - gon nn-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°.

From the previous slide, we have discovered that the sum of the measures of the interior angles of a convex n - gon is (n – 2) ∙ 180  This relationship can be used to find the measure of each interior angle in a regular n - gon because the angles are all congruent.

 Theorem 8.1 Polygon Interior Angles Sum If a convex polygon has n sides and S is the sum of its interior angles, then S = (n – 2)180. Example: m ∠ A+m ∠ B+m ∠ C+m ∠ D+m ∠ E = (5 – 2)180 = 540 A B C ED

Example 1 Use the Polygon Interior Angles Theorem Find the sum of the measures of the interior angles of a convex heptagon. SOLUTION A heptagon has 7 sides. Use the Polygon Interior Angles Theorem and substitute 7 for n. The sum of the measures of the interior angles of a convex heptagon is 900°. ANSWER (n – 2) · 180° = (7 – 2) · 180° Substitute 7 for n. = 5 · 180° Simplify. = 900° Multiply.

Example 2 Find the Measure of an Interior Angle SOLUTION The polygon has 6 sides, so the sum of the measures of the interior angles is: The measure of  A is 113°. ANSWER (n – 2) · 180° = (6 – 2) · 180° = 4 · 180° = 720°. Find the measure of  A in the diagram. Add the measures of the interior angles and set the sum equal to 720°. 136° + 136° + 88° + 142° + 105° + m  A = 720° The sum is 720°. 607° + m  A = 720° Simplify. m  A = 113° Subtract 607°.

Example 3 Interior Angles of a Regular Polygon SOLUTION The sum of the measures of the interior angles of any octagon is: The measure of an interior angle of a regular octagon is 135°. ANSWER (n – 2) · 180° = (8 – 2) · 180° = 6 · 180° = 1080°. Because the octagon is regular, each angle has the same measure. So, divide 1080° by 8 to find the measure of one interior angle ° = 135° Find the measure of an interior angle of a regular octagon.

Your Turn: In Exercises 1–3, find the measure of  G Find the measure of an interior angle of a regular polygon with twelve sides. 4. ANSWER 72° ANSWER 66° ANSWER 130° ANSWER 150°

Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. Step 2 Find the measure of one interior angle. (n – 2)180° (16 – 2)180° = 2520° Polygon  Sum Thm. Substitute 16 for n and simplify. The int.  s are , so divide by 16.

Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. (5 – 2)180° = 540° Polygon  Sum Thm. m  A + m  B + m  C + m  D + m  E = 540°Polygon  Sum Thm. 35c + 18c + 32c + 32c + 18c = 540Substitute. 135c = 540Combine like terms. c = 4Divide both sides by 135.

Example 4B Continued m  A = 35(4°) = 140° m  B = m  E = 18(4°) = 72° m  C = m  D = 32(4°) = 128°

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

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

Find the measure of each interior angle of parallelogram RSTU. Since the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon. Step 1Find x.

Sum of measures of interior angles Substitution Combine like terms. Subtract 8 from each side. Divide each side by 32.

Step 2Use the value of x to find the measure of each angle. Answer: m  R = 55, m  S = 125, m  T = 55, m  U = 125 m  R=5xm  R=5x =5(11) or 55 m  S=11x + 4 =11(11) + 4 or 125 m  T=5xm  T=5x =5(11) or 55 m  U=11x + 4 =11(11) + 4 or 125

A.x = 7.8 B.x = 22.2 C.x = 15 D.x = 10 Find the value of x.

The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon. Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides. S=180(n – 2)Interior Angle Sum Theorem (150)n=180(n – 2)S = 150n 150n=180n – 360Distributive Property 0=30n – 360Subtract 150n from each side.

Answer: The polygon has 12 sides. 360=30nAdd 360 to each side. 12=nDivide each side by 30.

A.12 B.9 C.11 D.10 The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.

Exterior angle is an angle formed by one side of a polygon and the extension of another side ∠ 1, ∠ 2 and ∠ 3 are exterior angles.

Interestingly, the measures of the exterior angles of a polygon is an even easier formula. Let’s look at the following example to understand it.

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°.

Theorem 8.2 Polygon Exterior Angles Sum If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360°. Example: m ∠ A+m ∠ B+m ∠ C+m ∠ D+m ∠ E+ m ∠ F+m ∠ G+m ∠ H+m ∠ J = 360 ˚

Example 7 Find the Measure of an Exterior Angle The value of x is 60. ANSWER 95° + 85° + 2x° + x° = 360° Polygon Exterior Angles Theorem x = 360 Combine like terms.Subtract 180 from each side. 3x = 180 Divide each side by 3. x = 60 Find the value of x. Using the Polygon Exterior Angles Theorem, set the sum of the measures of the exterior angles equal to 360°. SOLUTION

Your turn: Find the value of x. ANSWER 91 ANSWER 21 ANSWER 125 ANSWER

Example 8A: Finding Exterior Angle Measures and Sums in Polygons 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°. 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°. Polygon  Sum Thm. measure of one ext.  =

Example 8B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° Polygon Ext.  Sum Thm. 120b = 360Combine like terms. b = 3Divide both sides by 120.

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

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

Find the value of x in the diagram.

Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x. Answer:x = 12 5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) + (5x + 5)=360 (5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) (–12) ]=360 31x – 12=360 31x=372 x=12

Find the measure of each exterior angle of a regular decagon. A regular decagon has 10 congruent sides and 10 congruent angles. The exterior angles are also congruent, since angles supplementary to congruent angles are congruent. Let n = the measure of each exterior angle and write and solve an equation. 10n=360Polygon Exterior Angle Sum Theorem n=36Divide each side by 10. Answer:The measure of each exterior angle of a regular decagon is 36.

A.10 B.12 C.14 D.15 A. Find the value of x in the diagram.

A.72 B.60 C.45 D.90 B. Find the measure of each exterior angle of a regular pentagon.

Pg. 420 – 423 #1 – 25 odd, 26 – 28 all, 29 – 43 odd