CHAPTER 1: POINTS, LINES, PLANES.AND ANGLES Objective: Use symbols for lines, segments rays and distances Find distances State and Use the Ruler Postulate and the Segment Addition Postulate 1.3 Segments, Rays and Distance

SYMBOLS: LINE In the diagram, point B is between points A and C. Point B must lie on LINE AC or AC. A B C

SYMBOLS: RAY o Ray -- Line that has a definite starting point, called an endpoint, that goes on forever in one direction C A A C P Ray AC (AC) consists of points A and C and all points that are between A and C, as well as points beyond C (Such as Point P). Points A is the endpoint of AC, and that point is always named first

o Two rays that have the same starting point (ending point) and goes in opposite directions S T U TS and TU are called opposite rays if Tis between S and U OPPOSITE RAYS

SYMBOLS: SEGMENT o Segment -- part of a line with a definite beginning and end o Can be measured o Has length A C Segment AC ( AC or CA) consists of points A and C and all points that are between A and C. Points A and C are called the endpoints of AC

WRITE SYMBOLS: CHECKPOINT.. Ray AB Ray BA Segment AB Line AB

SEGMENT LENGTH On a Number line every point is paired with a number and every number is paired with a point. In the diagram, point J is paired with –3, the coordinate of J. To find the Measure of a segment: Measure of MJ is written MJ, which is the distance between point M and point J. To find length between two points: Count units between coordinates OR Take the absolute value of the difference (subtract) of the coordinates of its endpoints: MJ = 4 – (–3) = ___ 7

WARMUP - PRACTICE MJ JM KL LK

POSTULATES Using number lines involves the following basic assumptions. Statements such as these that are accepted without proof are called POSTULATES. THEOREMS: Statements proven using postulates, definitions, and theorems that have already been proven to be true. Ruler Postulate: The distance between two points is the absolute value of the difference of their coordinates. When you use absolute value, the order in which you subtract coordinates doesn’t matter. The length must be a positive number, you can take the absolute value or you subtract the lesser coordinate from the greater one. JL = | –3 – 2 | = | –5 | = 5 or JL = | 2 – (–3) | = | 5 | = 5

YOU TRY Remember..the length must be a positive number, you can take the absolute value or you subtract the lesser coordinate from the greater one. Find LM

POSTULATES Segment Addition Postulate: Two little segments added together equal one whole segment If B is between A and C, then ­­­­AB + BC = AC. Example 1: AC = ?? = AC 12 = AC

YOU TRY Remember..If B is between A and C, then ­­­­AB + BC = AC. Find AC, if AB = 5 and BC = 8

POSTULATES Segment Addition Postulate: Example 2 If B is between A and C, then ­­­­AB + BC = AC. AB = x BC = x + 6 AC = 24 Find value of x X X + 6 X + (X + 6) = 24 2x + 6 = 24 2x + 6 – 6 = 24 – 6 2x = 18 X = 9

POSTULATES Segment Addition Postulate: Example 3 If B is between A and C, then ­­­­AB + BC = AC. AB = 3x BC = 4x + 2 AC = 30 Find BC 3x 4X + 2 3X + (4X + 2) = 30 7x + 2 = 30 7x + 2 – 2 = 30 – 2 7x = 28 X = 4 Substitute 4(4) + 2 = BC = 18

CONGRUENT Same Size Same shape Congruent segments are segments that have equal lengths. To indicate that they have equal lengths, you write DE = FG. To indicate that they are congruent, you write DE  FG. (symbol read “ is congruent to ”). The definition tells us that the two statements are equivalent. We will use them interchangeably.

MIDPOINT of a segment Point in the middle of a line segment that divides the segment into two equal or congruent parts P is the midpoint of AB, therefore AP = PB and AP  PB A midpoint is EQUIDISTANT from the endpoints. There is only one midpoint for each segment

MIDPOINT of a segment Definition of a Midpoint AB = BC 4x = BC Use Segment Addition Postulate AB + BC = AC 4x + 4x = 32 8X = 32 x = 4 4x 32 4x

MIDPOINT of a segment Definition of a Midpoint AB = BC 3x + 2 = 4x + 1 3x -3x +2 = 4x - 3x +1 2 = x = x = x

APPLY TO RULER POSTULATE Point L is the midpoint of KM; KM is |4 – 0| = 4 Point L is equidistant from endpoints K and M Therefore divide KM (the distance of) by 2 4 ÷ 2 = 2 ; [ = 2 or 4 – 2 = 2 ] 2 is the coordinate of L

BISECTOR of a segment a line, segment, ray, or plane that intersects the segment at its midpoint. Line l is a bisector of AB. PQ and plane X also bisect AB.