The following diagram represents a segment. Segments are made up of points and are straight.

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1 Notes Page Notes Thursday, ugust 21, :14 PM Points: Points are name by using capital letters. Example: Point or Point E E F The diagram above represents a line. Lines are made up of points and are straight. The arrow at the end of the figure show that the lines extend infinitely far in both directions. Short-hand: E, F, EF E, F, FE The following diagram represents a segment. Segments are made up of points and are straight. M R Short-hand: MR, RM Rays, like lines and segments are made up of points and are straight. ray begins at the endpoint and then extends infinitely far in only one direction. Short-Hand: ngles: Two rays that have the same endpoint form an angle. efinition: n angle is made up of two rays with an common endpoint. This point is called the vertex of the angle. The rays are called Sides of the angle.

2 Notes Page 2 P S 3 Q R Short-hand:, 3,, Short-hand: PQR, RQP, never Q PQS, SQP SQR, RQS Examples of union( and intersection(. Set {1, 2, 3, 4, 5} Set {2, 4, 6, 8} More Examples E How many lines are shown. Name these lines How many different ways can you name these lines. E

3 Notes Page E

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5 Notes Page Notes lassifying angles by size: 0 cute angle <90 Right angle = < obtuse angle < 180 Straight angle = 180 Parts of a degree 60' = 1 (60 minutes is equal to 1 degree) 60" = 1' (60 seconds is equal to 1 minute) efinition: ( )ongruent angles are angles that have the same measure. efinition: ( )ongruent segments are segments that have the same length. Given : E EF E E F Examples:

6 Notes Page 6 1. Given: is acute = 2x What are the restrictions on What are the restrictions on x. 1. Given: is a right angle. = 3x + 4 = x + 6 Find: m = 2.. hange each of the following to degrees, minutes, and seconds. 61 1/ / onvert to degrees '

7 Notes Page ' 10" 5.. Find the angle formed by the hands of a clock at each time. 4:00. 4:30. 4:32

8 Notes Page Notes Friday, September 04, :32 PM efinition: Points that lie on the same line are called collinear. Points that do not lie on the same line are called noncollinear. E F 3. In order for us to say that a point is between two other points, all three points must be collinear. Triangle Inequality: For any three points, there are only two possibilities. 1. They are collinear. Two of the distances add up to the third. 2 7 Q R S 9 2. They are noncollinear. The three points determine a triangle. The sum of the lengths of any two sides of a triangle is always greater than the third. x + y > z x y

9 Notes Page 9 x y z ssumptions from diagrams You should assume Straight lines and angles Points are collinear etweenness of points Relative position of points You should not assume Right angles ongruent Segments ongruent angles Relative sizes of segments and angles Example: Given: iagram as shown. Question: What should we assume? E

10 Notes Page notes Friday, September 05, :21 PM theorem is a mathematical statement that can be proven. postulate is a mathematical statement that is assumed Theorem 1: If two angles are right angles, then they are congruent. Theorem 2: If two angles are straight angles, then they are congruent. Example 1: Prove theorem 1. Given: and are right 's Prove: Statements Reasons 1. and are right 's 1. Given m y def. of right m y definition of 's Example 2: Given: RST 50 TSV 40 X is a right angle Prove: RSV X Statements 1. RST 50 TSV 40 X is a right angle Reasons 1. Given 2. RSV = ddition property 3. X = y def. of right 4. RSV X 4. y def. of 's

11 Notes Page Notes Tuesday, September 09, :45 M efinition: point that divides a segment into two congruent segments bisects the segment. The bisection point is called the midpoint of the segment. efinition: Two points that divide a segment into three congruent segments, trisects the segment. The two points at which the segment is divided are called the trisection points. efinition: ray that divides an angle into 2 congruent angles bisect the angle. The dividing ray is called the bisector of the angle. efinition: Two rays that divide an angle into three congruent angles, trisect the angle. The two dividing rays are called trisectors of the angle. Example 1: Segment EH is divided by f and G in the ratio 5:3:2 from left to right. If EH = 30, Find FG. Example 2: Given: and E trisect E =120 30'24" Find : m

12 Notes Page Notes Wednesday, September 10, :44 M Given: iagram Shown Pr ove : E E ccording to the diagram, angle is a straight angle. Therefore, 2x+x=180 implies that x = 60. ngle and angle E both equal 60 degrees, therefore the angles are congruent.

13 Notes Page Notes Wednesday, September 10, :01 PM Note: One very important characteristics of definitions is that they are reversible and are written in the form "If P then Q". Example 1: If a point is the midpoint of a segment then the point divides the segment into two congruent segments. Reverse or onverse: If a point divides a segment into two congruent segments, then the point is the midpoint of the segment. Note: Theorems and postulates are not always reversible. Example 2: Theorem: If two angles are right angles, then they are congruent. onverse: If two angles are congruent, then they are right angles.

14 Notes Page Notes Wednesday, September 10, :17 PM Statements of Logic onditional Statement: If p then q onverse: If q then p Inverse : if p then q ontrapositive : if q then p Theorem 3: If a conditional Statement is true, then the contrapositive of the statement is also true. Note: statement and its contrapositive are logically equivalent. converse statement and the inverse are logically equivalent. hain Rule: P Q and Q R, then P R Example 1 raw a conclusion from the following statements P e t w t e

15 1.9 Notes Sunday, September 14, :21 PM favorable Pr obability total favorable length Pr obability total length Pr obability favorable angle measure total angle measure Example 1: There are 5 red marble, 4 blue marbles and 6 green marbles in a bag... Find the probability of randomly choosing a green marble. Find the probability of randomly choosing a red or a blue marble. Example 2: point q is randomly chosen on segment What is the probability that it is within 3 units of.. What is the probability that it is within 5 units of. Example 3: We are given the angles below. m 25 m 40 m 60 m 100 m E 110 Notes Page 15

16 Notes Page 16 m 25 m 40 m 60 m 100 m E If two out of the five angles are chosen at random, what is the probability that both angles are acute? If two out of the five angles are chosen at random, what is the probability that one angle is acute and the other angle is obtuse?