Lesson 2.1 Objectives

1. Lesson 2.1 Objectives • Sketch the basic components of geometrical figures.

2. Give your definition of the following Point Line Plane Not an airplane! These terms are actually said to be undefined, or have no formal definition. However, it is important to have a general agreement on what each word means. Start-Up

3. Point • A point has no dimension. • Meaning it takes up no space. • It is usually represented as a dot. • When labeling we designate a capital letter as a name for that point. • We may call it Point A. A

4. B A Line • A line extends in one dimension. • Meaning it goes straight in either a vertical, horizontal, or slanted fashion. • It extends forever in two directions. • It is represented by a line with an arrow on each end. • When labeling, we use lower-case letters to name the line. • Or the line can be named using two points that are on the line. • So we say Line n, or AB n

5. A C B Plane M • A plane extends in two dimensions. • Meaning it stretches in a vertical direction as well as a horizontal direction at the same time. • It also extends forever. • It is usually represented by a shape like a tabletop or a wall. • When labeling we use a bold face capital letter to name the plane. • Plane M • Or the plane can be named by picking three points in the plane and saying Plane ABC.

6. A B C Collinear • The prefix co- means the same, or to share. • Linearmeans line. • So collinear means that points lie on the sameline. We say that points A, B, and C are collinear.

7. A C B Coplanar • Coplanar points are points that lie on the same plane. M So points A, B, and C are said to be coplanar.

8. B A Line Segment • Consider the line AB. • It can be broken into smaller pieces by merely chopping the arrows off. • This creates a line segment or segment that consists of endpointsA and B. • This is symbolized as AB

9. B A Ray • A ray consists of an initial point where the figure begins and then continues in one direction forever. • It looks like an arrow. • This is symbolized by writing its initial point first and then naming any other point on the ray, . • Or we can say rayAB. AB

10. Opposite Rays • If C is between A and B on a line, then ray CA and ray CB are oppositerays. • Oppositerays are only opposite if they are collinear. C B A

11. A m n Intersections ofLines and Planes • Two or more geometric figures intersect if they have one or more points in common. • If there is no point or points shown, then the figures do not intersect. • The intersection of the figures is the set of points the figures have in common. • Two lines intersect at onepoint. • Two planes intersect at oneline.

12. Lesson 2.2 Segments • Segment Addition • Distance • Midpoint

13. Definition of a Postulate • A postulateis a rule that is accepted without a proof. • They may also be called an axiom. • Basically we do not need to know the reason for the rule when it is a postulate. • Postulates are used together to prove other rules that we call theorems.

14. 4 8 B A Postulate 1: Ruler Postulate • The points on a line can be matched to real numbers called coordinates. • The distance between the points, say A and B, is the absolute value of the difference of the coordinates. • Distancealways positive.

15. Length • Finding the distance between points A and B is written as • AB • Writing AB is also called the length of line segment AB.

16. A B C Betweenness • When three points lie on a line, we can say that one of them is between the other two. • This is only true if all three points are collinear. • We would say that B is between A and C.

17. A B C Postulate 2: Segment Addition Postulate • If B is between A and C, then • AB + BC = AC. • Also, the opposite is true. • If AB + BC = AC, then B is between A and C. BC AB AC

18. Segment Addition Postulate Review • Identify the unknown lengths given that BD=4, AE=17, AD=7, and BC=CD • BC • 2 • AB • 3 • AC • 5 • DE • 10 E C D A B

19. Distance Formula To find the distance on a graph between two points A(1,2) B(7,10) We use the Distance Formula AB = (x2 – x1)2 + (y2 – y1)2 Distance can also be found using the Segment Addition Postulate, which simply adds up each segment of a line to find the total length of the line.

20. Example • Using the Distance Formula, find the length of segment OK with endpoints • O(2,6) • K(5,10) (x2 – x1)2 + (y2 – y1)2 (5 – 2)2 + (10 – 6)2 32 + 42 9 + 16 25 = 5

21. Example • This is one part of the problem for #34 • Find the distance between points A and C. • A(-4,7) • C(3,-2) (x2 – x1)2 + (y2 – y1)2 (3 – -4)2 + (-2 – 7)2 72 + (-9)2 49 + 81 130

22. Congruent Segments • Segments that have the same length are called congruent segments. • This is symbolized by =. Hint: If the symbols are there, the congruent sign should be there. LE = NT LE = NT If you want to state two segments are congruent, then you write If you want to state two lengths are equal, then you write

23. H O J Y T Midpoint Congruence marks are used to show that segments are congruent. If there is more than one pair of congruent segments, then each pair should get a different number of congruence marks. • The midpoint of a segment is the point that divides the segment into two congruent segments. • The midpointbisects the segment, because bisect means to divide into two equal parts. • A segment bisector is a segment, ray, line, or plane that intersects the original segment at its midpoint. Now we can say line HT is a segment bisector of segment JY. We say that O is the midpoint of line segment JY.

24. A(1,2) B(7,10) ( ) , Midpoint Formula We can also find the midpoint of segment AB by using its endpoints in… The Midpoint Formula Midpoint of AB = (y1 + y2) (x1 + x2) 2 2 This gives the coordinates of the midpoint, or point that is halfway between A and B.

25. (x1 + x2) (y1 + y2) A(1,2) B(7,10) 2 2 ( ( ( ( ) ) ) ) 2 2 , , , , 2 2 Example • This is an example of how to determine the midpoint knowing the two endpoints. (1 + 7) (2 + 10) 8 12 6 4

26. (4,9) (8,5) Short Cut to Find Endpoint • Say the midpoint is (8,5) and one endpoint is (4,9). • Remember that the midpoint is half way between the endpoints. Add 4 to x Minus 4 from y Add 4 to x (12,1) Minus 4 from y

27. Lesson 2.3 Angles and Their Measures

28. What is an Angle? • An angle consists of two different rays that have the same initial point. • The rays form the sides of the angle. • The initial point is called the vertex of the angle. • Vertex can often be thought of as a corner.

29. Naming an Angle • All angles are named by using three points • Name a point that lies on one side of the angle. • Name the vertex next. • The vertex is always named in the middle. • Name a point that lies on the oppositeside of the angle. So we can call It WON Or NOW W N O

30. = m WON NOW WON NOW Congruent Angles • Congruent angles are angles that have the same measure. • To show that we are finding the measure of an angle • Place a “m” before the name of the angle. m = Congruent Angles Equal Measures

31. Types of Angles

32. Other Parts of an Angle • The interior of an angle is defined as the set of points that lie between the sides of the angle. • The exteriorof an angle is the set of points that lie outside of the sides of the angle. Exterior Interior

33. Postulate 4: Angle Addition Postulate • The Angle Addition Postulate allows us to add each smaller angle together to find the measure of a larger angle. What is the total? 49o 32o 17o

34. C T R A Adjacent Angles • Two angles are adjacent angles if they share a common vertex and side, but have no common interior points. • Basically they should be touching, but not overlapping. CAT and TAR areadjacent. CAR and TAR are notadjacent.

35. Angle Bisector • An angle bisector is a ray that divides an angle into two adjacent angles that are congruent. • To show that angles are congruent, we use congruence arcs.

36. Lesson 2.4 Angle Pair Relationships

37. 1 2 4 3 Vertical Angles • Two angles are vertical angles if their sides form two pairs of opposite rays. • Basically the two lines that form the angles are straight. • To identify the vertical angles, simply look straight across the intersection to find the angle pair. • Hint: The angle pairs do not have to be vertical in position. • Vertical Angle pairs are always congruent! 1 2 4 3

38. Linear Pair • Two adjacent angles form a linear pair if their non-common sides are opposite rays. • Simply put, these are two angles that share a straight line. • Since they share a straight line, their sum is… • 180o 1 2

39. Complementary anglesare two angles whose sum is 90o. Complementary angles can be adjacent or non-adjacent. Supplementary anglesare two angles whose sum is 180o. Supplementary angles can be adjacent or non-adjacent. Complementary v Supplementary

40. Important note… • A linear pair is formed by two angles that are supplementary and adjacent. These two angles are supplementary but they are not adjacent. Therefore, they are not a linear pair

41. Example 70 Ð 1 = 1 110 Ð 2 = 2 110 ° 70 Ð 3 = 3

42. Example 93 Ð 1 = 2 87 Ð 2 = 3 93 87 ° Ð 3 = 1

43. Example P O 53 Ð NQO = 127 Ð NQT = 37 ° N R Q S T

44. Lesson 2.5 • Lines cut by a transversal

45. Lines and Angle Pairs Alternate Exterior Angles – because they lie outside the two lines and on opposite sides of the transversal. Transversal 2 1 3 4 Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal. 5 6 Corresponding Angles – because they lie in corresponding positions of each intersection. 8 7 Alternate Interior Angles – because they lie inside the two lines andon opposite sides of the transversal.

46. Example 1 Determine the relationship between the given angles • 3 and 9 • Alternate Interior Angles • 13 and 5 • Corresponding Angles • 4 and 10 • Alternate Interior Angles • 5 and 15 • Alternate Exterior Angles • 7 and 14 • Consecutive Interior Angles

47. Parallel versus Skew • Two lines are parallel if they are coplanar and do not intersect. • Lines that are not coplanar and do not intersect are called skew lines. • These are lines that look like they intersect but do not lie on the same piece of paper. • Skew lines go in different directions while parallel lines go in the same direction.

48. Example 2 Complete the following statements using the words parallel, skew, perpendicular. • Line WZ and line XY are _________. • parallel • Line WZ and line QW are ________. • perpendicular • Line SY and line WX are _________. • skew • Plane WQR and plane SYT are _________. • parallel • Plane RQT and plane WQR are _________. • perpendicular • Line TS and line ZY are __________. • skew • Line WX and plane SYZ are __________. • parallel.

49. Parallel and Perpendicular Postulates:Postulate 13-Parallel Postulate • If there is a line and a point not on the line, then there is exactly one line through the point that is parallel to the given line.

50. Parallel and Perpendicular Postulates:Postulate 14-Perpendicular Postulate • If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.