2. Plane vs. Solid Geometry Plane GeometrySolid Geometry

3. Plane Geometry vs Solid Geometry Plane GeometrySolid Geometry Two-Dimensional - 2DThree-Dimensional - 3D

4. Plane Geometry Notes

5. LINE A STRAIGHT PATH OF POINTS WITH NO ENDPOINTS. A B AB

6. SEGMENT PART OF A LINE WITH 2 ENDPOINTS A B AB

7. Lines can be said to have three relationships with each other. Lines Intersecting Skew Parallel Perpendicular

8. Intersecting Lines – Lines that meet at a point. –Perpendicular Lines lines that intersect to form 90 degree angles. m n M N

9. Parallel Lines – Lines in the same plane that do not intersect. m n M N

10. Parallel Lines – Lines in the same plane that do not intersect. Skew Lines - Lines in different planes that do not intersect.

11. RAY PART OF A LINE WITH ONLY ONE(1) ENDPOINT A B AB

12. ANGLE TWO RAYS WITH A COMMON ENDPOINT VERTEX K C J <KJC or <CJK

13. TYPES OF ANGLES ACUTE RIGHT OBTUSE Less than 90° Exactly 90° Greater than 90°

14. Classify these ANGLES ACUTE RIGHT OBTUSE

15. End of Section 1

16. Plane vs. Solid Geometry Plane GeometrySolid Geometry

17. Today we will look at some angle relationships! Vertical Angles Adjacent Angles Complementary Angles Supplementary Angles It takes two to tango!

18. Special Angle Pairs: 1) Vertical Angles – the angles formed on opposite sides of intersecting lines. 1 2 4 3 <2 and <4 are vertical angles so we know m<2 = m<4 Vertical angles are always congruent

19. 2) Adjacent Angles – are angles that share a common vertex and a common side. 2 1 1 2 1 2

20. Complementary Angles – two angles whose sum is 90 degrees Supplementary Angles – two angles whose sum is 180 degrees

21. Complementary Angles 60 X 90 - 60 = X 30 = X

22. What is the complement of a 72 degree angle? What is the complement of a 35 degree angle? 18 degrees 55 degrees

23. Supplementary Angles 38 X X = 180-38 X = 142 degrees 360° 180°

24. See IXL.com Q1 and Q2

25. How many pairs of supplementary angles can you name ?

26. Angle Facts Starter: Name these angles: Acute Reflex Right Angle Obtuse

27. 90 Vertical Horizontal 360 o 1 2 3 4 Angles at a Point

28. a b c d Angles at a point add to 360 o Angle a + b + c + d = 360 0 Angles at a Point

29. 75 o 85 o 80 o a Example: Find angle a. 90 360 o Angle a = 360 - (85 + 75 + 80) = 360 - 240 = 120 o 85 75 80 + 240 Angles at a Point

30. a b c d Opposite Angles are equal Angle a + b = 180 0 because they form a straight line Angle c + d = 180 0 because they form a straight line Angle c + b = 180 0 because they form a straight line Angle d + a = 180 0 because they form a straight line So a = c and b = d Angles at a Point

31. Horizontal line Oblique line a b Angles a + b = 180 o 70 o b Angle b = 180 – 70 = 110 o x 35 o Angle x = 180 – 35 = 145 o 90 Angles on a straight line add to 180 o 180 o Angles on a Line

32. Angle Facts Now do these: f g g g 2h h 60 o 45 o 120 o 110 o 35 o a b 22 o 116 o c d e 135 o 80 o 148 o i a = 180 – 35 = 145 o b = 180 – (22+90) = 68 o Opposite angles are equal So c = 116 o d = 180 – 116 = 64 o e = 360 – (135+80) = 145 o f = 360 – (45+120+110) f = 360 - 275 = 85 o 3g = 360 – (90+60) = 210 g = 70 i = 180 - 148 = 32 o 3h = 180 – i = 148 h = 49.3.

33. Parallel lines remain the same distance apart. Transversal Draw a pair of parallel lines with a transversal and measure the 8 angles. Vertically opposite angles are equal. Corresponding angles are equal. Angles between Parallel Lines

34. Parallel lines remain the same distance apart. Transversal Draw a pair of parallel lines with a transversal and measure the 8 angles. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

35. Parallel lines remain the same distance apart. Transversal Draw a pair of parallel lines with a transversal and measure the 8 angles. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

38. Parallel lines remain the same distance apart. Transversal a b c e g h f Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

39. Parallel lines remain the same distance apart. Transversal a b c h g d f Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

40. Parallel lines remain the same distance apart. Transversal a b e h g d f Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

41. Parallel lines remain the same distance apart. Transversal a b e h g d f Name an angle alternate interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

42. Parallel lines remain the same distance apart. Transversal a b e h g d c Name an angle alternate to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

43. Parallel lines remain the same distance apart. Transversal a b e h g d c Name an angle interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Angles between Parallel Lines

44. Parallel lines remain the same distance apart. Transversal a b e h g d c Name an angle interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Angles between Parallel Lines

45. a f e h g d c Name an angle corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Angles between Parallel Lines

46. a f e b g d c Name an angle alternate to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Angles between Parallel Lines

47. h f e b g d c Name an angle interior to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

48. h f e b g d c Name in order, the angles that are alternate, interior and corresponding to the marked angle. Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Angles between Parallel Lines

49. a f e h g d c Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Name in order, the angles that are alternate, interior and corresponding to the marked angle. Angles between Parallel Lines

50. x Vertically opposite angles are equal. Corresponding angles are equal. Alternate Interior angles are equal. Finding unknown angles 100 o Find the unknown angles stating reasons, from the list below. y z 60 o  x =  y =  z = 80 o Int.  s 60 o vert.opp.  s 120 o Int.  s Angles between Parallel Lines

51. x Vertically opposite angles are equal.vert.opp.  s Corresponding angles are equal.corr.  s Alternate angles are equal.alt.  s Interior angles sum to 180 o.(Supplementary) Int.  s Finding unknown angles 105 o Find the unknown angles stating reasons, from the list below. y z  x =  y =  z = 105 o corr.  s 55 o alt.  s 125 o Int.  s 55 o Angles between Parallel Lines

52. x Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Finding unknown angles 95 o Find the unknown angles stating reasons, from the list below. y  x =  y = 85 o Int.  s 120 o Int.  s 60 o Unknown angles in quadrilaterals and other figures can be found using these properties. Angles between Parallel Lines

53. x Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Finding unknown angles Find the unknown angles stating reasons, from the list below. y 55 o z  x =  y =  z = What does this tell you about parallelograms? 125 o Int.  s 125 o 55 o Int.  s 55 o 125 o Int.  s 125 o Unknown angles in quadrilaterals and other figures can be found using these properties. Angles between Parallel Lines

54. 70 o a Vertically opposite angles are equal. Corresponding angles are equal. Alternate angles are equal. Interior angles sum to 180 o.(Supplementary) Find the unknown angles stating reasons, from the list below. There may be more than one reason. 58 o vert.opp.  s 32 o  s in tri 58 o 32 o alt.  s b d c Angle sum of a triangle (180 o ) Angle on a line sum to (180 o ) 58 o  s on line e 58 o corr.  s 52 o  s at a point fg h Base angles isosceles triangle equal. 64 o isos tri  a =  b =  c =  d =  e =  f =  g =  h = 64 o isos tri Angles at a point sum to 360 o Mixing it! Angles between Parallel Lines

55. Angle Facts Now do these: 65 o p 99 o 77 o 38 o 54 o 48 o 35 o z 130 o y x w t u s r q v Corresponding angles p = 65 o Alternate angles q = 38 o Corresponding angles r = 77 o Opposite angles (with r) or Alternate angles with 77 o s = 77 o t = 99 o v = 54 o u = 81 o w = 126 o x = 130 o y = 130 o z = 35 o + 48 o = 83 o

56. POLYGON A CLOSED PLANE FIGURE MADE UP OF THREE OR MORE LINE SEGMENTS

57. NAMES OF DIFFERENT POLYGONS TRIANGLE - 3 SIDES QUADRILATERAL - 4 SIDES PENTAGON - 5 SIDES HEXAGON - 6 SIDES

58. HEPTAGON - 7 SIDES OCTAGON - 8 SIDES NONAGON - 9 SIDES DECAGON - 10 SIDES

59. HENDECAGON - 11 SIDES DODECAGON - 12 SIDES TRISKAIDECAGON - 13 ICOSAGON - 20 SIDES

60. Triangle Polygon with three sides

61. Congruent – having equal measure This could be angles, lines, or polygons.

62. Classifying Triangles by the SIDES SCALENE No Congruent Sides ISOSCELES Two Congruent Sides EQUILATERAL Three Congruent Sides

63. Classifying Triangles by the ANGLES ACUTE All Angles Less Than 90° RIGHT Has one Right Angle OBTUSE Has one Obtuse Angle

64. The SUM of the measures of the angles of a triangles is always180° 90 45 60 60+60+60 = 180° 45+45+90 = 180°

65. Homework was p. 366 13. 80 14. 64 15. 40 16. 38 17. 93 18. 60 19. Acute 20. Right 21. Right 22. 64

66. REGULAR POLYGON POLYGONS THAT HAVE EQUAL SIDES AND ANGLES

67. Classwork/Homework is p. 365-366 (1-21) Begin now

68. Special Quadrilaterals A TRAPEZOID has only one set of parallel lines A PARALLELOGRAM has two sets of parallel lines.

69. Special PARALLELOGRAMS A RECTANGLE has four right angles A RHOMBUS has 4 congruent sides. A SQUARE has four 90° angles and 4 congruent sides. 6 6 6 6 4 4 4 4

70. Parallelogram Facts to Know: Opposite sides are the same length Opposite angles are the same measure.

71. The SUM of the Measures of the Angles of a Quadrilateral is 360° 90 70 110 90 +90 360° 110+70+110+70 = 360°

72. To find the sum of the angles of any polygon you can use the formula 180(n-2), where n is the number of sides of the polygon.

73. CONGRUENT POLYGONS POLYGONS THAT HAVE THE SAME SHAPE AND THE SAME SIZE

74. SIMILAR POLYGONS POLYGONS WITH THE SAME SHAPE BUT NOT THE SAME SIZE

75. SIMILAR POLYGONS Corresponding Angles are Equal 70 110

76. SIMILAR POLYGONS Sides are in proportion 20 X 24 16

77. LINE OF SYMMETRY A LINE THAT DIVIDES A FIGURE INTO TWO CONGRUENT PARTS THAT WILL FLIP TO MATCH UP.

78. COORDINATE PLANE - A grid formed by an X-axis running left to right and label like a number line including both positive and negative numbers; and a Y-axis running up and down and labeled like a number line including positive and negative numbers. The ORIGIN is the point both number lines equal zero. (0,0)

79. Coordinates are written in the form (x,y) where x is the value of the x-axis and y is the value of the y- axis.

80. The grid is separated into 4 areas known as QUADRANTS: Quadrant I has a positive X and a positive Y. Quadrant II has a negative Y and a positive Y Quadrant III has a negative X and a negative Y. Quadrant IV has a positive X and a negative Y

81. Quadrant I - (+,+) Quadrant II - (-,+) Quadrant III - (-,-) Quadrant IV - (+, -) In other words...

82. Finding Areas... Remember: Area of a rectangle is found by A = L x W L = 22 cm W = 10 cm So, A or Area = L x W A = 22 x 10 A = 220 cm 2

83. To find area of a triangle... H = 14 B = 12 cm Think of a triangle as 1/2 of a rectangle. So its area is 1/2 of the B x H. A = 1/2 x B x H A = 1/2 x 12 x 14 A = 6 x 14 A = 84 cm 2

84. Sample: Find the sum of the angles of a octagon. An octagon has 8 sides, so if we use the formula, 180(n-2); 180 (8-2) 180 x 6 1080° total in an Octagon