2. Lesson 1-1 Point, Line, Plane Modified by Lisa Palen

3. 1-1 Part A - Geometry : The Objects Point, Line, Plane, Segment, Ray, Angle

4. Undefined Terms Point Line Plane We describe these, rather than defining them.

5. Point A place in space. Has no actual size. How to Sketch: Use dots How to label: Use capital printed letters Never name two points with the same letter (in the same sketch). A B A C

6. B Line Straight figure, extends forever, has no thickness or width. How to Sketch: Use arrows at both ends How to label: (1) Use small script letters – line n (2) Use any two points on the line - Never name a line using three points. Never name two points with the same letter (in the same sketch). n A C

7. Plane Flat surface that extends forever in all directions. How to sketch: Use a parallelogram (four sided figure) How to name: 2 ways: (1) Use a capital script letter – Plane M (2) Use any 3 noncollinear points in the plane A B C Horizontal Plane M Vertical PlaneOther ABC ACB BAC BCA CAB CBA

8. More Objects Segment Ray Angle

9. Segment part of a line that includes two points (called the endpoints) and all points between them How to sketch: How to name: Definition: AB (without a symbol) means the length of the segment or the distance between points A and B. A B

10. Ray Definition: ( the symbol RA is read as “ray RA” ) How to sketch: How to name: Part of a line starting at one point (called the endpoint) And extending forever in one direction. C D R A Y What is ?

11. Angle vertex ray Definition:Angle - Figure formed by two rays with a common endpoint, called the vertex. The two rays are called sides of the angle

12. Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points:vertex must be the middle letter This angle can be named as Using 1 point:using only vertex letter * Use this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called. A B C

13. Naming an Angle - continued Using a number:A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as. * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present. 2 A BC

14. Example Therefore, there is NO in this diagram. There are. K is the vertex of more than one angle.

15. Lesson 1-1 Part B Vocabulary Collinear, Coplanar, Intersection, Intersect, Parallel, Perpendicular Modified by Lisa Palen

16. Collinear Points Collinear points are points that lie on the same line. (The line does not have to be visible.) Points A, B and C are noncollinear Points A, B and C are collinear. Definition:

17. Coplanar Coplanar objects (points or lines) are objects that lie on the same plane. (The plane does not have to be visible.) Definition: Q R S P P, Q, R and S are coplanar. A D C B A, B, C and D are noncoplanar.

18. Coplanar Coplanar objects (points or lines) are objects that lie on the same plane. (The plane does not have to be visible.) Definition: Are they coplanar? ABC ? yes ABCF ? NO HGFE ? yes DCEF ? yes AGF ? yes CBFH ? NO

19. Line g and line r intersect at point I. Intersection / Intersect Definition The intersection of two objects is the set of points in common to both objects. (where the objects touch.) Definition Two objects intersect if they have points in common. (if the objects touch.) r I g The intersection of line g and line r is point I.

20. Intersection of Two Lines If two lines intersect, what is their intersection? Otherwise, they are either parallel or skew. Intersection is a point. parallel skew

21. Two coplanar lines that don’t intersect Symbol: ║ means “is parallel to” Parallel Lines Parallel lines go in the same direction. w v v║wv║w

22. Perpendicular Lines lines that intersect at right angles Illustration: Symbol:  m eans “is perpendicular to” Key Fact: 4 right angles are formed. m n mnmn

23. Lesson 1-2 Segments and Rays Modified by Lisa Palen

24. Recall: What is a Segment? two points (called the endpoints) and all points between them How to sketch: How to name: Definition: A B

25. Measure (of a Segment) The length of the segment or the distance between the two endpoints Notation: Definition: A B Recall: The symbol is read as “segment A B”. AB (without a symbol) means the length of the segment or the distance between points A and B. The measure of is AB.

26. Congruent Segments Definition: Congruent segments are segments with equal measures (lengths). Mark congruent segments with.. dashes.. Congruent segments have the same number of dashes. EF H G Notation: The symbol  means “is congruent to”.

27. Congruent Segments Using the Notation: Numbers are equal. Objects are congruent. AB: the distance from A to B ( a number ) AB: the segment AB ( an object ) Correct notation: Incorrect notation:

28. Midpoint A midpoint is a point that divides a segment into two congruent segments. Definition: and DE = EF E is the midpoint of.

29. Segment Bisector A segment bisector is ANY object that divides a segment into two congruent segments. Definition:

30. Postulates Definition: a statement we accept as true without proof. Examples: Through any two points there is exactly one line. Through any three non-collinear points, there is exactly one plane.

31. Postulates If two lines intersect, then the intersection is a point. Examples: If two planes intersect, then the intersection is a line.

32. The Ruler Postulate The points on any line can be paired with the real numbers in such a way that: Any two chosen points can be paired with 0 and 1. The distance between any two points in a number line is the absolute value of the difference of the real numbers corresponding to the points.. The Ruler Postulate says you can use a ruler to measure the distance between any two points! (It also gives us a formula.)

33. The Ruler Postulate So, we can measure the distance between two points using a “ruler”. PK = (distance is always positive) | 3 - -2 | = 5 Formula: take the absolute value of the difference of the two coordinates a and b: │a – b │

34. Reminder The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points. Coordinates: -3, -2, -1, 0, 1, 2, 3, etc. Points: G, H, I, J, etc.

35. Another Example Find the distance between I and S. Coordinate of I: Coordinate of S: 6 -4 │ -4 - 6 │= │ - 10 │ = 10 Take the absolute value of the difference: │a – b │

36. Finding the Midpoint (of Two Points on a Number Line) The coordinate of a midpoint of a segment whose endpoints have coordinates a and b is

37. Example Find the coordinate of the midpoint of the segment PK. Now find the midpoint on the number line.

38. So what do we mean by between? Which picture shows, “C is between A and B”? So “C is between A and B” means that C is ON the segment. Okay, but this is not the definition. 

39. Between Definition: If C is between A and B, then AC + CB = AB. If AC + CB = AB, then C is between A and B. AC + CB = ABAC + CB > AB This is also called the Segment Addition Postulate. or The Segment Addition Postulate betweennot between

40. The Segment Addition Postulate (This is the same as “between.” ) In Other Words: The whole is the sum of the parts. Or: Part + Part = Whole These are the same length.

41. The Segment Addition Postulate Example:If C is between A and B, AC = 4 and CB = 8, then find AB. AC + CB = AB 4 + 8 = AB 8 4 AB Step 1: Draw. Step 2: Label. Step 3: Find equation. (Substitute) Step 4: Solve. Step 5: Make sure you answer the question. 12 = AB Part + Part = Whole

42. The Segment Addition Postulate Example:If E is between D and F, DE = 5 and DF = 15, then find EF. DE + EF = DF 5 + EF = 15 EF 5 15 Step 1: Draw. Step 2: Label. Step 3: Find equation. (Substitute) Step 4: Solve. Step 5: Make sure you answer the question. EF = 10 Part + Part = Whole

43. The Segment Addition Postulate Example:If C is between A and B, AC = x, CB = 2x and AB = 12, then find x, AC and CB. 2x x 12 x = 4 AC = 4 CB = 2*4 = 8 Step 1: Draw. Step 2: Label. Step 3: Find equation. (Substitute) Step 4: Solve. Step 5: Answer question. AC + CB = AB x + 2x = 12 3x = 12 x = 4 Part + Part = Whole

44. Midpoint If E is the midpoint of, and DE = 5, then find EF and DF. Example: Step 1: Draw. Step 2: Label. Step 3: Find equation. (Substitute) Step 4: Solve. Step 5: Answer question. 5 DE = EF 10 = DF Part = Part 5 = EF 5 + 5 = DF DE + EF = DF Part + Part = Whole 5

45. Midpoint If E is the midpoint of, and DE = x + 1 and DF = 3x - 6, then find EF and DF. Example: Step 1: Draw. Step 2: Label. Step 3: Find equation. (Substitute) Step 4: Solve. Step 5: Answer question. x+1 DE = EF EF = x + 1 Part = Part x + 1 = EF x + 1 + x + 1 = 3x - 6 DE + EF = DF Part + Part = Whole x+1 3x - 6 2x + 2 = 3x - 6 8 = x EF = 8 + 1 EF = 9 DF = 3x - 6 DF = 3*8 - 6 DF = 18

46. Lesson 1-4 Angles

47. Angle vertex ray Definition:Angle - Figure formed by two rays with a common endpoint, called the vertex. The two rays are called sides of the angle

48. Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points:vertex must be the middle letter This angle can be named as Using 1 point:using only vertex letter * Use this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called. A B C

49. Naming an Angle - continued Using a number:A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as. * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present. 2 A BC

50. Example Therefore, there is NO in this diagram. There are. K is the vertex of more than one angle.

51. Angles and Points Angles can have points in the interior, in the exterior, or on the angle. n Points A, B and C are on the angle, D is in the interior and E is in the exterior. n B is the vertex.

52. Interior / Exterior of an Angle Definition (you don’t need to memorize this.) A point is in the interior of an angle if it does not lie on the angle itself and it lies on a segment whose endpoints are on the sides of the angle. A, B, and C are on the angle. An exterior point is a point that is neither on the angle nor in the interior of the angle. Interior Point Exterior Point

53. The Protractor Postulate Given a ray AB and a number r between 0 and 180, there is exactly one ray with endpoint A extending to either side of AB, such that the measure of the angle formed is r degrees. You don’t need to memorize this! The Protractor Postulate says you can use a protractor to measure angles!

54. The Ruler and Protractor Postulates The Ruler Postulate lets us use a ruler to measure the distance between two points. The Protractor Postulate lets us use a protractor to measure an angle.. Protractor Applet

55. Measuring Angles lJust as we can measure segments, we can also measure angles. lWe use units called degrees to measure angles. – A circle measures _____ – A half-circle measures _____ – A quarter-circle measures _____ – One degree is the angle measure of 1/360th of a circle. ? ? ? 360º 180º 90º

56. Measure (of an Angle) The size of the angle Notation: Definition: The measure of  ABC is m  ABC Angles are measured using units called degrees (in this class.) A C B

57. 4 Types of Angles Lesson 1-4: Angles56 Acute Angle: an angle whose measure is less than 90 . Right Angle: an angle whose measure is exactly 90 . Obtuse Angle: an angle whose measure is greater than 90  and less than 180 . Straight Angle: an angle that measures exactly 180 . A B C D

58. Congruent Angles Lesson 1-4: Angles57  2   4. 2 4 Definition: Congruent angles - angles that have equal measures Congruent angles are marked with the same number of “arcs”. The symbol for congruence is  Example:

59. is an angle bisector. Since  3   5, bisects  ABC. Angle Bisector / Bisect An angle bisector is a ray that splits the angle into two congruent angles. The ray bisects the angle. Lesson 1-4: Angles58 5 3 Example 1: A B C D Example 2:

60. Example 1 Angle Bisector If is an angle bisector of  PMY and m  PML = 68 , then find: m  PMY = _______ m  LMY = _______

61. Example 2 Angle Bisector If is an angle bisector of  PMY and m  PMY = 86 , then find: m  PML = _______ m  LMY = _______

62. Adding Angles lWhen you want to add angle measures, use the notation m  1, meaning the measure of  1. lIf you add m  1 + m  2, what is your result? m  ADC = 36  + 22  m  ADC = 58  How did you know to add???

63. Angle Addition Postulate That last example is an example of The Angle Addition Postulate: If D is in the interior of  ABC, then m< ____ + m< ____ = m< _____ ABD DBC ABC If m  ABD + m  DBC = m  ABC, then D is in the interior of  ABC.

64. Angle Addition Postulate A simpler way to remember this postulate: _______ + _______ = _________ part whole part whole

65. Example Angle Addition 3x + x + 6 = 90 4x + 6 = 90 – 6 = –6 4x = 84 x = 21 K is interior to  MRW, m  MRK = (3x) , m  KRW = (x + 6)  and m  MRW = 90º. Find m  MRK. 3x x+6 Are we done? m  MRK = 3x = 3*21 = 63º First, draw it!

66. Lesson 1-5: Pairs of Angles65 Lesson 1-5 Pairs of Angles

67. Lesson 1-5: Pairs of Angles66 Adjacent Angles A pair of coplanar angles with a common (shared) vertex and common side that do not have overlapping interiors. vertexside  1 and  2 are adjacent.  3 and  4 are not.  1 and  ADC are not adjacent. Adjacent Angles( a common side )Non-Adjacent Angles 4 3 Definition: Examples:

68. Lesson 1-5: Pairs of Angles67 Complementary Angles A pair of angles whose sum of measures is 90˚Definition: Examples:  1 and  2 are adjacent complementary angles. ( have a common side )  1 and  2 are complementary but not adjacent angles. ( don’t have a common side )

69. Lesson 1-5: Pairs of Angles68 Supplementary Angles A pair of angles whose sum of measures is 180˚Definition: Examples:  1 and  2 are adjacent supplementary angles.  1 and  2 are supplementary but not adjacent angles.

70. Opposite Rays opposite rays not opposite rays

71. Opposite Rays Definition: Two rays with the same endpoint, that together form a line. Or (better): Two rays with the same endpoint that together form a straight angle. AX and AY are opposite rays.  XAY is a straight angle

72. Lesson 1-5: Pairs of Angles71 Linear Pair A linear pair is a pair of adjacent angles whose non-adjacent rays form opposite rays. Definition:  1 and  2 are a linear pair. A linear pair is a pair of adjacent supplementary angles. Another “Definition”:

73. Lesson 1-5: Pairs of Angles72 Vertical Angles A pair of non-adjacent angles formed by intersecting lines. Definition: Examples: Another Definition: A pair of angles whose sides form opposite rays. The pairs of opposite rays are and

74. Postulates vs. Theorems Definition: A postulate is a statement we accept as true without proof. Examples: Segment Addition Postulate and Angle Addition Postulate Definition: A theorem is a statement we use logic to show is true. Examples: Linear Pair Theorem and Vertical Angles Theorem (next slides)

75. Lesson 1-5: Pairs of Angles Theorem (Linear Pairs) A linear pair is supplementary.  1 and  2 are supplementary. Given: Prove:  1 and  2 are a linear pair. Statements Reasons 1.  1 &  2 are linear pair. 2. and are opposite rays. 3.  AQC is a straight angle. 4. m  AQC = 180 5. m  1 + m  2 = m  AQC 6. m  1 + m  2 = 180 7.  1 and  2 are supplementary. 1. Given 2. Defn. linear pair 3. Defn. opposite rays 4. Defn. straight angle 5. Angle Addition Postulate 6. Substitution Property 7. Defn. supplementary

76. Lesson 1-5: Pairs of Angles75 Vertical Angles Theorem Vertical angles are congruent.Theorem

77. Theorem: Vertical angles are congruent. 1.  1 &  2 and  2 &  3 are linear pairs 2.  1 &  2 and  2 &  3 are suppl. 3. m  1 + m  2 = 180, m  2 + m  3 = 180 4. m  1 + m  2 = m  2 + m  3 5. m  1 = m  3 6.  1   3 1. Defn. linear pair/diagram 2. Linear pairs are supplementary. 3. Defn. supplementary 4. Substitution Property 5. Subtraction Property 6. Defn. Congruent Angles The diagramGiven: Prove: StatementsReasons

78. Lesson 1-5: Pairs of Angles77 What’s “Important” in Geometry? 360˚ 180˚ 90˚ 4 things to always look for !... and Congruence Most of the rules (theorems) and vocabulary of Geometry are based on these 4 things.

79. Lesson 1-5: Pairs of Angles78 Algebra and Geometry ( ) = ( ) ( ) + ( ) = ( ) ( ) + ( ) = 90˚ ( ) + ( ) = 180˚ Common Algebraic Equations used in Geometry: If the problem you’re working on has a variable (x), then consider using one of these equations.

80. Lesson 1-5: Pairs of Angles79 Example: If m  4 = 67º, find the measures of all other angles. 67º Step 1: Mark the figure with given info. Step 2: Write an equation.

81. Lesson 1-5: Pairs of Angles80 Example: If m  1 = 23 º and m  2 = 32 º, find the measures of all other angles. Answers:

82. Lesson 1-5: Pairs of Angles81 Example: If m  1 = 44º, m  7 = 65º find the measures of all other angles. Answers: