Lesson 8-1 Angles of Polygons
5-Minute Check on Lesson 7-4 Transparency 7-5 5-Minute Check on Lesson 7-4 Use a graphing calculator to find tan 54°. Round to the nearest ten-thousandth. 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? 36° 18.5 x x° x 13 59° 9 24 Standardized Test Practice: 13 12 --- 5 12 --- 13 5 --- 12 5 --- 13 Θ A B C D 5 Click the mouse button or press the Space Bar to display the answers.
5-Minute Check on Lesson 7-4 Transparency 7-5 5-Minute Check on Lesson 7-4 Use a graphing calculator to find tan 54°. Round to the nearest ten-thousandth. 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? 1.3764 26.8° 36° 18.5 x x° x 13 59° 9 24 14.1 30.8 46.2° Standardized Test Practice: 13 12 --- 5 12 --- 13 5 --- 12 5 --- 13 Θ A B C D 5 Click the mouse button or press the Space Bar to display the answers.
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 1 2 3 4 5 6 7 8 Octagon n = 8 8 triangles @ 180° - 360° (center angles) = (8-2) • 180 = 1080 Sum of Interior angles = (n-2) • 180
Sum of Interior Angles: so each interior angle is (n – 2) * 180 Angles in a Polygon 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 Interior Angle Sum of Exterior Angles: 360 so each exterior angle is 360 n Interior Angle + Exterior Angle = 180 Exterior Angle Octagon Sum of Exterior Angles: 360 Sum of Interior Angles: 1080 One Interior Angle: 135 One Exterior Angle: 45
Polygons Sides Name Sum of Interior Angles One Interior Angle Sum Of Exterior Angles One Exterior Angles 3 Triangle 180 60 360 120 4 Quadrilateral 90 5 Pentagon 540 108 72 6 Hexagon 720 7 Heptagon 900 129 51 8 Octagon 1080 135 45 9 Nonagon 1260 140 40 10 Decagon 1440 144 36 12 Dodecagon 1800 150 30 n N - gon (n-2) * 180 180 – Ext 360 ∕ n =
Interior Angle Sum Theorem 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. Example 1-1a
Interior Angle Sum Theorem 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. Interior Angle Sum Theorem Distributive Property Subtract 135n from each side. Add 360 to each side. Divide each side by 45. Answer: The polygon has 8 sides. Example 1-2a
SHORT CUT!! The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Exterior angle = 180 – Interior angle = 45 360 360 n = --------- = ------- = 8 Ext 45
Answer: The polygon has 10 sides. 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. Example 1-2b
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 Example 1-3a
Subtract 8 from each side. 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: Example 1-3b
Find the measure of each interior angle. Answer: Example 1-3d
At each vertex, extend a side to form one exterior angle. 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. e = measure of each exterior angle 9e = 360 e = 40 Divide each side by 9. 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. Example 1-4a
Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF. Answer: 60; 120 Example 1-4c
Polygon Hierarchy Polygons Quadrilaterals Parallelograms Kites Trapezoids Isosceles Trapezoids Rectangles Rhombi Squares
Quadrilaterals Venn Diagram Trapezoids Parallelograms Isosceles Trapezoids Rhombi Kites Squares Rectangles
Quadrilateral Characteristics Summary Convex Quadrilaterals 4 sided polygon 4 interior angles sum to 360 4 exterior angles sum to 360 Parallelograms Trapezoids Bases Parallel Legs are not Parallel Leg angles are supplementary Median is parallel to bases Median = ½ (base + base) Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Rectangles Rhombi Isosceles Trapezoids All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Angles all 90° Diagonals congruent Legs are congruent Base angle pairs congruent Diagonals are congruent Squares Diagonals divide into 4 congruent triangles
Summary & Homework Summary: Homework: 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