5.7 Find Angle Measure in Polygons

Theorem 5.16: Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a convex n-gon is (n – __ ) . _____.2 180

________21 nnmmm 2 180

1

23

456 6n

Corollary to Theorem 5.16: Interior Angles of a Quadrilateral

The sum of the measures of the interior angles of a quadrilateral is _____.360

5.7 Find Angle Measure in Polygons

Theorem 5.16: Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is _____.360

_____21 nmmm 360

1

2 3

4

55n

5.7 Find Angle Measure in Polygons

Example 1 Find the sum of angle measures in a polygonFind the sum of angle measures in a polygon

Solution

______ n

Find the sum of the measure of the interior angles of a convex octagon.

An octagon has ____ sides. Use the Polygon Interior Angles Theorem.

8Substitute __ for n.2 180 ________ 8 2 180 8

Subtract._______ 6 180Multiply._____ 1080

The sum of the measures of the interior angles of an octagon is _______.1080

5.7 Find Angle Measure in PolygonsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

1.1. Find the sum of the measure of the Find the sum of the measure of the interior angles of the convex decagon.interior angles of the convex decagon.

1802n ________ 10 2 180_______ 8 180

_____ 1440

5.7 Find Angle Measure in Polygons

Example 2 Find the number of sides of a polygonFind the number of sides of a polygon

Solution

____________ n

The sum of the measures of the interior angles of a convex polygon is 1260o. Classify the polygon by the number of sides.

Use the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to find the number of sides.

Polygon Interior Angle Theorem.2 180 1260Divide each side by ___.______ n 2 7Add __ to each side.___n 9

The polygon has ___ sides. It is a _________.9

180

2

nonagon

5.7 Find Angle Measure in Polygons

Example 3 Find an unknown interior angle measureFind an unknown interior angle measure

Solution

________________ o x

Find the value of x in the diagram shown.

The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation.

Corollary to Theorem 5.16.135 112 71

Combine like terms.

______ x 318 360

Subtract ____ from each side.

___x 42 318

x

135 112

71

360

5.7 Find Angle Measure in PolygonsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

2.2. The sum of the measures of the interior angles of a The sum of the measures of the interior angles of a convex polygon is 1620convex polygon is 1620. Classify the polygon by the . Classify the polygon by the number of sides.number of sides.

n 2 180 1620

2n 9180 180

n 11gon-11

105

68 109J

K L

M

N

5.7 Find Angle Measure in PolygonsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

3. Use the diagram at the right.

1802 n 540

. and Find KmKm

x x5

________________oo xx 68 105 109 540

x2 282 540x2 258x o129

5.7 Find Angle Measure in Polygons

Example 4 Find unknown exterior angle measuresFind unknown exterior angle measures

Solution

________________ o x

Find the value of x in the diagram shown.

Use the Polygon Exterior Angles Theorem to write and solve an equation.

Polygon Exterior Angle Theorem.x2 89 85Combine like terms.

________ x 174 360Solve for x.___3 x 186

360

oxo89

o2xo85

3

___x 62

5.7 Find Angle Measure in PolygonsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 4. A convex pentagon has exterior angles with

measures 6677, 82, and 62. What is the measure of an exterior angle at the fifth vertex.

xx 287 360

x o73

66 77 82 62 360

5.7 Find Angle Measure in PolygonsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 5. Find the measure of (a) each interior angle and

(b) each exterior angle of a regular nonagon.

1802 a. n 126099

o140

x9 b. 360x o40

5.7 Find Angle Measure in Polygons

Pg. 312, 5.7 #1-40