6-1 The Polygon Angle-Sum Theorems

Quick Review

The sum of the measures of the interior angles of an n-gon is ( n − 2 ) 180 . open n minus 2 close 180 .  The measure of one interior angle of a regular n-gon is ( n − 2 ) 180 n . fraction open n minus 2 close 180 , over n end fraction . .  The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

Example

Find the measure of an interior angle of a regular 20-gon.

Measure = ( n − 2 ) 180 n Corollary to the Polygon Angle-Sum Theorem   = ( 20 − 2 ) 180 20 Substitute   = 18 ⋅ 180 20 Simplify.   = 162   table with 4 rows and 3 columns , row1 column 1 , cap measure , column 2 equals . fraction open n minus 2 close 180 , over n end fraction , column 3 cap corollarytothecap polygoncap angleminuscap sumcap theorem , row2 column 1 , , column 2 equals . fraction open 20 minus 2 close 180 , over 20 end fraction , column 3 cap substitute , row3 column 1 , , column 2 equals . fraction 18 dot 180 , over 20 end fraction , column 3 cap simplify. , row4 column 1 , , column 2 equals 162 , column 3 , end table

The measure of an interior angle is 162.

Exercises

Find the measure of an interior angle and an exterior angle of each regular polygon.

5. hexagon
6. 16-gon
7. pentagon
8. What is the sum of the exterior angles for each polygon in Exercises 5–7?

Find the measure of the missing angle.

6-2 Properties of Parallelograms

Quick Review

Opposite sides and opposite angles of a parallelogram are congruent. Consecutive angles in a parallelogram are supplementary. The diagonals of a parallelogram bisect each other. If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Example

Find the measures of the numbered angles in the parallelogram.

Since consecutive angles are supplementary, m ∠ 1 = 180 − 56 , m angle , 1 equals . 180 minus 56 comma  or 124. Since opposite angles are congruent, m ∠ 2 = 56 m angle , 2 equals 56  and m ∠ 3 = 124 . m angle . 3 equals 124 .

Exercises

Find the measures of the numbered angles for each parallelogram.

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14. 

Find the values of x and y in ▱ white parallelogram ABCD.

15. A B = 2 y , B C = y + 3 , C D = 5 x − 1 , D A = 2 x + 4 eh b equals 2 y comma b c equals . y plus 3 comma . c d equals . 5 x minus 1 comma . d eh equals . 2 x plus 4
16. A B = 2 y + 1 , B C = y + 1 , C D = 7 x − 3 , D A = 3 x eh b equals . 2 y plus 1 comma . b c equals . y plus 1 comma . c d equals . 7 x minus 3 comma . d eh equals 3 x

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