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Journal Chapter 6 .

Polygons and Quadrilaterals. Journal Chapter 6. Kirsten Erichsen 9-5. INDEX: 6-1: Properties and Attributes of Polygons 6-2: Properties of Parallelograms 6-3: Conditions for Parallelograms 6-4: Properties of Special Parallelograms

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Journal Chapter 6 .

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  1. Polygons and Quadrilaterals Journal Chapter 6. Kirsten Erichsen 9-5 INDEX: 6-1: Properties and Attributes of Polygons 6-2: Properties of Parallelograms 6-3: Conditions for Parallelograms 6-4: Properties of Special Parallelograms 6-5: Conditions for Special Parallelograms 6-6: Properties of Kites and Trapezoids

  2. POLYGONS.

  3. WHAT IS A POLYGON? • Definition: a closed figure formed by 3 or more segments, whose endpoints have to touch another 2 endpoints. • Types of Polygons (sides): • Triangle = 3 • Quadrilateral = 4 • Pentagon = 5 • Hexagon = 6 • Heptagon = 7 • Octagon = 8 • Nonagon = 9 • Decagon = 10 • Dodecagon = 12 • For the rest of the polygons with more sides than 12 (including 11) you just place the number and add “gon” to it = n-gon.

  4. HOW DO YOU KNOW IT’S A POLYGON? • A polygon must not have curved sides, they all have to be straight. • The sides of the polygon must not intersect, meaning they must not intersect the other sides of the polygon.

  5. Example 1. • What type of polygon is this, based on its sides? HEXAGON

  6. Example 2. • What type of polygon is this, based on its sides? DODECAGON

  7. Example 3. • What type of polygon is this, based on its sides? PENTAGON

  8. Example 4. • What type of polygon is this, based on its sides? IT IS NOT A POLYGON BECAUSE IT HAS A CURVED SIDE.

  9. PARTS OF A POLYGON. • Side of a Polygon: each one of the segments that forms the sides of any polygon. • Vertex of a Polygon: it is the common point where any of the 2 endpoint of the polygon meet. • Diagonal: a segment that connects 2 non-consecutive vertices. Diagonal Vertex Side

  10. Example 1. • Tell me the parts of the polygon in this shape. Diagonal Vertex Side

  11. Example 2. • Tell me the parts of the polygon in this shape. Side Diagonal Vertex

  12. Example 3. • Tell me the parts of the polygon in this shape. Vertex Side Diagonal

  13. CONVEX AND CONCAVE. • Convex Polygons: If the polygon contains all of the angles facing the outside (exterior), then it is considered convex. • Concave Polygons: If any of the angles in the polygon face the inside of the shape.

  14. Example 1. • Identify the polygon as a convex or concave and name the polygon by its sides. CONCAVE, 11-gon

  15. Example 2. • Identify the polygon as a convex or concave and name the polygon by its sides. CONCAVE, DODECAGON

  16. Example 3. • Identify the polygon as a convex or concave and name the polygon by its sides. CONVEX, QUADRILATERAL

  17. REGULAR POLYGONS. • Regular Polygon: a polygon that is both equilateral or equiangular. • EQUILATERAL: All of the sides are congruent in the polygon. • EQUIANGULAR: All of the angles are congruent in the polygon. Each side is 4 inches. Each angle is 90°.

  18. Example 1. • Classify the polygon as equilateral or equiangular. Equiangular, because each angle measures 90°.

  19. Example 2. • Classify the polygon as equilateral or equiangular. 5 cm. 5 cm. 5 cm. Equilateral, because each side measures 5 centimeters. 5 cm. 5 cm. 5 cm.

  20. Example 3. • Classify the polygon as equilateral or equiangular. 6 cm. Both equilateral and equiangular because the sides measure 6 centimeters and the angles measure 90°. 6 cm. 6 cm. 6 cm.

  21. INTERIOR ANGLES THEOREM. • This theorem states that the sum of the interior angle measures of a regular, convex polygon follows the equation of (n-2)180°. • After using the equation above, you have to divide by the number of sides of the polygon to get the angle measures. • It is also called Theorem 6-1-1 or Polygon Angle Sum Theorem.

  22. Example 1. • Find the interior angle measures of a regular decagon using the Interior angles theorem. (n − 2)180° (10 − 2)180° 8 × 180° = 1440 1440 ÷ 10 = 144° Each angle measures 144°.

  23. Example 2. • Find the interior angle measures of a regular dodecagon using the Interior angles theorem. (n − 2)180° (12 − 2)180° 10 × 180° = 1800 1800 ÷ 12 = 150° Each angle measures 150°.

  24. Example 3. • Find the interior angle measures of a regular dodecagon using the Interior angles theorem. (n − 2)180° (8 − 2)180° 6 × 180° = 1080 1080 ÷ 8 = 135° Each angle measures 135°.

  25. Example 4. • Find the interior angle measures of a regular dodecagon using the Interior angles theorem. (n − 2)180° (5 − 2)180° 3 × 180° = 540 540 ÷ 5 = 108° Each angle measures 108°.

  26. EXTERIOR ANGLES THEOREM. • This theorem states that the sum of the exterior angles of a regular, convex polygon is always going to be 360°. • To find the angle measurements you just divide 360° by the number of sides. • This theorem is also called 6-1-2 (Polygon Exterior Angle Sum Theorem).

  27. Example 1. • Find the exterior angle measures of a regular decagon using the exterior angles theorem. 360° ÷ number of sides 360° ÷ 10 = 36° Each angle measures 36°.

  28. Example 2. • Find the exterior angle measures of a regular hexagon using the exterior angles theorem. 360° ÷ number of sides 360° ÷ 6 = 60° Each angle measures 60°.

  29. Example 3. • Find the exterior angle measures of a regular dodecagon using the exterior angles theorem. 360° ÷ number of sides 360° ÷ 12 = 30° Each angle measures 30°.

  30. PARALLELOGRAM THEOREMS. Parallelogram: a quadrilateral with two pairs of opposite sides.

  31. THEOREM ONE (6-2-1) • If a quadrilateral is a parallelogram, then its opposite sides are congruent. • CONVERSE: If its opposite sides are congruent, then the quadrilateral is a parallelogram. • This means that the quadrilateral has to have a pair of congruent sides to be considered a parallelogram. • It is considered a parallelogram if the sides are congruent and parallel.

  32. Example 1. • This is a parallelogram because the opposite sides are congruent.

  33. Example 2. • This is a parallelogram because the opposite sides are congruent.

  34. Example 3. • This is not a parallelogram because we don’t know it the opposite sides are congruent.

  35. THEOREM TWO (6-2-2) • If a quadrilateral is a parallelogram, then its opposite angles are congruent. • CONVERSE: If the opposite angles are congruent, then the quadrilateral is considered a parallelogram. • This theorem is used only when the parallelogram is proved to be when it has 2 pairs of congruent angles.

  36. Example 1. • This is a parallelogram because the opposite angles are congruent.

  37. Example 2. • This is a parallelogram because the opposite angles are congruent. 55° 125° 55° 125°

  38. Example 3. • This is not a parallelogram because the opposite angles are not congruent and they do not add up to 360°. 55° 125° 80° 130°

  39. THEOREM THREE (6-2-3) • If a quadrilateral is a parallelogram, then its consecutive angles or same side angles are supplementary. • CONVERSE: If the consecutive or same side angles are supplementary, then it is considered a parallelogram. • It is used when the same side angles add up to 180° to make up a linear pair.

  40. Example 1. • This is a parallelogram because the same side angles add up to 180° and both pairs add up to 360°. 120° 60° 60° 120°

  41. Example 2. • This is a parallelogram because the consecutive angles add up to 180° and both linear pairs add up to 360°. 55° 125° 55° 125°

  42. Example 3. • This is not considered a parallelogram because one pair of consecutive angles does not add up to 180°. 55° 125° 55° 130°

  43. THEOREM FOUR (6-2-4) • If a quadrilateral is a parallelogram, then its diagonals bisect each other. • CONVERSE: If the diagonals bisect each other, then it is considered to be a parallelogram. • The diagonals have to bisects right through the middle of reach other (meaning the midpoint).

  44. Example 1. • This is a parallelogram because the diagonals bisects at the midpoints. 5 cm. 7 cm. 7 cm. 5 cm.

  45. Example 2. • This is a parallelogram because the diagonals bisect exactly at the midpoints. 8 cm. 9 cm. 9 cm. 8 cm.

  46. Example 3. • This is not considered a parallelogram because both of the diagonals do not intersect at the midpoint. 9.5 cm. 7 cm. 7.5 cm. 9 cm.

  47. PROOVING PARALLELOGRAMS

  48. REMEMBER ABOUT PARALLELOGRAMS. • Both pairs of opposite sides are parallel. • One pair of opposite sides are parallel and congruent. • Both pairs of opposite sides are congruent. • Both pairs of angles are congruent. • One angle is supplementary to both of the consecutive angles. • The diagonals bisect each other.

  49. THEOREM 6-3-1 Yes • If one pair of opposite sides of a quadrilateral are parallel and congruent, then it is a parallelogram. • Tell me if it is a parallelogram and why. Yes No

  50. THEOREM 6-3-2 • If both pairs of opposite sides of a quadrilateral are congruent, then it is considered a parallelogram. • Tell me if it is a parallelogram and why. Yes Yes No

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