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Review Interior Angle Sum New: Exterior Angle Sum QUIZ: Prove that the diagonal connecting the vertex angles of a kite cut the kite into two congruent triangles. 1 Interior Angle Sum Formula: Some Problems (solve for x): 37x 141˚ 3x + 90 30x - 20 153˚ 116˚ x˚ 4x + 101 27x + 50 20x + 15 103˚ 147˚ 172˚ 2 Working Backwards If the sum of the angles in a polygon is 1620˚, how many sides does it have? If the sum of the angles in a polygon is 1260˚, how many sides does it have? If the sum of the angles in a polygon is 540˚, how many sides does it have? 3 Each Interior Angle in a Regular Polygon Formula: Find each interior angle in each polygon: Regular Hexagon: Regular Decagon: Regular Octagon: Regular Quadrilateral: 4 Working Backwards If each interior angle in a regular polygon is 150˚, how many sides does it have? If each interior angle in a regular polygon is 144˚, how many sides does it have? If each interior angle in a regular polygon is 135˚, how many sides does it have? 5 Exterior Angles Definition/Diagram: Property Compared to Interior Angles: 6 Triangle: Calculate the sum of the exterior angle measures of a triangle by completing each step. Step 1: Draw a triangle and extend each side to locate an exterior angle at each vertex. Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a triangle. 7 Quadrilateral 2. Calculate the sum of the exterior angle measures of a quadrilateral by completing each step. Step 1: Draw a quadrilateral and extend each side to locate an exterior angle at each vertex. Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a quadrilateral. 8 8 Pentagon Calculate the sum of the exterior angle measures of a pentagon by completing each step. Step 1: Draw a pentagon and extend each side to locate an exterior angle at each vertex. Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a pentagon. 9 Hexagon Calculate the sum of the exterior angle measures of a hexagon by completing each step. Step 1: Draw a hexagon and extend each side to locate an exterior angle at each vertex. Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a hexagon. 10 Putting it Together 5. Complete the table. Number of sides of the polygon 3 4 5 6 7 15 Number of linear pairs formed Sum of measures of linear pairs Sum of measures of interior angles Sum of measures of exterior angles 6. When you calculated the sum of the exterior angle measures in the 15-sided polygon, did you need to know anything about the number of linear pairs, the sum of the linear pair measures, or the sum of the interior angle measures of the 15-sided polygon? Explain. 7. If a polygon has 100 sides, calculate the sum of the exterior angle measures. Explain how you calculated your answer. 8 11 7. If a polygon has 100 sides, calculate the sum of the exterior angle measures. Explain how you calculated your answer. 8 8. What is the sum of the exterior angle measures of an n-sided polygon? 9. If the sum of the exterior angle measures of a polygon is 360!, how many sides does the polygon have? Explain how you got this answer. Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 463 12 OBLEM 2 Regular Polygons Regular Polygons 1. Calculate the measure of each exterior angle of an equilateral triangle. Explain your reasoning. 2. Calculate the measure of each exterior angle of a square. Explain your reasoning. 4. Calculate the measure of each exterior angle of a regular hexagon. Explain your reasoning. 5. Complete the table shown to look for a pattern. Number of sides of a regular polygon 3 4 5 6 7 15 2010 Carnegie Learning, Inc. 3. Calculate the measure of each exterior angle of a regular pentagon. Explain your reasoning. 13 5. Complete the table shown to look for a pattern. Number of sides of a regular polygon 3 4 5 6 7 15 © 2010 Carnegie Learning, Regular Polygons Sum of measures 6. When you calculated the measure of each exterior angle in the 15-sided regular of exterior angles polygon, did you need to know anything about the measure of each interior angle? Explain. Measure of each interior angle Measure of each exterior angle 7. If a regular polygon has 100 sides, calculate the measure of each exterior angle. Explain how you calculated your answer. Quadrilaterals 8. What is the measure of each exterior angle of an n-sided regular polygon? 9. If the measure of each exterior angle of a regular polygon is 18!, how many sides does the polygon have? Explain how you calculated your answer. 8 14 Name _____________________________________________ Date ____________________ Home Work Exterior and Interior Angle Measurement Interactions Sum of the Exterior Angle Measures of a Polygon 8 Use the figure below to answer each question. 2 4 1 5 3 6 1. What is the sum of the measures of angles 1 and 4? Explain your reasoning. © 2010 Carnegie Learning, Inc. 2. What is the sum of the measures of angles 2 and 5? Explain your reasoning. 3. What is the sum of the measures of angles 3 and 6? Explain your reasoning. 4. What is the sum of the measures of angles 1, 2, 3, 4, 5, and 6? Explain your reasoning. 15 Home Work 5. What is the sum of the measures of angles 1, 2, and 3? Explain your reasoning. 6. What is the difference of the sum of the measures of angles 1, 2, 3, 4, 5, and 6 and the sum of the measures of angles 1, 2, and 3? What does this demonstrate? 7. If a regular polygon has 30 sides, what is the measure of each exterior angle? Explain your reasoning. , Inc. 8. The degree measure of each exterior angle of a regular octagon is represented by the expression 7x ! 4. Solve for x. 16