Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides...

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Lesson 8-1 Angles of Polygons • Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = 180(n – 2) • Theorem 8.2 Exterior Angle Sum Theorem If a polygon is convex, then the sum of the measures of the

Transcript of Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides...

Page 1: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Lesson 8-1 Angles of Polygons

• Theorem 8.1 Interior Angle Sum TheoremIf a convex polygon has n sides and S is the sum

of the measures of its interior angles, then S = 180(n – 2)• Theorem 8.2 Exterior Angle Sum TheoremIf a polygon is convex, then the sum of the

measures of the exterior angles, one at each vertex, is 360.

Page 2: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of measures of the interior angles of the pentagon.

Since a pentagon is a convex polygon, we can use the Angle Sum Theorem.

Page 3: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Interior Angle Sum Theorem

Simplify.

Answer: The sum of the measures of the angles is 540.

Page 4: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon.

Answer: 1080

Page 5: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

The measure of an interior angle of a regular polygon is 135. 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.

Answer: The polygon has 8 sides.

Interior Angle Sum Theorem

Distributive Property

Subtract 135n from each side.

Add 360 to each side.

Divide each side by 45.

Page 6: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.

Answer: The polygon has 10 sides.

Page 7: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Find the measure of each interior angle.

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.

Page 8: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Sum of measures of angles

Substitution

Combine like terms.

Subtract 8 from each side.

Divide each side by 32.

Page 9: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Use the value of x to find the measure of each angle.

Answer:

Page 10: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Find the measure of each interior angle.

Answer:

Page 11: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ.

At each vertex, extend a side to form one exterior angle.

Page 12: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

The sum of the measures of the exterior angles is 360. A convex regular nonagon has 9 congruent exterior angles.

Divide each side by 9.

Answer: The measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140.

Page 13: Lesson 8-1 Angles of Polygons Theorem 8.1 Interior Angle Sum Theorem If a convex polygon has n sides and S is the sum of the measures of its interior angles,

Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF.

Answer: 60; 120


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