Problem 3 Using the Polygon Angle-Sum Theorem

What is m ∠ Y bold italic m angle y in pentagon TODAY?

Use the Polygon Angle-Sum Theorem for n = 5. n equals 5.

m ∠ T + m ∠ O + m ∠ D + m ∠ A + m ∠ Y = ( 5 − 2 ) 180   110 + 90 + 120 + 150 + m ∠ Y = 3 · 180 Substitute . 470 + m ∠ Y = 540 Simplify . m ∠ Y = 70 Subtract 470 from each side . table with 4 rows and 3 columns , row1 column 1 , m angle t plus m angle o plus m angle d plus m angle eh plus m angle y , column 2 equals open 5 minus 2 close 180 , column 3 , row2 column 1 , 110 plus 90 plus 120 plus 150 plus m angle y , column 2 equals 3 middle dot 180 , column 3 cap substitute . . , row3 column 1 , 470 plus m angle y , column 2 equals540 , column 3 cap simplify , . , row4 column 1 , m angle y , column 2 equals70 , column 3 cap subtract470fromeachside . . , end table

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3. What is m ∠ G angle g in quadrilateral EFGH?

Take note Theorem 6-2: Polygon Exterior Angle-Sum Theorem

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

For the pentagon, m ∠ 1 + m ∠ 2 + m ∠ 3 + m ∠ 4 + m ∠ 5 = 360 . m angle , 1 plus , m angle , 2 plus , m angle , 3 plus , m angle , 4 plus , m angle , 5 equals 360 , .

You will prove Theorem 6-2 in Exercise 39.

Problem 4 Finding an Exterior Angle Measure

What is m ∠ 1 angle 1 in the regular octagon below?

By the Polygon Exterior Angle-Sum Theorem, the sum of the exterior angle measures is 360. Since the octagon is regular, the interior angles are congruent. So their supplements, the exterior angles, are also congruent.

m ∠ 1 = 360 8 Divide  360  by  8 , the number of sides in an octagon.   = 45 Simplify. table with 2 rows and 3 columns , row1 column 1 , m angle 1 , column 2 equals , 360 over 8 , column 3 cap divide . 360 , by , 8 comma . the number of sides in an octagon. , row2 column 1 , , column 2 equals 45 , column 3 cap simplify. , end table

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4. What is the measure of an exterior angle of a regular nonagon?