This document discusses various concepts in geometry including conditional statements, counterexamples, definitions, perpendicular lines, bi-conditional statements, deductive reasoning, the laws of logic, algebraic proofs using properties of equality, segment and angle properties, writing two-column proofs, the linear pair postulate, congruent and supplementary angle theorems, and the vertical angles theorem. Examples are provided to illustrate each concept.

1. Geo Journal 2 By: Esteban Lara

2. 1. What’s a conditional statement? A conditional statement, or if then statement, is a statement that says that if something happens, then something else will happen. It is compound by a hypothesis (P) and a conclusion (Q). It should be written like P->Q but in some cases it’s written Q->P, this would be the converse. Examples: If I do my homework, then I go to the movies. If I don’t get a good grade in math, then I don’t go to the party. If I play xbox, then I have a good time.

3. 2. What is a counterexample? A counterexample is when you give one example to prove that a conditional statement is false. You only need one. Examples: a. If I don’t go to the movies, then I didn’t do my homework. b. If I don’t go to the party, then I didn’t got good grades. c. If I’m having a good time, then I’m playing xbox. Counterexamples: Maybe I did my homework but didn’t had time for movies. Maybe I didn’t went to the party because I had an accident. I could have a good time playing playing outside.

4. 4. What is a definition? A definition is a description of what something means. Examples: A car is a vehicle, which you can drive, with four wheels and a internal combustion motor. A computer is an electronic machine that can calculate and be programmed for different purposes. A cell phone is a small electronic devise which is used to communicate persons through a microphone connected to an antenna, a network and finally to the other person’s speaker.

5. 4. What is a perpendicular line and lines perpendicular to a plane? A perpendicular line is a line that forms right angles with another line. Lines perpendicular to a plane are lines that extend from a plane and forms right angles with it. Examples:

6. 5. What is a bi-conditional statement? A bi-conditional statement is like a conditional statement except that both the converse and normal form are true. It’s written with an “if and only if” in it. They’re important because you can use it in any format you want and are used if you want to say something that is completely true. Examples: I get burned iff I touch something hot. I go to church iff my parents tell me to go. I study math iff Mr. Turner tells us there’s an exam.

7. 6. Deductive Reasoning It’s when you know what will happen next by following a certain pattern, you could use this as logic to complete the puzzle or to know what the conclusion will be. It will lead you to the conclusion from a hypothesis. In deductive reasoning we can use something called symbolic notation, which is a way of writing words (like if and only if, iff; then, ->; equals, =) in symbols to represent what you mean.

8. 6. Examples: Everyone who is five years old knows how to ride a bike; Michael is five, therefore he knows how to ride a bike. Everyone who does homework is a good student; Esteban does his homework, therefore he’s a good student. My dog barks when he’s angry; my dog is barking, therefore he’s angry.

9. 7. The laws of logic There are two laws of logic: The law of Detachment states that if P and Q is a true statement and if P happens (true), then Q must also happen (true). The law of Syllogism states that if P->Q and Q->R are both true statements, then if P is true R is also true.

10. 7. Examples: Detachment: If you bisect an obtuse angle, then you get acute angles; I bisected an obtuse angles so I got acute angles. If you go under a roof, then the sun doesn’t get you; you are under a roof so you got protected from the sun. If you are born in Guatemala, then you’re a Guatemalan; you were born in Guatemala therefore you’re Guatemalan. Syllogism: If I get drunk, then I don’t remember last night. If I don’t remember last night, then I don’t know where I am. I got drunk therefore I don’t know where I am. If it’s the weekend, then I can play xbox; if I play xbox, then I have a good time. It’s the weekend so I’m having a good time. If it’s raining, then I’m sad. If I’m sad, then I cry. It’s raining so I’m crying.

11. 8. How to do an algebraic proof with the properties of equality To do an algebraic proof using the algebraic properties of equality you must first know them. They are the addition property, subtraction property, multiplication property, division property, reflexive property, symmetric property, transitive property and substitution property. Now you must see which one of these applies for the work you did to prove a given question.

12. 8. Examples: Prove that 3x=9 We use division property to divide 3 from 3x and 9. X=3 now that we finished this step. Prove that x+3=6 We use subtraction property to subtract 3 from both sides. X=3 after this property has taken place. Prove that x=5 and 5=x We use symmetric property to infer that the statement is correct (true).

13. 9. Segment and Angle properties The segment and angle properties of equality are used to prove that something is true by giving the definition of what it does and you can compare it to the problem you have to prove. Examples: The symmetric property of congruence tells you that A≅B therefore B≅A. The addition property tells us that if A=B then A+C=B+C. The reflexive property tells us that A=A.

14. 10. Writing a two column proof To write a two column proof you must write two columns, one for the statements and another for the reasons. In the statements column you put your steps and in the reasons column you put the properties that allow you to make that happen.

15. 10. Examples: Given: <a congr. <d Prove: <b is congr. <c Statements Reasons <a≅<dGiven <a≅<b, <c≅<dVertical < theo. <b≅<d, <a≅<cTransitiveprop. OMG Given: AC is a segment Prove: AB+BC=AC Statements Reasons AC Given AB+BC Addition prop.  AC

16. 10. Example: Given: AB≅BC Prove: B is midpoint of AC Statements Reasons AB≅BC Given mAB=mBCDef. Congruence B is midoint of AC Def. Midpoint ELL

17. 11. Segment properties of congr. The segment properties of congruence are used to define the step you took. The segment properties are: Reflexive Property of Congruence and Transitive Property of Congruence. Examples: EL≅EL, reflexive property CB≅AB and AB≅XY, then CB≅XY. Transitive property AC≅WS and WS≅DC, then AC≅DC. Transitive property

18. 12. Angle Property of congr. The angle property of congruence is the Symmetric Property of Congruence. Examples: <DC≅<AC, then <AC≅DC <LO≅<OL, then <OL≅<LO <OP≅<CH, then <OP≅<CH

19. 13. Linear Pair Postulate LPP!!! The linear pair postulate states that two angles that form a line have a measure of 180°. Examples:

20. 14. Congruent and Supplements thm. If two angles are complementary or supplementary to the same angle, then they’re congruent to each other. Examples: <ABC+<ELL=180, <OMG+<ELL=180, then <ABC≅<OMG <OPQ+<RST add up to 180 and so do <RST+<FUN, so <FUN≅<OPQ <123≅<456 because if you add either one to <890, they add up to 180° <FFO+<KOF=90° and <MAL+<KOF=90°, then <FFO≅<MAL <KIA≅<YOU because if you add them up to <MIA they add up to 90° <SDF and <FDS add 90, and <ASD+<FDS too, <SDF≅<ASD

21. 15. Vertical Angles Theorem This theorem says that if we have two lines intersect and form an X, then the angle on top and on the bottom are congruent and the angle on the right is congruent to the one on the left. Examples: