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Polygons

Salvador Amaya 9-5. Polygons. Polygons. A polygon is a closed figure that has no curved sides . Therefore, it needs to have at least three sides. Not a polygon. Yes, a polygon. Yes, a polygon. Parts of a Polygon. Exterior Angle. Interior angle. Vertex. Diagonal. Side.

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Polygons

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  1. Salvador Amaya 9-5 Polygons

  2. Polygons • A polygon is a closed figure that has no curved sides. Therefore, it needs to have at least three sides. Not a polygon Yes, a polygon Yes, a polygon

  3. Parts of a Polygon Exterior Angle Interior angle Vertex Diagonal Side

  4. Parts of a Polygon (ctd) • Interior angle: angle in the inside of a polygon • Exterior angle: angle in the outside of a polygon • Vertex: point where two lines meet • Diagonal: segment that connects two vertices of a polygon • Side: segment in a polygon

  5. Concave vs. Convex • Concave polygon: polygon that has a vertex that points in • Convex polygon: polygon that has all vertices pointing out.

  6. Concave vs. Convex (ctd) • Concave • Convex

  7. Equilateral vs. Equiangular • Equilateral: All sides measure the same. • Equiangular: All the angles measure the same. • Equilateral and Equiangular = Regular Polygon

  8. Equilateral vs. Equiangular (ctd) • Equilateral • Equiangular • Regular

  9. Interior Angles Theorem for Polygons • The sum of the interior angles of a polygon is (n-2) x 180, being n the number of sides in the polygon.

  10. Interior Angles Theorem for Polygons (ctd) • What is the sum of interior angles for a nonagon? • (n-2) x 180 • (9-2) x 180 • 7 x 180 • 1260

  11. Interior Angles Theorem for Polygons (ctd) • What is the measure of an angle in a regular hexagon? • [(n-2) x 180] / n • [(6-2) x 180] / 6 • (4x 180) / 6 • 720 / 6 • 120

  12. Interior Angles Theorem for Polygons (ctd) • What is the sum of the interior angles for a pentagon? • (n-2) x 180 • (5-2) x 180 • 3 x 180 • 540

  13. Theorems for Parallelograms: Sides • If a quadrilateral is a parallelogram, then opposite sides are congruent.

  14. Examples AC is cong. To BD, AB is cong. to CD, EF is cong. to GM, EG is cong. to FM, HI is cong. to KJ, HK is cong. to IJ. If BACD, KHIJ, and FEGH are parallelograms, then…. a c e d b g h f i k j m

  15. Theorems for Parallelograms , converse : Sides • If opposite sides in a quadrilateral are congruent, then the quadrilateral is a parallelogram.

  16. Examples If AC is cong. To BD, AB is cong. to CD, EF is cong. to GM, EG is cong. to FM, HI is cong. to KJ, HK is cong. to IJ, then…. a c 68 e 13 13 68 d 27 b 22 g h 4 8 f i 22 27 k 4 8 BACD, KHIJ, and FEGH are parallelograms j m

  17. Theorems for Parallelograms: Angles • If a quadrilateral is a parallelogram, then opposite angles are congruent.

  18. Examples <a is cong. to <d, <c is cong. to <b, <e is cong. to <m, <f is cong. to < g, <h is cong. to <j, <k is cong. to <i If BACD, KHIJ, and FEGH are parallelograms, then…. a c e d b g h f i k j m

  19. Theorems for Parallelograms, converse: Angles • If opposite angles are congruent in a quadrilateral, then the quadrilateral is a parallelogram.

  20. Examples If <a is cong. to <d, <c is cong. to <b, <e is cong. to <m, <f is cong. to < g, <h is cong. to <j, <k is cong. to <i, then…. a c 128 52 e 36 52 128 d b g 144 h 97 144 f i 83 83 k 36 97 BACD, KHIJ, and FEGH are parallelograms j m

  21. Theorems for Parallelograms: Angles 2 • If a quadrilateral is a parallelogram, then the consecutive angles add up to 180.

  22. Examples If BACD is a parallelogram, then…. a c <a+<b=180 <b+<d=180 <d+<c=180 <a+<c=180 b d

  23. Theorems for Parallelograms, converse: Angles 2 • If the consecutive angles in a quadrilateral add up to 180, then the quadrilateral is a parallelogram

  24. Examples If <a+<b=180, <a+<c=180, <b+<c=180, <d+<c=180, <e+<g=180, <e+<f=180, <g+<m=180, <m+<f=180, <h+<i=180, <h+<k=180, <k+<j=180, <j+<i=180, then…. a c 135 65 e 24 65 135 d b g 156 h 103 156 f i 77 77 k 24 103 BACD, KHIJ, and FEGH are parallelograms j m

  25. Theorems for Parallelograms: Diagonals • If a quadrilateral is a parallelogram, then the diagonals bisect each other.

  26. Examples If CABD, and GEFH are parallelograms, then…. AL=LD BL=LC EK=KH FK=KG a b 45 f e 48 29 l 32 k 29 45 32 48 g h c d

  27. How to prove that a quadrilateral is a parallelogram • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • Its consecutive angles are supplementary. • Its diagonals bisect each other. • It has a pair of congruent parallel sides. • Both pairs of opposite sides are parallel.

  28. Examples, opposite sides All are parallelograms because both their opposite sides are congruent. 6 2 2 6

  29. Examples, opposite angles 147 33 96 94 94 33 147 96 All are parallelograms because both their opposite angles are congruent.

  30. Examples, consecutive angles All are parallelograms because their consecutive angles add up to 180. 132 48 48 132 91 99 34 146 146 91 99 34

  31. Examples, diagonals 7 5 7 5 All are quadrilaterals because their diagonals bisect each other.

  32. Examples, pair of sides All are parallelograms because they have a pair of congruent parallel sides. 45 45

  33. Examples, parallel sides

  34. Rhombus • Rhombus: parallelogram with 4 congruent sides. Its diagonals are perpendicular to each other.

  35. Rhombus Theorems • If a quadrilateral is a rhombus, then it is a parallelogram.

  36. Examples Since these are rhombuses because they are equilateral sides, then they are also parallelograms. 28 28 28 28

  37. Rhombus Theorems (ctd) • If a parallelogram is a rhombus, then its diagonals are perpendicular to each other.

  38. Examples Since these are rhombuses, then their diagonals are perpendicular to each other

  39. Rhombus Theorems (ctd) • If a parallelogram is a rhombus, then their diagonals bisect the opposite angles.

  40. Examples Since this is a rhombus, then… <BAE is cong. to <CAE<EBA is cong. to <EBD <BDE is cong. to <CDE <DCE is cong. to <ACE a e b c d

  41. Rhombus Theorems (ctd) • If a pair of consecutive sides is congruent, then the quadrilateral is a rhombus.

  42. Examples Since these are parallelograms, because they have congruent consecutive sides, then they are rhombuses. 98 98

  43. Rhombus Theorems (ctd) • If the diagonals in a parallelogram are perpendicular, then it is a rhombus.

  44. Examples Since the diagonals are perpendicular, then they are rhombuses

  45. Rhombus Theorems (ctd) • If a diagonal bisects a pair of opposite angles, then it is a rhombus.

  46. Examples Since…. <ZXB is cong. to <CXB, then CXZV is a rhombus <XZB is cong. to <VZB, then CXZV is a rhombus <ZVB is cong. to <CVB, then CXZV is a rhombus <XCB is cong. to <VCB, then CXZV is a rhombus z b x v c

  47. Rectangle • Rectangle: parallelogram with 4 right angles . Its diagonals are congruent.

  48. Rectangle Theorems • If it is a rectangle, then it is a parallelogram.

  49. Examples Since these are rectangles (4 rt. angles), then they are also parallelograms.

  50. Rectangle Theorems (ctd) • If it is a rectangle, then the diagonals are congruent.

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