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Daniela Morales Leonhardt 9-5

JOURNAL CHAPTER 6. Daniela Morales Leonhardt 9-5. Describe what a polygon is. Include a discussion about the parts of a polygon. Also compare and contrast a convex with a concave polygon. Compare and contrast equilateral and equiangular. Give 3 examples of each. TOPIC 1: Polygons.

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Daniela Morales Leonhardt 9-5

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  1. JOURNAL CHAPTER 6. Daniela Morales Leonhardt 9-5

  2. Describe what a polygon is. Include a discussion about the parts of a polygon. Also compare and contrast a convex with a concave polygon. Compare and contrast equilateral and equiangular. Give 3 examples of each. TOPIC 1: Polygons

  3. What is a polygon? • A polygon is a closed plane figure that contains only straight sides. • All polygons contain interior angles, exterior angles, sides, and diagonals. • Diagonals are segments that connect one of the vertices of a polygon with another (non adjacent) vertex in that same shape. • Examples of polygons (with diagonals in blue):

  4. Concave vs. Convex • Concave: A concave polygon is a polygon with one or more of its vertices pointing inward. • Convex: A polygon with all of its vertices pointing outward.

  5. Equilateral Vs. Equiangular • An equilateral polygon is a polygon with all equal (equi-) sides (lateral). • An equiangular polygon is a polygon with all equal (equi-) angles (angular).

  6. Int. Angles Theorem for Polygons TOPIC 2: Explain the Interior angles theorem for polygons. Give at least 3 examples.

  7. Interior Angles Theorem For Polygons • The Interior Angles Theorem states that • A convex polygon with “n” sides has angle measures equal to (n-2) 180/n

  8. Example 1. • How much do the interior angles of a nonagon measure? • (9-2)180/9 • 7 x 180 = 1260 • 1260/9 • 140°

  9. Example 2. • What is the measure of the interior angles of a regular hexagon? 120° 120° • Apply the Steps: • (6-2) 180/6 • 4 x 180 = 720/6 • 120 120° 120° 120° 120°

  10. Example 3. • Find the measure of the interior angles of a square. • Apply the Steps • (4-2) 180 / 4 • 2 x 180= 360/4= 90°

  11. TOPIC 3: Parallelograms Describe the 4 theorems of parallelograms and their converse and explain how they are used. Give at least 3 examples of each.

  12. Theorem 1 • If a quadrilateral has two pairs of parallel sides, then it’s a parallelogram. • If a quadrilateral is a parallelogram, then it has two pairs of parallel sides. C A ll B D ll C Polygon ABCD is a Parallelogram (Ex. 1) A B D

  13. Example 2 & 3. D ll F E ll G Quadrilateral FED is a parallelogram E F D X G Quadrilateral XYZ is a rectangle. Rectangles are parallelograms. We can infer that sides W and Y are parallel, as are X and Z. Y W Z

  14. Theorem 2 & Example 1. • If a quadrilateral’s two pairs of opposite angles are congruent, then it’s a parallelogram. • If a quadrilateral is a parallelogram, then its two opposite angles are congruent.

  15. Examples 2, 3, 4, & 5.

  16. Theorem 3 & Example 1. • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. • If the consecutive angles of a quadrilateral are consecutive, then it’s a parallelogram. 1 3 m. Angle 1 + m. Angle 1 = 100° m. Angle 3 + m. Angle 4 = 100 ° 2 4

  17. Examples 2, 3, 4. 76 104 104 76 90 90 24 156 90 90 156 24

  18. Theorem 4 & Example 1. • If a quadrilateral is a parallelogram, then its diagonals bisect each other. • If a quadrilateral’s diagonals bisect each other, then it’s a parallelogram.

  19. Examples 2, 3, and 4.

  20. TOPIC 4: Quadrilaterals Parallelograms Describe how to prove that a quadrilateral is a parallelogram. Include an explanation about theorem 6.10. Give at least 3 examples of each.

  21. If a Quadrilateral… • If a quadrilateral’s opposite sides are congruent, then it’s is a parallelogram. • … has opposite sides that are parallel, then it’s a parallelogram. • … has congruent opposite angles, then it’s a parallelogram. • … has supplementary consecutive angles then it’s a parallelogram • If a quadrilateral’s diagonals bisect each other, then it’s a parallelogram.

  22. Theorem 6.10 • Theorem 6.10 states that if a quadrilateral has a set of sides that are both congruent and parallel, then it must be a parallelogram.

  23. Examples 1-5.

  24. Examples 6-10. Supp. Supp. Supp. Supp. Supp. Supp. Supp. Supp. Supp. Supp.

  25. TOPIC 5: Rectangles,Rhombuses, And Squares. Compare and contrast a rhombus with a square with a rectangle. Describe the rhombus, square and rectangle theorems. Give at least 3 examples of each.

  26. Rectangles • A parallelogram with: • 4 right angles. • Congruent diagonals. • If a parallelogram has a right angle then it’s a rectangle. • If a parallelogram has congruent diagonals, then it’s a rectangle. Theorems:

  27. Examples

  28. Theorems Rhombuses • A parallelogram with 4 congruent sides. • Diagonals are perpendicular. • If a quadrilateral has two congruent consecutive sides, then it is a rhombus. • If a quadrilateral’s diagonals are perpendicular, then it’s a parallelogram.

  29. Examples

  30. Squares • A parallelogram that has 4 right angles and four congruent sides. • It’s both a rhombus and a rectangle, so all their properties apply.

  31. Examples

  32. TOPIC 6: Trapezoids Describe a trapezoid. Explain the trapezoidal theorems. Give at least 3 examples of each.

  33. Trapezoids • A quadrilateral with only one pair of congruent opposite sides. • The sides that are parallel are called the bases of a trapezoid, the non-congruent ones are called legs. • If both legs are congruent, then the trapezoid is isosceles. • The midsegment of a trapezoid is parallel to both bases and it’s half the sum of the bases. • The base angles in an isosceles trapezoid are congruent. • An isosceles trapezoid’s diagonals are congruent. Isosceles Trapezoids

  34. Examples base leg leg base Isosceles Trapezoid

  35. TOPIC 7: Kites Describe a kite. Explain the kite theorems. Give at least 3 examples of each.

  36. Kites • A quadrilateral with two sets of congruent (and consecutive) sides. (Opposite sides NOT congruent) • The included angles between the congruent sides are called vertex angles. • One diagonal bisects the other. • If a quadrilateral is a kite, then its diagonals are perpendicular. • If a quadrilateral is a kite, then 1 pair of opposite angles is congruent.

  37. Examples Vertex Angle Vertex Angle Diagonals

  38. Examples

  39. _____(0-10 pts.) Describe what a polygon is. Include a discussion about the parts of a polygon. Also compare and contrast a convex with a concave polygon. Compare and contrast equilateral and equiangular. Give 3 examples of each. • _____(0-10 pts.) Explain the Interior angles theorem for polygons. Give at least 3 examples. • _____(0-10 pts.) Describe the 4 theorems of parallelograms and their converse and explain how they are used. Give at least 3 examples of each. • _____(0-10 pts.) Describe how to prove that a quadrilateral is a parallelogram. Include an explanation about theorem 6.10. Give at least 3 examples of each. • _____(0-10 pts.) Compare and contrast a rhombus with a square with a rectangle. Describe the rhombus, square and rectangle theorems. Give at least 3 examples of each. • _____(0-10 pts.) Describe a trapezoid. Explain the trapezoidal theorems. Give at least 3 examples of each. • _____(0-10 pts.) Describe a kite. Explain the kite theorems. Give at least 3 examples of each. • _____(0-5 pts.) Neatness and originality bonus • _____Total points earned (90 possible)

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