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Section 6.1 The Polygon Angle-Sum Theorem

Section 6.1 The Polygon Angle-Sum Theorem. Students will be able to: Find the sum of the measures of the interior angles of a polygon. Find the sum of the measures of the exterior angles of a polygon. Lesson Vocabulary Equilateral polygon Equiangular polygon Regular polygon.

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Section 6.1 The Polygon Angle-Sum Theorem

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  1. Section 6.1 The Polygon Angle-Sum Theorem Students will be able to: Find the sum of the measures of the interior angles of a polygon. Find the sum of the measures of the exterior angles of a polygon. Lesson Vocabulary Equilateral polygon Equiangular polygon Regular polygon

  2. Section 6.1 The Polygon Angle-Sum Theorem List the names of all of the polygons with 3 sides to 13 sides: 3 sided: ____________ 8 sided: ____________ 4 sided: ____________ 9 sided: ___________ 5 sided: ____________ 10 sided: ___________ 6 sided: ____________ 11 sided: ___________ 7 sided: ____________ 12 sided: ___________ 13 sided: ___________

  3. Section 6.1 The Polygon Angle-Sum Theorem A diagonal is a segment that connects two nonconsecutive vertices in a polygon!

  4. Section 6.1 The Polygon Angle-Sum Theorem The Solve It is related to a formula for the sum of the interior angle measures of a CONVEX polygon.

  5. Section 6.1 The Polygon Angle-Sum Theorem Essential Understanding: The sum of the interior angle measures of a polygon depends on the number of sides the polygon has. By dividing a polygon with n sides into (n – 2) triangles, you can show that the sum of the interior angle measures of any polygon is a multiple of 180.

  6. Section 6.1 The Polygon Angle-Sum Theorem Problem 1: Finding a Polygon Angle Sum What is the sum of the interior angle measures of a heptagon?

  7. Section 6.1 The Polygon Angle-Sum Theorem Problem 1b: Finding a Polygon Angle Sum What is the sum of the interior angle measures of a 17-gon?

  8. Section 6.1 The Polygon Angle-Sum Theorem Problem 1c: The sum of the interior angle measures of a polygon is 1980. How can you find the number of sides in the polygon? Classify it!

  9. Section 6.1 The Polygon Angle-Sum Theorem Problem 1d: The sum of the interior angle measures of a polygon is 2880. How can you find the number of sides in the polygon? Classify it!!!

  10. Section 6.1 The Polygon Angle-Sum Theorem

  11. Section 6.1 The Polygon Angle-Sum Theorem

  12. Section 6.1 The Polygon Angle-Sum Theorem Problem 2: What is the measure of each interior angle in a regular hexagon?

  13. Section 6.1 The Polygon Angle-Sum Theorem Problem 2b: What is the measure of each interior angle in a regular nonagon?

  14. Section 6.1 The Polygon Angle-Sum Theorem Problem 2c: What is the measure of each interior angle in a regular 100-gon? Explain what happens to the interior angles of a regular figure the more sides the figure has? What is the value approaching but will never get to?

  15. Section 6.1 The Polygon Angle-Sum Theorem Problem 3: What is m<Y in pentagon TODAY?

  16. Section 6.1 The Polygon Angle-Sum Theorem Problem 3b: What is m<G in quadrilateral EFGH?

  17. Section 6.1 The Polygon Angle-Sum Theorem You can draw exterior angles at any vertex of a polygon. The figures below show that the sum of the measures of exterior angles, one at each vertex, is 360.

  18. Problem 4: What is m<1 in the regular octagon below?

  19. Problem 4b: What is the measure of an exterior angle of a regular pentagon?

  20. Problem 5: What do you notice about the sum of the interior angle and exterior angle of a regular figure?

  21. Problem 6: If the measure of an exterior angle of a regular polygon is 18. Find the measure of the interior angle. Then find the number of sides the polygon has.

  22. Problem 6b: If the measure of an exterior angle of a regular polygon is 72. Find the measure of the interior angle. Then find the number of sides the polygon has.

  23. Problem 6c: If the measure of an exterior angle of a regular polygon is x. Find the measure of the interior angle. Then find the number of sides the polygon has.

  24. Section 6.2 – Properties of Parallelograms Students will be able to: Use relationships among sides and angles of parallelograms Use relationships among diagonals of parallograms Lesson Vocabulary: Parallelogram Opposite Angles Opposite Sides Consecutive Angles

  25. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Essential Understanding: Parallelograms have special properties regarding their sides, angles, and diagonals.

  26. In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side.

  27. Angles of a polygon that share a side are consecutive angles. In the diagram, <A and <B are consecutive angles because the share side AB.

  28. Problem 1: What is <P in Parallelogram PQRS?

  29. Problem 1b: Find the value of x in each parallelogram.

  30. Problem 2: Solve a system of linear equations to find the values of x and y in Parallelogram KLMN. What are KM and LN?

  31. Problem 2b: Solve a system of linear equations to find the values of x and y in Parallelogram PQRS. What are PR and SQ?

  32. Problem 3:

  33. Extra Problems: Find the value(s) of the variable(s) in each parallelogram.

  34. Extra Problems: Find the measures of the numbered angles for each parallelogram.

  35. Extra Problems:

  36. Extra Problems:

  37. Extra Problems:

  38. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram Students will be able to: Determine whether a quadrilateral is a parallelogram

  39. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram Essential Understanding: You can decide whether a quadrilateral is a parallelogram if its sides, angles, and diagonals have certain properties. In Lesson 6-2, you learned theorems about the properties of parallelograms. In this lesson, you will learn the converses of those theorems. That is, if a quadrilateral has certain properties, then it must be a parallelogram.

  40. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

  41. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

  42. Section 6.3 – Proving That a Quadrilateral Is a Parallelogram

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