Lesson 8-1

Angles of Polygons

5-Minute Check on Lesson 7-45-Minute Check on Lesson 7-45-Minute Check on Lesson 7-45-Minute Check on Lesson 7-4 Transparency 7-5

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

1. Use a graphing calculator to find tan 54°. Round to the nearest ten-thousandth.

2. Find mB to the nearest tenth of a degree if cos B = 0.8926 and B is an acute angle.

Find x. Round the nearest tenth.

3. 4. 5.

6. What is the value of tan Θ?

13x°

9

36°

24x

59°

18.5 x

Standardized Test Practice:

A CB D

13Θ

5

12--- 5

12---13

5---12

5---13

5-Minute Check on Lesson 7-45-Minute Check on Lesson 7-45-Minute Check on Lesson 7-45-Minute Check on Lesson 7-4 Transparency 7-5

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

1. Use a graphing calculator to find tan 54°. Round to the nearest ten-thousandth.

2. Find mB to the nearest tenth of a degree if cos B = 0.8926 and B is an acute angle.

Find x. Round the nearest tenth.

3. 4. 5.

6. What is the value of tan Θ?

13x°

9

36°

24x

59°

18.5 x

Standardized Test Practice:

A CB D

13Θ

5

12--- 5

12---13

5---12

5---13

1.3764

26.8°

14.146.2°

30.8

Objectives

• Find the sum of the measures of the interior angles of a polygon– Sum of Interior angles = (n-2) • 180– One Interior angle = (n-2) • 180 / n

• Find the sum of the measures of the exterior angles of a polygon– Sum of Exterior angles = 360– One Exterior angle = 360/n– Exterior angle + Interior angle = 180

Vocabulary

• Diagonal – a segment that connects any two nonconsecutive vertices in a polygon.

Angles in a Polygon

Octagon n = 8

1

2

3

4

5

6

7

8

8 triangles @ 180° - 360° (center angles) = (8-2) • 180 = 1080

Sum of Interior angles = (n-2) • 180

Angles in a Polygon

Exterior Angle

Interior Angle

Sum of Interior Angles:

(n – 2) * 180 where n is number of sides

so each interior angle is (n – 2) * 180 n

Octagon n = 8

Sum of Exterior Angles: 360

so each exterior angle is 360 n

Interior Angle + Exterior Angle = 180OctagonSum of Exterior Angles: 360Sum of Interior Angles: 1080One Interior Angle: 135One Exterior Angle: 45

Polygons

Sides NameSum of Interior Angles

One Interior Angle

Sum OfExteriorAngles

OneExterior Angles

3 Triangle 180 60 360 120

4Quadrilateral

360 90 360 90

5 Pentagon 540 108 360 72

6 Hexagon 720 120 360 60

7 Heptagon 900 129 360 51

8 Octagon 1080 135 360 45

9 Nonagon 1260 140 360 40

10 Decagon 1440 144 360 36

12 Dodecagon 1800 150 360 30

n N - gon (n-2) * 180 180 – Ext 360 360 ∕ n =

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.

Interior Angle Sum Theorem

Simplify.

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

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.

The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.

SHORT CUT!!

Exterior angle = 180 – Interior angle = 45

360 360 n = --------- = ------- = 8 Ext 45

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.

Find the measure of each interior angle.

Since n = 4 the sum of the measures of the interior angles is 180(4 – 2) or 360°. Write an equation to express the sum of the measures of the interior angles of the polygon.

Sum of interior angles

Substitution

Combine like terms.

Subtract 8 from each side.

Divide each side by 32.

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

Answer:

Find the measure of each interior angle.

Answer:

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.

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.

9e = 360

e = 40

e = measure of each exterior angle

Answer: 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.

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

Answer: 60; 120

Polygon Hierarchy

Polygons

Squares

RhombiRectangles

Parallelograms Kites Trapezoids

IsoscelesTrapezoids

Quadrilaterals

Quadrilaterals Venn Diagram

Quadrilaterals

Parallelograms

Rectangles

IsoscelesTrapezoids

Trapezoids

Rhombi

Squares Kites

Quadrilateral Characteristics SummaryConvex Quadrilaterals

Squares

RhombiRectangles

Parallelograms Trapezoids

IsoscelesTrapezoids

Opposite sides parallel and congruentOpposite angles congruentConsecutive angles supplementaryDiagonals bisect each other

Bases ParallelLegs are not ParallelLeg angles are supplementary Median is parallel to basesMedian = ½ (base + base)

Angles all 90°Diagonals congruent

Diagonals divide into 4 congruent triangles

All sides congruentDiagonals perpendicularDiagonals bisect opposite angles

Legs are congruent Base angle pairs congruent Diagonals are congruent

4 sided polygon4 interior angles sum to 3604 exterior angles sum to 360

Summary & Homework

• Summary:– If a convex polygon has n sides and sum of the

measures of its interior angles is S, then S = 180(n-2)°

– The sum of the measures of the exterior angles of a convex polygon is 360°

– Interior angle + Exterior angle = 180 (linear pair)

• Homework: – pg 407-408; 13-15, 22-24, 27-30, 35, 36