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Properties of Polygons: Polygons, Parallelograms, and Rhombuses

Learn about polygons, parallelograms, and rhombuses, including their definitions, properties, and theorems. Explore how to identify and prove these shapes using coordinates and angle measures.

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Properties of Polygons: Polygons, Parallelograms, and Rhombuses

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  1. Chapter 6 Notes

  2. 6.1 – Polygons

  3. Poly  Many A polygon is a plane figure that meets the following conditions: 1) It is formed by three or more segments called sides such that no two sides with a common endpoint are collinear. 2) Each side intersects exactly two other sides, one at each endpoint.

  4. Names of polygons # of sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 8 Octagon 10 decagon 12 dodecagon n n-gon When naming polygons, you list the vertices in order! R P T E L B Hexagon PRTBLE Or Hexagon TBLEPR Or other names

  5. Convex  A polygon such that no line containing a side of a polygon contains a point in the interior of the polygon.

  6. Equilateral  Sides are the same. Equiangular  Angles are the same. Regular  Both

  7. Is it a polygon? If so, name it and say if it is convex or concave. Diagonal – Segment that joins two nonconsecutive vertices

  8. Interior angles of a Quadrilateral The sum of the measures of the interior angles of a quadrilateral is 360o. Solve for x xo xo x+20o x+20o 2xo 2xo 100o 100o

  9. (3x + 2)o (2x – 10 )o (x + 5)o (2x – 7)o

  10. 6.2 – Properties of Parallelograms

  11. Definition of a parallelogram  Both pairs of opposite sides are parallel. A D M C B

  12. Find all information A D M C B

  13. x + 24 z 65 y 4x – 12

  14. zo 7y wo 9q + 4 3y + 28 xo 30o 30o write

  15. Proving Theorem 6.2 A D B C

  16. C B F A D E

  17. X C B 1 E F Y 2 A D Z

  18. 6.3 – Proving Quadrilaterals are Parallelograms

  19. Opposite sides are parallel, then it’s a ||-gram by def. S 3 T 2 1 R 4 Q THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram. THRM 6.8, If an angle of a quad is supplementary to both its consecutive angles, then the quadrilateral is a ||- gram THRM 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a ||-gram THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram

  20. Let’s discuss how to prove two of these theorems S 3 T 2 1 R 4 Q THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram. M THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram

  21. THRM 6.10, If one pair of opposite sides are both CONGRUENT and PARALLEL, then the quadrilateral is a ||-gram S 3 T 2 M 1 R 4 Q Yes parallelogram, not a parallelogram, and why?

  22. Prove that the following coordinates make a parallelogram by the given theorems\definitions using slope formula and distance formula. A B (-1, 3) (3, 2) D C (-4, -1) (0, -2) Opposite sides are parallel, then it’s a ||-gram by def. One pair of opposite sides are both CONGRUENT and PARALLEL. The diagonals of a quad bisect each other

  23. Do these points make a parallelogram? (0,2) (-3,1) (-2, 3) (1,4) Do these points make a parallelogram? (-1,-3) (-2, 1) (2, 2) (1, -3)

  24. 6.4 – Rhombuses, Rectangles, and Squares

  25. Rectangle – Quad with 4 rt angles. Rhombus – Quad with 4 congruent sides Square – 4 rt angles AND 4 congruent sides, It’s a rectangle AND a rhombus!! Why are these parallelograms?

  26. World O’ Parallelograms Answer with always, sometimes, or never A Rhombus is a Square Always Sometimes Never A Square is a Parallelogram Always Sometimes Never

  27. Corollaries Rhombus Corollary A quadrilateral is a rhombus iff it has four congruent sides. Rectangle Corollary A quadrilateral is a rectangle iff it has four right angles. Square Corollary  A quadrilateral is a square iff it is a rhombus and a rectangle.

  28. A D M B C A B D C Thrm 6.11  A ||-gram is a rhombus iff its diagonals are perpendicular. Thrm 6.12  A ||-gram is a rhombus iff each diagonal bisects a pair of opposite angles.

  29. Thrm 6.13  A ||-gram is a rectangle iff its diagonals are congruent. A D B C Prove this theorem. Stuwork

  30. Which shape could it be? • What can be true about it?

  31. Solve for x and y. Given the figure is a rectangle, Solve for x and y. 2y + 35 70o xo 2y + 16 (2x + 5)o 4y – 10 5y – 10 Stuwork

  32. Solve for x and y. Given the figure is a square, Solve for x and y. 55o A B xo 10 yo 5 y D C AC = 4x – 10 BD = 2x + 2 Stuwork

  33. A ||-gram is a rectangle iff its diagonals are congruent. We’ll now prove this using coordinate proof A D B C ( , ) ( , ) ( , ) ( , ) Stuwork

  34. Well do some coordinate proof stuff with rhombus, rectangles, and squares added on.

  35. 6.5 – Trapezoids and Kites

  36. A quadrilateral with EXACTLY one pair of parallel sides is called a TRAPEZOID. The parallel sides are called BASES. The other sides are LEGS Trapezoids have two pairs of base angles. ISOSCELES TRAPEZOID – LEGS are CONGRUENT! KITE – A quadrilateral with two pairs of consecutive sides, BUT opposite sides aren’t congruent.

  37. Theorem 6.14  Base angles of an isosceles trapezoid are congruent. B A X Y Z ONLY TRUE FOR ISOS TRAPEZOID, NOT REGULAR TRAPEZOID! Congruent, opp sides ||-gram congruent. Transitive to work all sides congruent. Corresponding, base angles thrm, transitive. Same side interior angles, measure, supp, subtraction.

  38. Theorem 6.15 If a trapezoid has one pair of congruent base angles, then it’s an isosceles trapezoid. A B D C A B D C Theorem 6.16 A trapezoid is isosceles iff its diagonals are congruent.

  39. A B b1 E midsegment F D C b2 Just like in a triangle, the midsegment will go through the midpoint of the legs.

  40. A A B B 10 x E F D D C C 18 12 15 E F y

  41. A B z 11 E F D C A B 14 D C 14 2x+2 7y-10 5y+10

  42. 13 Write in x 19 y z A B R D C AD = 15 AR = 6 BC = _____ BR = __ RC = ____ RD = __

  43. 55 y 22 44 x 45 z

  44. C Theorem 6.18 If a quadrilateral is a kite, then its diagonals are perpendicular B D A Theorem 6.18 If a quadrilateral is a kite, then EXACTLY one pair of opposite angles are congruent. C B D A

  45. C B D A C B D A x 12 5 60o 140o zo yo

  46. C B D A C B D A 25 24 x zo 100o 70o yo

  47. Go Over HW • Verify Quadrilateral while HW checked • Properties of shapes of Parallelogram, Rhombus, Rectangle, Square • Discuss what’s on quiz

  48. 6.6 – Special Quadrilaterals

  49. Quad-Tree-Laterals

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