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Understanding Regular Polygons: Properties and Angle Relationships

Dive into the fascinating world of regular polygons, where all sides and angles are equal. This presentation covers essential properties of regular polygons, including types like triangles, quadrilaterals, and hexagons. Learn how to calculate interior and exterior angles using fundamental formulas. Discover tips for solving problems related to regular polygons, such as finding side counts based on angle measures. Strengthen your geometry skills with practical examples and engaging insights into the behavior of these unique shapes.

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Understanding Regular Polygons: Properties and Angle Relationships

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  1. Regular Polygons Geometry Chapter 3 A BowerPoint Presentation

  2. Regular polygon • A regular polygon is • Equilateral (all sides are ) AND • Equilangular (all angles are )

  3. Regular polygon • A regular polygon is • Equilateral (all sides are ) AND • Equilangular (all angles are ) A regular triangle

  4. Regular polygon • A regular polygon is • Equilateral (all sides are ) AND • Equilangular (all angles are ) A regular quadrilateral (also known as a square)

  5. Regular polygon • A regular polygon is • Equilateral (all sides are ) AND • Equilangular (all angles are ) A regular hexagon

  6. Interior angles of regular polygons • What is the measure of each interior angle of this regular hexagon? There are six congruent sides & six congruent angles

  7. Interior angles of regular polygons • What is the measure of each interior angle of this regular hexagon? Do you remember (n – 2) 180 ? [n is the # of sides]

  8. Interior angles of regular polygons • What is the measure of each interior angle of this regular hexagon? (6 – 2) 180 = (4) 180 = 720° is the sum for int ∡s

  9. Interior angles of regular polygons • What is the measure of each interior angle of this regular hexagon? 120° All interior ∡s are equal, so 720° / 6 = 120°

  10. Exterior angles of regular polygons • What is the measure of each exterior angle of this regular hexagon? Do you remember what the sum of the exterior ∡ s of ANY convex polygon must = ?

  11. Exterior angles of regular polygons • What is the measure of each exterior angle of this regular hexagon? The sum of the exterior ∡ s of ANY convex polygon must = 360° !

  12. Exterior angles of regular polygons • What is the measure of each exterior angle of this regular hexagon? 60° All the exterior ∡ s are equal, so 360° / 6 = 60° !

  13. Interior ∡ + Exterior ∡ = 180° • Notice that at each vertex, the interior ∡ + the exterior ∡ = 180°

  14. Interior ∡ + Exterior ∡ = 180° • Notice that at each vertex, the interior ∡ + the exterior ∡ = 180° For regular polygons… if you know either the interior ∡ or the exterior ∡ , it is easy to find the other one! (How?)

  15. A BowerPowerful TIP • When you are working on regular polygon problems, your BEST FRIEND is the…

  16. A BowerPowerful TIP • When you are working on regular polygon problems, your BEST FRIEND is the… EXTERIOR ∡

  17. A BowerPowerful TIP • When you are working on regular polygon problems, your BEST FRIEND is the… EXTERIOR ∡ 1) We know all the exterior angles of a regular polygon are equal

  18. A BowerPowerful TIP • When you are working on regular polygon problems, your BEST FRIEND is the… EXTERIOR ∡ • We know all the exterior angles of a regular polygon are equal, and • We know all the exterior angles of a regular polygon have a sum of 360°

  19. Example #1- You know the exterior ∡ Each exterior angle of a certain regular polygon has a measure of 10°. • How many sides does this regular polygon have? • What is the measure of each interior ∡ ?

  20. Example #1- You know the exterior ∡ Each exterior angle of a certain regular polygon has a measure of 10°. • How many sides does this regular polygon have? • What is the measure of each interior ∡ ? We are in great shape, because we know each exterior ∡ = 10° !

  21. Example #1- You know the exterior ∡ Each exterior angle of a certain regular polygon has a measure of 10°. • How many sides does this regular polygon have? Let’s start with the # of sides (which is the same as the # of angles) 360° / 10 = 36 sides

  22. Example #1- You know the exterior ∡ Each exterior angle of a certain regular polygon has a measure of 10°. • What is the measure of each interior ∡ ? We know exterior ∡+interior ∡= 180° 10° +interior ∡= 180° each interior ∡ = 170°

  23. YOU TRY- You know the exterior ∡ Each exterior angle of a certain regular polygon has a measure of 20°. • How many sides does this regular polygon have? • What is the measure of each interior ∡ ?

  24. YOU TRY- You know the exterior ∡ Each exterior angle of a certain regular polygon has a measure of 20°. • How many sides does this regular polygon have? 18 sides • What is the measure of each interior ∡ ? 160°

  25. Example #2- You DON’T know the exterior ∡ Each interior angleof a certain regular polygon has a measure of 179°. • How many sides does this regular polygon have? • What is the measure of each exterior ∡ ?

  26. Example #2- You DON’T know the exterior ∡ Each interior angleof a certain regular polygon has a measure of 179°. • How many sides does this regular polygon have? • What is the measure of each exterior ∡ ? Our best friend is the exterior ∡ - let’s find it first!

  27. Example #2- You DON’T know the exterior ∡ Each interior angleof a certain regular polygon has a measure of 179°. • What is the measure of each exterior ∡ ? exterior ∡+interior ∡= 180° exterior ∡+179°= 180° exterior ∡ = 1°

  28. Example #2- You DON’T know the exterior ∡ Each interior angleof a certain regular polygon has a measure of 179°. • How many sides does this regular polygon have? We can use exterior ∡ = 1° to help us find the number of sides.

  29. Example #2- You DON’T know the exterior ∡ Each interior angleof a certain regular polygon has a measure of 179°. • How many sides does this regular polygon have? Since each exterior ∡ = 1° , 360° / 1 = 360 sides

  30. YOU TRY- You DON’T know the exterior ∡ A certain regular polygon has 10 sides. • What is the measure of each interior ∡ ? • What is the measure of each exterior ∡ ? [Remember which ∡ is your best friend!!!]

  31. YOU TRY- You DON’T know the exterior ∡ A certain regular polygon has 10 sides. • What is the measure of each interior ∡ ? 144° • What is the measure of each exterior ∡ ? 36° [Did you find the exterior ∡ first?]

  32. BowerPower.net

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