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Polygons 6-1 to 6-5

Polygons 6-1 to 6-5. Find and use the sum of the measures of the interior angles of a polygon. Find and use the sum of the measures of the exterior angles of a polygon. A polygon is an enclosed plane figure that is made up of segments. Diagonal.

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Polygons 6-1 to 6-5

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  1. Polygons6-1 to 6-5 Find and use the sum of the measures of the interior angles of a polygon. Find and use the sum of the measures of the exterior angles of a polygon.

  2. A polygon is an enclosed plane figure that is made up of segments.

  3. Diagonal • The diagonals of a polygon are the segments that connect any two nonconsecutive vertices.

  4. Polygons • 3 sided Triangle • 4 sided Quadrilateral • 5 sided Pentagon • 6 sided Hexagon • 7 sided Heptagon • 8 sided Octagon • 9 sided Nonagon • 10 sided Decagon • 11 sided hendecagon • 12 sided Dodecagon • n sided n-gon

  5. FYI • Names of Polygons • 13 triskaidecagon 14 tetrakaidecagon, tetradecagon 15 pentakaidecagon, pentadecagon 16 hexakaidecagon, hexadecagon 17 heptakaidecagon 18 octakaidecagon 19 enneakaidecagon • 20 icosagon 21 icosikaihenagon, icosihenagon 22 icosikaidigon 23 icosikaitrigon 24 icosikaitetragon 25 icosikaipentagon 26 icosikaihexagon 27 icosikaiheptagon 28 icosikaioctagon 29 icosikaienneagon • 30 triacontagon 31 triacontakaihenagon 32 triacontakaidigon 33 triacontakaitrigon 34 triacontakaitetragon 35 triacontakaipentagon 36 triacontakaihexagon 37 triacontakaiheptagon 38 triacontakaioctagon 39 triacontakaienneagon • 40 tetracontagon 41 tetracontakaihenagon 42 tetracontakaidigon 43 tetracontakaitrigon 44 tetracontakaitetragon 45 tetracontakaipentagon 46 tetracontakaihexagon 47 tetracontakaiheptagon 48 tetracontakaioctagon 49 tetracontakaienneagon • 50 pentacontagon ... 60 hexacontagon ... 70 heptacontagon ... 80 octacontagon ... 90 enneacontagon ...

  6. Polygons Quadrilateral Kite Parallelogram Trapezoid Isosceles trapezoid Rhombus Rectangle Square

  7. Formulas • The sum of the interiors angle of a convex polygon is (n-2)180. • The measure of each interior angle of a regular n-gon is (n-2)180/n • The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex is 360. • The measure of each exterior angle of a regular n-gon is 360/n.

  8. Example 1 a • Find the sum of the measures of the interior angles of a convex heptagon. • (n – 2)180 = ( 7 -2)180 • = 5*180 • = 900

  9. 1 b) Find the measure of each interior angle of quadrilateral ABCD. 3x x

  10. Find the sum of the measures of the interior angles of a convex octagon. • (n – 2)180 • (8-2)180 • 6*180 • 1080

  11. Find the measures of eachinterior angle of pentagonHJKLM. 142 2x 2x 3x+14 3x+14

  12. The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.

  13. The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.

  14. Find the value of x. • 2x-5 +5x +2x +6x-5 +3x+10 = 360 • 18x = 360 • X = 20 2x-5 3x+10 5x 2x 6x-5

  15. Find the measure of each exterior angle of a regular nonagon. • 360 = 360 = 40° • n 9

  16. Parallelogram6-2 • A parallelogram is a four-sided figure with both pairs of opposite sides parallel.

  17. Quadrilaterals • Quadrilaterals are four-sided polygons. • <A + <B + <C + <D = 360° A B D C

  18. Properties of a Parallelogram • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • Consecutive angles are supplementary.

  19. D C A B • 1. AB // DC, AD // BC • 2. AB =DC, AD = BC • 3. <A = <C and <B = <D • 4. AM = MC and MD = MB • 5. <A + <B = 180 and <B + <C = 180 • <C + <D = 180 and <D + <A = 180

  20. WXYZ is a parallelogram, m<ZWX = b, and m<WXY = d. Find the values of a, b, c, and d. 2c W X 15 a 18° 31° Y Z 22

  21. Ch = • GF // • <DCG = • DC = • <DCG is supplementary to __ • ∆HGC = C G H D F

  22. In parallelogram ABCD, AB = 2x +5, m<BAC = 2y, m<B = 120, m<CAD = 21, and CD= 21. Find the values of x and y.

  23. Quadrilateral WXYZ is a parallelogram with m<W = 47. Find the measure of angles X, Y, and Z.

  24. Assignment • Class work on page 407 • problems 1-5, 15-17 • Homework page 408, problems 18-20, 33-36

  25. 6-3 Tests for Parallelogram • A Quadrilateral is a parallelogram if any of the following is true. • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • Diagonals bisect each other. • A pair of opposite sides is both parallel and congruent.

  26. Determine whether the quadrilateral is a parallelogram. Justify your answer. • Yes, one pair of sides • are congruent and parallel 14 in 100 80 14 in

  27. 12 5 85 5 85 12 no yes

  28. Find x and y so that each quadrilateral is a parallelogram. 56 5y-26 7x 4y+4

  29. 4y-9 3x+4 5x-2 2y+5

  30. Polygons Quadrilateral Kite Parallelogram Trapezoid Isosceles trapezoid Rhombus Rectangle Square

  31. Rectangle • A rectangle is a quadrilateral with four right angles.

  32. Properties of a Rectangle • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • Consecutive angles are supplementary • All angles are congruent • The diagonals are congruent

  33. 1. Explain why a rectangle is a special type of parallelogram. • All rectangles are parallelograms, but not all parallelograms are rectangles.

  34. Ex. 2 A rectangular park has two walking paths as shown. If PS = 180 meters and PR = 200 meters, find QT. Q P • 1A If TS = 120m, find PR • If m<PRS =64, find m<SQR S R

  35. Ex. 3 Quadrilateral MNOP is a rectangle. Find the value of x. • MO = 2x – 8; NP = 23 • MO = 4x – 13; PC = x + 7 N M O P

  36. Ex. 4 Use rectangle KLMN and the given information to solve each problem. • M<1 = 70. Find m<2, M<5, M<6 K L 8 1 7 2 C 9 10 6 3 4 5 N M

  37. Ex. 5 Quadrilateral JKLM is a rectangle. If m<KJL = 2x +4 and m<JLK = 7x + 5, find x. J K P L M

  38. 6-4 Rhombus • A rhombus is a quadrilateral with four congruent sides.

  39. Assignments6-4 Rectangles • Class work on page 426, problems 10-19 • Homework – problems 26-31

  40. Properties of a Rhombus • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • Consecutive angles are supplementary • All sides are congruent • The diagonals are perpendicular • The diagonals bisect the opposite angles

  41. Rhombus A B D C

  42. Use rhombus BCDE and the given information to find each missing value. C • If m<1 = 2x + 20 and m<2 = 5x – 4, • find the value of x. • If BD = 15, find BF. • If m<3 = y2 + 26, find y. 1 2 3 B D F E

  43. Square • A square is a quadrilateral with four right angles and four congruent sides.

  44. Properties of a Square • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • Consecutive angles are supplementary • All angles are congruent. • The diagonals are congruent. • All sides are congruent • The diagonals are perpendicular. • The diagonals bisect the opposite angles.

  45. Assignment 6-5 • Page 435 • Class work – problems 7-12 • Homework – 23-33

  46. Polygons Quadrilateral Kite Parallelogram Trapezoid Isosceles trapezoid Rhombus Rectangle Square

  47. 6-6Trapezoids and Kites • Properties of a trapezoid • A trapezoid is a quadrilateral with exactly one pair of parallel sides. • The angles along the legs are supplementary.

  48. base leg leg base

  49. Trapezoid • AB // DC • M<A + m<D = 180 • M<B + m<C = 180 A B D C

  50. Isosceles Trapezoid Properties • The legs are congruent • Both pairs of base angles are congruent • The diagonals are congruent • Angles along the legs are supplementary.

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