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Pre-AP Bellwork 10-19

Pre-AP Bellwork 10-19. 30 . (8 + 6x) . 3) Solve for x. (4x + 2)°. Pre-AP Bellwork 10-24. 5) Find the values of the variables and then the measures of the angles. z°. x°. w°. 30°. y°. (2y – 6)°. 3-4 Polygon Angle-Sum Theorem. Geometry. Definitions:. SIDE.

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Pre-AP Bellwork 10-19

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  1. Pre-AP Bellwork 10-19 30 (8 + 6x) 3) Solve for x.. (4x + 2)°

  2. Pre-AP Bellwork 10-24 5) Find the values of the variables and then the measures of the angles. z° x° w° 30° y° (2y – 6)°

  3. 3-4 Polygon Angle-Sum Theorem Geometry

  4. Definitions: SIDE • Polygon—a plane figure that meets the following conditions: • It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear. • Each side intersects exactly two other sides, one at each endpoint. • Vertex – each endpoint of a side. Plural is vertices. You can name a polygon by listing its vertices consecutively. For instance, PQRST and QPTSR are two correct names for the polygon above.

  5. State whether the figure is a polygon. If it is not, explain why. Not D – has a side that isn’t a segment – it’s an arc. Not E– because two of the sides intersect only one other side. Not F because some of its sides intersect more than two sides/ Example 1: Identifying Polygons Figures A, B, and C are polygons.

  6. Polygons are named by the number of sides they have – MEMORIZE

  7. Convex or Concave??? A convex polygon has no diagonal with points outside the polygon. A concave polygon has at least one diagonal with points outside the polygon

  8. Measures of Interior and Exterior Angles • You have already learned the name of a polygon depends on the number of sides in the polygon: triangle, quadrilateral, pentagon, hexagon, and so forth. The sum of the measures of the interior angles of a polygon also depends on the number of sides.

  9. Measures of Interior and Exterior Angles • For instance . . . Complete this table

  10. Pre-AP Bellwork 10 - 24 6) Find the sum of the interior angles of a dodecagon.

  11. Measures of Interior and Exterior Angles • What is the pattern? (n – 2) ● 180. • This relationship can be used to find the measure of each interior angle in a regular n-gon because the angles are all congruent.

  12. Find the value of x in the diagram shown: Ex. 1: Finding measures of Interior Angles of Polygons 142 88 136 105 136 x

  13. S(hexagon)= (6 – 2) ● 180 = 4 ● 180 = 720. Add the measure of each of the interior angles of the hexagon. 142 88 136 105 136 x SOLUTION:

  14. 136 + 136 + 88 + 142 + 105 +x = 720. 607 + x = 720 X = 113 SOLUTION: • The measure of the sixth interior angle of the hexagon is 113.

  15. The sum of the measures of the interior angles of a convex n-gon is (n – 2) ● 180 COROLLARY: The measure of each interior angle of a regular n-gon is: Polygon Interior Angles Theorem ● (n-2) ● 180 or

  16. EX.2 Find the measure of an interior angle of a decagon…. n=10

  17. Ex. 2: Finding the Number of Sides of a Polygon • The measure of each interior angle is 140. How many sides does the polygon have? • USE THE COROLLARY

  18. Solution: = 140 Corollary to Thm. 11.1 (n – 2) ●180= 140n Multiply each side by n. 180n – 360 = 140n Distributive Property Addition/subtraction props. 40n = 360 n = 90 Divide each side by 40.

  19. Copy the item below.

  20. EXTERIOR ANGLE THEOREMS 3-10 3-10

  21. Ex. 3: Finding the Measure of an Exterior Angle

  22. Ex. 3: Finding the Measure of an Exterior Angle 3-10 Simplify.

  23. Ex. 3: Finding the Measure of an Exterior Angle 3-10

  24. Using Angle Measures in Real LifeEx. 4: Finding Angle measures of a polygon

  25. Using Angle Measures in Real LifeEx. 5: Using Angle Measures of a Regular Polygon

  26. Using Angle Measures in Real LifeEx. 5: Using Angle Measures of a Regular Polygon

  27. Using Angle Measures in Real LifeEx. 5: Using Angle Measures of a Regular Polygon Sports Equipment: If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of: • 135°? • 145°?

  28. Using Angle Measures in Real LifeEx. : Finding Angle measures of a polygon

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