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S: Geometry <3

FC: Prove That Triangle, Right Euclid ? | By: Dazia Enriquez & Krichell Blair | Euclid -"Father of Geometry" | Period: 3 MS. SALKE

1: * TABLE OF CONTENTS Geometry BASICS ..............................................................................................2 Angle-Angle Theorem & Real World Example ..........................................3 Parallel Lines & Triangles .................................................................................4 Railroad Tracks ...................................................................................................5 About a Transversal ..........................................................................................6 Triangle Congruence Proofs ...........................................................................7 Tips On How to do a Proof ...............................................................................8 Answers to an Example ...................................................................................9 Lawyers and Proofs .........................................................................................10 Similarity ...............................................................................................................11 Similarity Theorems .........................................................................................12 Similarity in Action ...........................................................................................13 Logic ......................................................................................................................14 Real World Logic & Negations.......................................................................15 Rules of Logarithm ...........................................................................................16 Newton's Law of Cooling ...............................................................................17 Logarithm ...........................................................................................................18 Rules For Exponential Functions ................................................................19 Scale Factor .......................................................................................................20 Negative Exponent Rule .................................................................................21 Real World Examples of Exponential Function .............................. 22-23

2: Geometry BASICS | Intersecting lines are always occuring in Geometry problems. These lines form vertical angles. | Lines are ongoing and sum up to 180 degrees. | Concurrent Lines ... The concept of concurrence is that 3 or more lines share the same points and intersect at the same point. In this case is G

3: Angle-Angle Theorem : * This theorem proves that because two triangles have 2 congruent angles, they are similar. | REAL WORLD => *You can figure out how long a tree's shadow is by applying the rules of Geometry.

4: Parallel Lines & Triangles | Triangles are eminent in Geometry. They are used for proofs, similarities with other triangles, congruence w/ other triangles, and inequalities. | Parallel Lines never intersect and when made into a transversal, alternate interior angles are formed with corresponding and vertical angles. | *When parallel lines divide a triangle, the sides become proportional.

5: REAL WORLD EXAMPLE ; Railroad Tracks are examples of parallel lines.

6: * Transversal | A transversal intersects a set of parallel lines. It creates corresponding angles, alternate interior angles, and alternate exterior angles. |

7: Triangle Congruence Proofs | -Making a formal proof for a triangle can be difficult, but here's a few things to remember ; * Always use ALL the GIVENS. * Make sure you always use relevant statements. -Ways to Prove Triangles Congruent; *SSS *SAS *AAS *ASA *HL (Only for Right Triangles)

8: 1.GIVEN STATEMENTS 1. GIVEN 2. USE GIVENS 2. POSTULATE/THEOREM 3. FIND CONGRUENT 3. REFLEXIVE/POSTULATE/ SIDES/ANGLES THEOREM/DEFINITION 4.COME TO A CONCLUSION 4. POSTULATE/THEOREM/DEFINITION 5. PROVEN 5. POSTULATE/THEOREM | Note: If you get stuck, asses what needs to be proven and concoct a plan to prove it.

9: REASON ; 1. GIVEN | STATEMENT ; 1. BC= CD AC Bisects

10: *Proving triangles congruent in a formal proof builds a student's ability to prove their thoughts by vindication. This a good quality to have in order to be a lawyer. *

11: Similarity | * Triangles are Geometry's Pet Shape ;-) | - REMEMBER ! When a line is drawn across a triangle, if it is parallel to the base, the two triangles are similar. [ <--- EXAMPLE ] | Similar Triangles contain proportionate sides. In order to prove two triangles similar, you can use the Angle-Angle Theorem or state that the line drawn across the triangle is parallel to the base of a triangle. There is also Side-Angle-Side Theorem.

12: Side-Angle-Side Similarity Theorem ; *When a triangle has an angle congruent to another triangle, and the two sides are proportionate in the same order, the triangles are similar. | Angle-Angle Theorem When two triangles contain 2 congruent angles, they are similar.

13: Real World Example of Similar Triangles ; In order to figure out the height that the tennis ball must be hit so that it will past over the net and land 6 meters away from it, you could use a similar triangle diagram and accompanying theorems to solve this dilemma.

14: Logic | Logic statements contains inverses, converses, contrapositives, and negations.There are various types of statements; disjunction, conjunction, conditional, and biconditional | NOTE: Theorems used for proving logic are Law of Modus Tollens, Disjunctive Inference, Law of Contrapositive, Law of Detachment, Chain Rule, and DeMorgan's Law.

15: A negated statement changes everything inside the parentheses. For Example: ~(~p v q) --> p ^ ~q | REAL WORLD : If I get a 95 on my math quiz, then my dad will buy me an Ipod Touch.

16: Logarithmic Functions | Rules of Logarithms *

17: REAL WORLD ^ *Using logarithms, Newton could predict how the body will be affected (after the induction of any given thing) at any given time. | Sir Isaac Newton

18: Logarithm | * Remember , Logs are the INVERSES of exponentials. | LogGraph y = log2(x) | Exponential y = 2^x | -->

19: Exponential Function | * rules for exponential functions | * When multiplying, if your bases are the same, add exponents. Ex: (2^7)(2^3) = 2^10 *When dividing, if the bases are the same, subtract the exponents. Ex: (2^10)/(2^5) = 2^5 *When multiplying unlike bases with exponents, you must; 1. Find common base by factoring Ex: (16^4)(64^3) ---> ((4^4)^4) ((4^3)^3) *When there is an exponent in and outside of the parenthesis, multiply the exponents. Ex: (2^3)^3 = 2^9 | REMEMBER ! -When a base has an exponent of 1, the answer is the same number as the base. Ex: 10,000^1 = 10,000 -When the base has an exponent of 0, the answer is ALWAYS 1 !!! Ex: 10,000^0 = 1

20: Scale Factor * | *Scale Factors are a type of exponential function | Y=2^x | EXPONENTIAL GROWTH

21: * Negative Exponent Rule | -When a base has a negative exponent, you must put the base over one with the now positive exponent. For Example: 2^-3 = 1/2^3 = 1/8

22: * REAL WORLD EXAMPLES OF EXPONENTIAL FUNCTION | Banks use this compound interest formula for the accumulation of money (interest) --> | * Exponential Functions are most commonly used for : 1. Population Growth 2. Exponential Decay 3. Compound Interest

23: * The rate of population growth at any instant is given by the below equation dN/dt = rN where r - rate of natural increase t - time, N - is the number of individuals in the population at a given instant. d - given country being analyzed. | Birth rate (b) - death rate (d) = rate of natural increase (r).

**By:**Krichell L.**Joined:**over 6 years ago**Published Mixbooks:**1

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**Title:**Geometry Scrapbook PROJECT- by Krichell Blair & Dazia Enriquez
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