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Lesson 6.1 Polygons

Lesson 6.1 Polygons. Today, we will learn to… > identify, name, and describe polygons > use the sum of the interior angles of a quadrilateral. triangle. quadrilateral. pentagon. hexagon. heptagon. octagon. nonagon. decagon. dodecagon. Theorem 6.1 Interior Angles of a Quadrilateral.

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Lesson 6.1 Polygons

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  1. Lesson 6.1 Polygons Today, we will learn to… > identify, name, and describe polygons > use the sum of the interior angles of a quadrilateral

  2. triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon dodecagon

  3. Theorem 6.1Interior Angles of a Quadrilateral The sum of the measures of the interior angles of a quadrilateral is ______ 360°

  4. Section 6.1 Vocabulary Convex Concave Equilateral Equiangular Regular Diagonal

  5. UD TU UY TY DY ST SU SD TD YS T S U Y D S, T, U, D, Y Sides: Vertices: Diagonals:

  6. T S U Y D There are 10 possible names of this pentagon. STUDY SYDUT DYSTU TUDYS DUTSY TSYDU YSTUD UDYST YDUTS UTSYD

  7. How many diagonals can be drawn from N? M N O R Q P

  8. Starting with N, give 2 names for the hexagon. M N O R Q P NRQPOM NMOPQR

  9. Is this a polygon? If not, explain. If so, is it convex or concave? Yes, it’s a convex pentagon

  10. Is this a polygon? If not, explain. If so, is it convex or concave? No, polygons must be made of segments

  11. Is this a polygon? If not, explain. If so, is it convex or concave? Yes, it’s a concave dodecagon

  12. Is this a polygon? If not, explain. If so, is it convex or concave? No, polygons must be closed figures

  13. Find x. 90 + 87 + 93 + x = 360 x = 90

  14. Find x. 3x + 3x + 2x + 2x = 360 x = 36

  15. Lesson 6.2 Properties of Parallelograms RULERS AND PROTRACTORS Today, we will learn to… > use properties of parallelograms

  16. parallelogram A quad is a parallelogram if and only if two pairs of opposite sides are parallel

  17. Draw a Parallelogram. Measure each angle. Measure each side in centimeters.

  18. Theorems 6.2-6.5 If a quadrilateral is a parallelogram, then… 1) 6.2 2) 6.3 3) 6.4 4) 6.5

  19. congruent … opposite sides are __________

  20. congruent … opposite angles are __________.

  21. 4 3 1 2 supplementary … consecutive angles are __________.

  22. bisect … diagonals __________ each other.

  23. 75˚ 6 75˚ 105˚ 10 ABCD is a parallelogram. Find the missing angle and side measures. 1. A 10 B 105˚ 6 D C

  24. 8 5 ABCD is a parallelogram. Find AC and DB. 2. A B 8 5 D C DB = 16 AC = 10

  25. 3. In ABCD, m C = 115˚. Find mA and mD. 4. Find x in JKLM. J K (3x+18)˚ (4x-9)˚ M L mA = 115˚ mD = 65˚ x = 27

  26. ABCD is a parallelogram. EC = 5 AD = 8 70° mBCD = mADC = 110°

  27. The figure is a parallelogram. 2x – 6 = 4 2y = 8 x = 5 y = 4

  28. The figure is a parallelogram. 4x + 2x = 180 2y + 3 = y + 9 x = y = 30 6

  29. y y The figure is a parallelogram. 2y – 1 = y + 5 3x + 1 = 10 x = y = 3 6

  30. The figure is a parallelogram. 3x – 9 = 2x + 31 4y + 5 = 2y + 21 x = y = 40 8

  31. Lesson 6.3 Proving that Quadrilaterals are Parallelograms Today, we will learn to… > prove that a quadrilateral is a parallelogram What is a converse?

  32. Theorem 6.6 If both pairs of opposite sides are __________, then it is a parallelogram. congruent

  33. Theorem 6.7 congruent If both pairs of opposite angles are __________, then it is a parallelogram.

  34. B A D C Is ABCD a parallelogram? Explain. 1. 2. A 10 B 6 6 D C 10 yes no

  35. 1 3 m1 + m3 = 180˚ m1 + m2 = 180˚ 2 Theorem 6.8 supplementary If an angle is _______________ to both of its consecutive angles, then it is a parallelogram.

  36. A B E D C Theorem 6.9 bisect each other If the diagonals __________________, then it is a parallelogram. AE = EC and DE = EB

  37. Is ABCD a parallelogram? Explain. 3. 4. B A A B 104˚ 104˚ 86˚ D C D C no yes

  38. Theorem 6.10 congruent If one pair of opposite sides are ___________ and __________, then it is a parallelogram. parallel

  39. 7. 5. 8. Yes No 6. Yes No

  40. 9. List 3 ways to prove that a quadrilateral is a parallelogram 1) prove that both pairs of opposite sides are __________ parallel 2) prove that both pairs of opposite sides are __________ congruent 3) prove that one pair of opposite sides are both ________ and ________ parallel congruent

  41. Prove that this is a parallelogram… 4 slope of AB is slope of BC is slope of CD is slope of AD is -2/5 4 -2/5 4.1 AB = BC = CD = AD = 5.4 4.1 5.4 2 4 6 A ( , ) B ( , ) C ( , ) D ( , ) 0 3 -2 2 -3

  42. Lesson 6.4 Special Parallelograms Today, we will learn to… > use properties of a rectangle, a rhombus, and a square

  43. four right angles. four congruent sides. four congruent sides four right angles A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles.

  44. parallelograms rhombuses rectangles squares

  45. Sometimes, always, or never true? 1. A rectangle is a parallelogram. 2. A parallelogram is a rhombus. 3. A square is a rectangle. 4. A rectangle is a rhombus. 5. A rhombus is a square. always true sometimes true always true sometimes true sometimes true

  46. AB = 4.48 cm A Geometer’s Sketchpad B AD = 4.48 cm BC = 4.48 cm E CD = 4.48 cm D Ð ° m AEB = 90 C What do we know about the diagonals in a rhombus?

  47. Theorem 6.11 The diagonals of a rhombus are _____________. perpendicular

  48. Ð ° m EAB = 40 A Ð ° m EAD = 40 B Ð ° m EDA = 50 Ð ° m EDC = 50 Ð ° m ECD = 40 E Ð ° m ECB = 40 D C Ð ° m EBC = 50 Ð ° m EBA = 50 What do we know about the diagonals in a rhombus?

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