Issue 22: Geometry...

Sorry a little numbering problems
Ok... a little more Geometry
By Miyu.
A = C
B     D

Extremes: A and D
Means: C and B
Cross Products: The product of the means and the product of the extremes. In other words, if A/B=C/D
then ad=bc
Similar: Two Figures that have the same shape, but not necessarily the same size, are similiar.
Similarity Statement: A statement indicating that two polygons are similar by listing their vertices in order of correspondence. (In the same order)
Theorems:
1.Through a line and a point not on the line, there exist exactly one perpendicular line to the given line.
2.The perpendicular segment from a point to a line is the shortest segment from the point to the line.
3.The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.
4.If two lines are parallel, then all points on one line are equidistant from the other line.
5.If a diameter is perpendicular to a chord, then it bisects the chord and its arcs
6.If a diameter bisects a chord other than another diameter then it is perpendicular to the chord.
7.The perpendicular bisector of a chord contains the center of the circle.
8.In a circle or congruent circles:
Chords equidistant from the center are congruent.
Congruent chords are equidistant from the center of the circle.
9.If two polygons are similar, then the ration of their perimeters is equal to the ratio of their corresponding sides.
Postulate:(AA)(ANGLE ANGLE):If two angles of one triangle are congruent to two angles of another triangle, then the triangle are similar
Theorem: SSS Similarity Theorem: If the length of the sides of a triangle are proportional to the lengths of the sides of another triangles are similar
SAS Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar
Inscribed Angles: An angle whose vertex is on a circle and whose sides contain chords of the circle.
The arc formed by an inscribed angle is the intercepted arc of that angle.
The measure of an inscribed angle is equal to half the measure of its intercepted arc.
If an inscribed angle intercepts a semicircle, then it is a right angle.
If two inscribed angles intercept the same arc, then they are congruent.
If a quadrilateral is inscribed in a circle, then it has supplementary opposite angles.
(49)

Issue 21: Geometry

By Miyu:
WELL... It is the hot humid summer but we still have to keep studying to keep up the good work. More Geometry （ｏ。ｏ；）☆⌒(*＾-°)v
So... ISSUE 21: Geometry. 
Arc Length: Distance along an arc measured in linear units
 <<< Here's a picture I found on google that could help you. (This isn't a photo made by me, just a good photo to use)

Sector of a circle: The region inside a circle bounded by two radii of the circle and their intercepted
arc.
 <<<More good pictures and information.
Incenter of the triangle: When all 3 angels of a triangle are bisected, the point of concurrency.
If a line bisects an angle of a triangle,  then it divides the opposite side proportionally to the other 2 sides of the triangle.
If perpendicular bisectors are drawn for every side of a triangle, the point of concurrency is the circumcenter of the triangle.
The circumcenter lies at the center of the circle that contains the three vertices of the triangle. Any circle that contains all the vertices of a polygon is called a circumscribed circle. Any Polygon with each vertex on a circle is an inscribed polygon
Theorem 39-1: If one side of a triangle is longer than the other side, then the angle opposite the first side is larger than the angle opposite the first side is larger than the angle opposite the second side.

Theorem 39-2: If one angle of a triangle is larger than another angle, then the side opposite the first angle is longer than the side opposite the second angle.

Theorem 39-3: Exterior Angle Inequality Theorem: The measure of an exterior angle is greater than the measure of either remote interior angle.

Theorem 39-4: The sum of the length of any two sides of a triangle must be greater than the length of the third side.

Postulate 19: If two polygons are congruent, then they have the same area.
Postulate 20: The area of a region is equal to the sum of the areas of its nonoverlapping parts.
Theorem 40-1: Hinge Theorem- If two sides of one triangle are congruent to two sides of another triangle and the included angle of the first triangle is greater than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
Theorem 40-2: Converse of the Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle and  the third side of the first triangle is longer than the third side of the second triangle.

**EDIT**
Measure of central angle = Measure of arc. 
(A friend assured me... LOL)

Math Vocab Issue 13

• Math  3/21
• Vocabulary
• CHAPTER 1:
• A collinear set of points is a set of points all of which lie on the same straight line
• A noncollinear set of points is a set of three or more points that do not all lie on the same straight line
• The distance between two points on the real number line is the absolute value of the difference of the coordinates of the two points
• B is between A and C if and only if A, B, and C are distinct collinear points and AB + BC = AC
• A line segment, or a segment, is a set of points consisting of two points on a line, called endpoints, and all of the points on the line between the endpoints
• The length or measure of a line segment is the distance between its endpoints
• Congruent segments are segments that have the same measure
• The midpoint of a line segment is a point of that line segment that divides the segment into two congruent segments
• The bisector of a line segment is any line, or subset of a line, that intersects the segment at its midpoint
• A line segment, RS, is the sum of two line segments, RP and PS, if P is between R and S
• Two points, A and B, are on one side of a point P if A, B, and P are collinear and P is not between A and B
• A half-line is a set of points on one side of a point
• A ray is a part of a line that consists of a point on the line, called an endpoint, and all the points on one side of the endpoint
• Opposite rays are two rays of the same line with a common endpoint an no other point in common
• An angle is a set of points that is the union of two rays having the same endpoint
• A straight angle is an angle that is the union of opposite rays and whose degree measure is 180
• An acute angle is an angle whose degree measure is greater than 0 and less than 90
• A right angle is an angle whose degree measure is 90
• An obtuse angle is an angle whose degree measure is grater than 90 and less than 180
• Congruent angles are angles that have the same measure
• A bisector of an angle is a ray whose endpoint is the vertex of the angle, and that divides that angle into two congruent angles
• Perpendicular lines are two lines that intersect to form right angles
• The distance from a point to a line is the length os the perpendicular from the point to the line
• If point P is a point on the interior of Angle RST and Angle RST is not a straight angle, or if P is any point not on straight angle RST, then Angle RST is the sum of two angles, Angle RSP and Angle PST
• A polygon is a closed figure in a plane that is the union of line segments such that the segments intersect only at their endpoints and no segments sharing a common endpoint are collinear
• A triangle is a polygon that has exactly three sides
• A scalene triangle is a triangle with no congruent sides
• An isosceles triangle is a triangle that has two congruent sides
• An equilateral triangle is a triangle that has three congruent sides
• An acute triangle is a triangle that has three acute angles
• A right triangle is a triangle that has a right angle
• An obtuse triangle is a triangle that has an obtuse angle
• An equiangular triangle is a triangle that has three congruent angles
• CHAPTER 2:
• Logic is the study of reasoning
• In logic, a mathematical sentence is a sentence that contains a complete thought and can be judged to be true or false
• A phrase is an expression that is only part of a sentence
• An open sentence is any sentence that contains a variable
• The domain or replacement set is the set of numbers that can replace a variable
• The solution set or truth set is the set of all replacements that will change an open sentence to true sentences
• A statement or a closed sentence is a sentence that can be judged to be true or false
• A closed sentence is said to have a truth value, either true (T) or false (F)
• The negation of a statement has the opposite truth value of a given statement
• In logic, a compound sentence is a combination of two or more mathematical sentences formed by using the connectives not, and, or, if...then, or if and only if
• A conjunction is a compound statement formed by combining two simple statements, called conjuncts, with the word and. The conjunction P and Q is written symbolically as P ^ Q
• A disjunction is a compound statement formed by combining two simple statements, called disjuncts, with or. The disjunction P or Q is written symbolically as P \/ Q
• A truth table is a summary of all possible truth values of a logic statement
• A conditional compound statement formed by using the words if...then to combine two simple statements. The conditional if P then Q is written symbolically as P -> Q
• A hypothesis, also called a premise or antecedent, is an assertion that begins an argument. The hypothesis usually follows the word if
• A conclusion, also called a consequent, is an ending or a sentence that closes an argument. The conclusion usually follows the word then
• The inverse of a conditional is formed by negating the hypothesis and conclusion
• The converse of a conditional is formed by interchanging the hypothesis and the conclusion
• The contrapositive of a conditional if formed by interchanging and negating, both the hypothesis and conclusion
• Two statements are logically equivalent or logical equivalents if they always have the same truth value
• A biconditional is a compound statement formed by the conjunction ( P->Q) and its converse (Q->P)
• A valid argument uses a series of statements called premises that have known truth values to arrive at a conclusion
• CHAPTER 3:
• Postulates - 3.1 The Reflexive property of equality: a=a
• 3.2 The Symmetric Property of Equality: if a=b, then b=a
• 3.3 The Transitive Property of Equality: if a=b and b=c, then a=c
• 3.4 A quantity may be substituted for its equal in any statement of equality
• 3.5 A whole is equal to the sum of all its parts
• 3.5.1 A segment is congruent to the sum of all its parts
• 3.5.2 A angle is congruent to the sum of all its parts
• 3.6 If equal quantities are added to equal quantities, the sums re equal
• 3.6.1 If congruent segments are added to congruent segments, the sums are congruent
• 3.6.2 If congruent angles are added to congruent angles, the sums are congruent
• 3.7 If equal quantities are subtracted from equal quantities, the differences are equal
• 3.7.1 If congruent segments are subtracted from congruent segments, the differences are congruent
• 3.7.2 If congruent angles are subtracted from congruent angles, the differences are congruent
• 3.8 If equals are multiplied by equals, the products are equal
• 3.9 Doubles of equal quantities are equal
• 3.10 If equals are divided by nonzero equals, the quotients are equal
• 3.11 Halves of equal quantities are equal
• 3.12 The squares of equal quantities are equal
• 3.13 The positive square roots of equal quantities are equal
• CHAPTER 4:
• Adjacent angles are two angles in the same plane that have a common vertex and a common side but do not have any interior points in common
• Complementary angles are two angles the sum of whose degree measures is 90
• Supplementary angles are two angles the sum of whose degree measures 180
• A linear pair of angles are two adjacent whose sum is a straight angle
• Vertical angles are two angles in which the sides of one angle are opposite rays to the sides of the second angle
• Two polygons are congruent if and only if there is one-to-one correspondence between their vertices such that corresponding angles are congruent and corresponding sides are congruent
• -Corresponding parts of congruent polygons are congruent
• -Corresponding parts of congruent polygons are equal in measure
• Postulates - 4.1 A line segment can be extended to any length in either direction
• 4.2 Through two given points, one and only one line can be drawn (two points determine a line)
• 4.3 Two lines cannot intersect in more than one point 
• 4.4 One and only one circle can be drawn with any given point as center and 
• 4.5 At a given point on a given line, one and only one perpendicular can be drawn to the line
• 4.6 From a given point not on a given line, one and only one perpendicular can be drawn to the line
• 4.7 For any two distinct points, there is only one positive real number that is the length of the line segment joining the two points (Distance Postulate)
• 4.8 The shortest distance between two points is the length of the line segment joining these two points
• 4.9 A line segment has one and only one midpoint
• 4.10 An angle has one and only one bisector
• 4.11 Any geometric figure is congruent to itself ( Reflexive Property )
• 4.12 A congruence can be expressed in either order ( Symmetric Property )
• 4.13 Two geometric figures congruent to the same geometric figure are congruent to each other ( Transitive Property )
• 4.14 Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of the other ( SAS )
• 4.15 Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of the other ( ASA )
• 4.16 Two triangles are congruent if the three sides of one triangle are congruent, respectively, to the three sides of the other
• Theorems - 4.1 If two angles are right angles, then they are congruent
• 4.2 If two angles are straight angles, then they are congruent
• 4.3 If two angles are complements of the same angle, then they are congruent
• 4.4 If two angles are congruent, then their complements are congruent
• 4.5 If two angles are supplements of the same angle, then they are congruent
• 4.6 If two angles are congruent, then their supplements are congruent
• 4.7 If two angles form a linear pair, then they are supplementary
• 4.8 If two lines intersect to from congruent adjacent angles, then they are perpendicular
• 4.9 If tow lines intersect, then the vertical angles are congruent
• ‎(c) VLAD 2012
• 
  
• Made by Vlad, :D THANK YOU!!! XD...
•  lol

Test Issue 10

https://docs.google.com/document/d/12mcDyVOZg4fjiCXYkjOEgQRbeAznuBLU9-cXtXU9V0g/edit
3.01.12 ISSUE 10: Math Definitions Geometry
Made by: Miyu

Generalization: A reference from using a few tries
Inductive Reasoning: A method of reasoning in which a series of particular example leads to a conclusion
Counterexamples: A false example of an generalization
Conjecture: Statements that are likely to be true but not yet been proven true by deductive proof
Deductive Reasoning: Using the logic to combine definitions and general statements that we know to be true to reach a valid conclusion
Given: The information known to be true
Prove: The conclusion
Two- Column Proof: The left column, you write Statements that we know to be true and in the right column, you write Reasons why the statement is true
Paragraph Proof: Each statement must be justified by stating a definition or another statement that has been accepted or proved to be true.
Direct Proof: A proof that starts with the given statements and uses the laws of logic to arrive at the statement to be proved
Indirect Proof: A proof that starts with the negation of the statement to be proved and uses the laws of logic to show that it is false. This is also known as Proof By Contradiction
Postulate/ Axiom: Statements that seems so obvious that we accept them without proof
Theorem: A statement that is proved by deductive reasoning
Reflexive Property of Equality: A quantity equal to itself
Symmetric Property of Equality: An equality may be expressed in either order
Transitive Property of Equality: Quantities equal to the same quantity are equal to each other
Equivalence Relation: A relation for which these postulates are said to be true
Substitution Postulate: A quantity may be replaced for its equal in any statement of equality
Partition Postulate: A whole is equal to the sum of all its parts
Postulate 3.5.1: A segment is congruent to the sum of all its parts
Postulate 3.5.2: A angle is congruent to the sum of all its parts
Addition Postulate: If a=b and c=d, then a+c=b+d. If equal quantities are added to equal quantities, the sums are equal
Postulate 3.6.1: If congruent segments are added to congruent segments, the sums are congruent.
Angle Addition Postulate: If congruent angles are added to congruent angles, the sums are congruent
Subtraction Postulate: a=b and c=d, then a-c=b-d, If equal quantities are subtracted from equal quantities, the differences are equal.
Postulate 3.7.1: If congruent statement are subtracted from congruent segments, the differences are congruent
Postulate 3.7.2: If congruent statement are subtracted from congruent angles, the difference are congruent
Multiplication Postulate: If a=b, and c=d, then ac=bd. If equals are multiplied by equals, the products are equal
Postulate 3.9: Doubles of equal quantities are equal.
Division Postulate: If a=b, and c=d, then a/c = b/d. (c=/= 0 and d=/= 0) If equals are divided by nonzero equals, the quotients are equal
Postulate 3.11: Halves of equal quantitiies are equal.
Power Postulate: If a=b, then a^2 =b^2. The squares of equal quantities are equal.
Root Postulate: If a=b and a>0, then radical a =radical b
Postulate 3.13: Postivie Square Roots of postive equals quantities are equal

Issue 8 Math Quiz Logic

https://docs.google.com/document/d/13R-CfU-GQNWEpxq3ZQrsOOTl81KfO48mxic20x9ZQ80/edit

2.13.12 ISSUE 8: Math Quiz//Logic
Made By:  Miyu     Em        {@}>;--
Logic is the study of reasoning
Mathematical Sentence: A sentence is a sentence that contains a complete thought and can be judged to be true and false
Phrase: An expression that is part of a sentence
Open Sentence: Any sentence that contains a variable
Domain/ Replacement Set: Set of numbers that can replace a variable
Solution Set/ True Set: Set of all replacement that will change an open sentence or true sentence
Statement/ Close Sentence: A sentence that can be judged to be true or false
Truth Value: A close sentence has this. Either True or False
Negation: Statement with an opposite truth value of an given statement
Compound Sentence: A combination of two or more mathematical sentences formed by using the connectives not, and, or, if...then, or if and only if.
Conjunction: A compound statement formed by combining two simple statements, called conjuncts, with the word AND. The conjunction is usually formed by using the symbols
p ^ q.
Disjunction: A compound statement formed by combining two simple statements, called disjuncts, with the word OR. The disjunction is usually formed by using the symbols p v q.
Truth Table: A summary of all possible truth values of a logic statement.
Conditional: A compound statement formed by using the words if...then to combine two simple statements. The conditional if p then q is writing symbolically as p --> q.
Hypothesis/Premise/ Antecedent: an assertion that begins an argument. The hypothesis usually follows the word if.
Conclusion/ Consequent: An ending or a sentence that closes the argument. The conclusion usually follow the word then.
Tautology: Statement that is always true
Contradiction: A statement that is always false
Beginning with a statement (p→q), the inverse (~p→~q)is formed by negating the hypothesis and negating the conclusion.
Beginning with the statement (p→q), the converse (q→p) is formed by interchanging the hypothesis and the conclusion
Beginning with the statement (p→q), then the contrapositive (~q→ ~p), formed by interchanging the resulting negation.
Two statements are logically equivalent-or logical equivalents- if they always have the truth value.
A biconditional (p↔q) is a compound statement formed by the conjunction of the conditional p→q and its converse q→p.
A valid argument uses a series of statements called premises that have known truth value to arrive at a conclusion.
The Law of Detachment states that when p→q is true and p is true, then q must be true.
The Law of Disjunctive Inference states that when p۷q is true and p is false, then q is true.

Issue 6// Math

https://docs.google.com/document/d/1NSyTz0lsNkv2fiU6guvIxBgf--HtIONYgQUxrdo6klY/edit

Made by: Miyu, Em        ^     ^
So many words in just one week.... O.o            =( >  . < )=
1.1                                          >epic cat>                  o     o
Geometry: The branch of mathematics that defines and relates the basic properties and measurement of line segment and angles
Undefine Terms: Terms that we will accept w/o definition
4 terms we will accept w/o definitions are Set, Point, Plane and Line
Set: Collection of Objects such that is possible to determine whether a given object belongs to the collection or not
Point: A Geometric Point has no length, width, or thickness, only indicating a place or position
Line: A set of Points, Curved or Straight
Straight Line: A set of points that is straight (Aderf.) Unless otherwise stated, a line is straight
Plane: A set of Points that form a flat surface extending indefinitely in all direction
1.2
Number Line: A line where every point on the number line corresponds to a real number
Coordinate: Number that corresponds to a point in the line
Graph: A point which a number corresponds
Numerical Operation: Assign a real number to every pair of real number
Closure Property of Addition: a+b is a real number
Closure Property of Multiplication: a· b is a real number
Commutative Property of Addition: a+b=b+a
Commutative Property of Multiplication: a·b=b·a
Associative Property of Addition: a+(b+c)=(a+b)+c
Associative Property of Multiplication: a(bc)=(ab)c
Additive Identity: a+0=a and 0+a=a
Multiplicative Identity: 1a=a and a1=a
Additive Inverse: a+(-a)=0
Multiplicative Inverse: a· (1/a)=1
Distributive Property : a(b+c)= (ab+ac)
Multiplication Property of Zero: ab=0 if only a or b equal to 0
1.3
Collinear Set of Points:set of points all which lie on the same straight line
Noncollinear Set of Points: a set of 3 or more points that do not all lie on the same straight line
Definition:statement of the meaning of the term
Distance between 2 points on the real number line: the absolute value the difference of the coordinates of the 2 points AB=|a-b|-|b-a|
Betweeness: B is in between A and C
Line Segments/ Segment: A set of points consisting of 2 points on a line called endpoints and all of the points on the line between the endpoints
Length/ Measure of a line segment: Distance between its endpoints
Congruent Segments: Segments that have the same measure
1.4
Midpoint: Point on the Line that divides the segment into 2 rays (a+b)/2
Bisector of Line Segment: Any line or subset of line that intersects segment at midpoint
1.5
Half-line: Every point on a line divides the line into 2 opposite sets of the points
Rays:Point on line and all points on one side of the point of direction
Endpoints: Dividing Points
Opposite Rays: 2 rays of the same line w/ a common endpoint and no other points in common
Angle: Set of Points that is the union of 2 rays having same endpoints
Sides:  Lines that form a angle
Vertex: Endpoint of Ray
Straight Angle: Angle that is Union of Opposite Rays
Interior of Angle: One region consisting all points
Exterior of Angle: All the other points except the points of the angle itself
Acute Angle: An angle whose degree measure is greater than 0 and less than 90
Right Angle: An angle whose degree measure is 90
Obtuse Angle: An angle whose degree measure is greater than 90 and less than 180
Congruent Angle: Angles with the same measure
Bisector of an Angle: A ray whose endpoint is the vertex of an angle and that divides that angle into two congruent angle
Perpendicular Lines: Two lines that intersect to form right angles
Distance from a Point to a Line: The length of the perpendicular from the point to the line
Foot: The Point which the perpendicular meets the line
Sum of Two Angle: (IF YOU DONT KNOW THIS I’M GOING TO KILL YOU TOMORROW)
Polygon: A close figure in a plane that is the union of line segments such that the segments intersect only at their endpoint and no segments sharing a common endpoint are collinear
Side:  The polygon consist of 3 or more line segments. The line segments are sides
Triangle: A polygon that has exactly 3 sides
Scalene Triangle: A triangle that has no congruent sides
Isosceles: A triangle that has 2 congruent sides
Equilateral Triangle: A triangle with 3 congruent sides






Acute Triangle: A triangle with 3 acute angles
Right Triangle: A triangle with an right angle
Obtuse Triangle: A triangle that has an obtuse angle
Equiangular Triangle: Triangle that has 3 congruent angles