Download 8.1 Find Angle Measures in Polygons

Geometry Chapter 8: Quadrilaterals 8.1-­‐ Find Angle Measures in Polygons SWBAT: find interior and exterior angle measures in polygons. Common Core: G.CO.11, G.CO.13, G.SRT.5 “Do Now” Fill in the blank. A _____________________ of a polygon is a segment that joins two nonconsecutive vertices. Polygon Investigation 1. How many degrees are in a triangle? _______________________ The number of triangles that can be drawn within the polygon can determine the sum of the degrees in any polygon. Complete the table below and look for a pattern to find the sum of the degrees in any polygon. Interior Angles Sum Polygon Picture # of Sides # of Triangles Sum of Interior ∠′𝒔 Triangle Quadrilateral Geometry Chapter 8: Quadrilaterals Pentagon Hexagon Heptagon Octagon Nonagon Decagon X X X X Sum of the Interior Angles of a Polygon If 𝒏 represents the number of sides of a polygon, then the sum of the interior angles(s) can be found using the formula: Geometry Chapter 8: Quadrilaterals 2. What does it mean for a polygon to be regular? ___________________________________________________________ Regular Polygon Sum of Interior Angles Measure of each Interior Angle Regular Polygon Sum of Interior Angles Measure of each Interior Angle Triangle Heptagon Quadrilateral Octagon Pentagon Nonagon Hexagon Decagon Measure of the Interior Angle of a Regular Polygon A regular polygon is one in which all the sides are equal; therefore all the ______________________ are equal. To find the measure of an interior angle in a regular polygon use: Example 1: Interior Angles 1. Find the sum of the interior angles of a 15-­‐gon. 2. Find the sum of interior angles of a 21-­‐gon. Geometry Chapter 8: Quadrilaterals 3. What is the measure of each interior angle of a regular dodecagon? 4. What is the measure of each interior angle of a regular 18-­‐gon? 5. The measure of an interior angle of a regular polygon is 165°, how many sides does this polygon have? 6. The measure of an interior angle of a regular polygon is 156°, how many sides does this polygon have? Polygon Exterior Angles: The sum of the measures of the exterior angles of a convex polygon is Geometry Chapter 8: Quadrilaterals Finding the measure of exterior angle of a regular polygon. •
Divide 360° by the number of sides to get the measure of each exterior angle. Example 2: Exterior Angles 7. What is the measure of each exterior angle of a regular hexagon? 8. What is the measure of each exterior angle of a regular nonagon? 9. What is the measure of each exterior angle of a regular 24-­‐gon? 10. What is the measure of each exterior angle of a regular 30-­‐gon? 11. If the exterior angle of a regular polygon measures 12°, how many sides does the polygon have? 12. If the exterior angle of a regular polygon measures 40°, how many sides does the polygon have? 13. The measure of the exterior angles of a convex quadrilateral are 90°, 10𝑥°, 5𝑥°, and 45°. What is the measure of the largest exterior angle? Geometry Chapter 8: Quadrilaterals 14. The measure of the interior angles of a convex octagon are 45𝑥°, 40𝑥°, 155°, 120°, 155°, 38𝑥°, 158°, and 41𝑥°. What is the measure of the smallest interior angle? Closure: Interior & Exterior Angles of a Polygon Ø Sum of Interior Angles of a Polygon: ______________________________________________________ Ø Interior Angles of a Regular Polygon: ______________________________________________________ Ø Sum of Exterior Angles of a Polygon: ______________________________________________________ Ø Exterior Angles of a Regular Polygon: ______________________________________________________ Ø Number of Sides of a Regular Polygon: ______________________________________________________ Homework: 1. What is the sum of the measuresof the interior 2. What is the measure of each interior angle of a angles of a 27-­‐gon? regular 20-­‐gon? 3. The sum of the interior angles of a polygon is 3960°. How many sides does the polygon have? 4. The measures of seven angles in a nonagon measure 138°, 154°, 145°, 132°, 128°, 147°, and 130°. If the two remaining angles are equal in measure, what is the measure of each angle? Geometry Chapter 8: Quadrilaterals 5. What is sum of the measures of the exterior angles of a heptagon? 6. What is the measure of each exterior angle of a 20-­‐gon? 7. If the exterior angle of a regular polygon measures 9°, how many sides does the polygon have? 8. If the interior angles of a regular polygon measure 108°, how many sides does the polygon have? 9. Find the value of x. 10. Find the value of x. 11. Find the value of x. Geometry Chapter 8: Quadrilaterals 12. 𝐴𝐵𝐶𝐹 ≅ 𝐸𝐷𝐶𝐹. Find the value of x. 13. Find the value of x. 14. Find 𝑚∠𝐵𝐴𝐶. 15. Find the value of x.