geometry quater 1

8/6/2019 Geometry Quater 1

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GEOMETRY 1

Logic - basing conclusions on premises- science of correct reasoning

Reasoning - drawing of conclusions/inferences from known/assumed facts

•KINDS OF REASONING

1.Inductive Reasoning- observing figures, patterns and examples- specific case to a general statement- usually used in science- cannot be proven

2. Deductive Reasoning- showing that a particular statement is the solution of a given

statement- concluding from general to specific- rules are applied- draw, analyze, then solve

3. Analogy- comparison between two things

- basing on some othe aspects4. Intuition- reaching a conclusion by mere guessing, common sense and

experience- can be 'entirely right', 'barely right', or 'entirely wrong'

•UNDEFINED TERMS1. Points

i. Interpretation in Algebra-> Notation: P(x,y)-> Indicates location or position on the xy-plane-> Determined by an ordered pair-> Cannot be located in two places at the same time

ii. Notation in Geometry

-> Represented by dots-> Named by capital letters-> •G "point G"

iii. Description-> Dimensionless spot in space [no length, width, thickness]-> Unique indicator of position in space

2. Linesi. Interpretation in Algebra

-> Graph of a linear function-> Union of infinitely many points that satisfy an equation

ii. Notation in Geometry-> <-----> m [line m]-> A <-----> B [line AB]-> usually named by any two letters, each representing a point on the

lineiii. Description

-> set of infinitely many points arranged in a straight path-> set of continuous points that extends infinitely-> no points, no thickness

3. Planesi. Interpretation in Algebra

-> Cartesian Plane-> Coordinate Plane



ii. Notation in Geometry

-> Plane ABC/Plane Qiii. Description

->Set of points that forms a flat surface which extends infinitely

Collinear Points - are points that lie on the same lineCoplanar Points - are points that lie on the same plane

POSTULATES[Postulate - a statements that is assumed to be true even without proof]

•POSTULATE 1: The Distance Postulate- To every pair of different points, there corresponds a unique positive

number

•POSTULATE 2: The Ruler Postulate- The points of a line can be placed in correspondence with the real

numbers in a way that:(a) to every point of the line, there corresponds exactly one real

number(b) to every real number, there corresponds exactly one point of the line

(c) the distance between any two points is the absolute value of the difference of the corresponding numbers.

•POSTULATE 3: The Ruler Placement Postulate- Given two points P and Q of a line, the coordinate system can be

chosen in such a way that the coordinate of P is zero and thecoordinate of Q is positive.

•POSTULATE 4: The Line Postulate- For every two different points, there is exactly one line that contains

both points.

•POSTULATE 5: The Plane-Space Postulate

- Every plane contains at least three different noncollinear points- Every space contains at least four different noncoplanar points

•POSTULATE 6: The Flat Plane Postulate- If two points on a line lie in a plane, then the line lies in the same

plane

•POSTULATE 7: The Plane Postulate- Any three points lie in at least one plane, and any three noncollinear

points determine a plane

•Case 1: Collinear-> Infinite number of planes

•Case 2: Non-collinear-> Exactly one plane

•POSTULATE 8: Intersection of Planes Postulate

- If two different planes intersect, then their intersection is a line

•POSTULATE 9: The Plane Separation Postulate- Given a line and a plane containing it, the points of the plane that do

not lie on the line form two sets such that:(a) each of the sets is convex(b) if P is in one of the sets and Q is in the other, then the

segment PQ intersects the line



•POSTULATE 10: The Space Separation Postulate- The points of a space that do not lie in a given plane form two sets

such that(a) each of the sets is convex(b) if P is in one of the sets and Q is in the other, then the

segment PQ intersects the plane

•

POSTULATE 11: The Angle Measurement Postulate- To every angle BAC, there corresponds a real number between o and180

•POSTULATE 12: The Angle Construction Postulate- Let ray AB be a ray on the edge of the half-plane H. For every number

r between 0 and 180 there is exactly one ray AP, with P in H,such that measure of angle PAB is r

•POSTULATE 13: The Angle Addition Postulate- If D is in the interior of angle BAC, then the measure angle BAC is

equal to the measure of angle BAD + measure of angle DAC

•POSTULATE 14: The Supplement Postulate- If two angles form a linear pair, then they are supplementary

•POSTULATE 15: The SAS Postulate

- Every SAS correspondence is a congruence•POSTULATE 16: The ASA Postulate

- Every ASA correspondence is a congruence

•POSTULATE 17: The SSS Postulate- Every SSS correspondence is a congruence

THEOREMS[Theorem - a statement that needs to be proven]

• THEOREM 2-1- If a-b > 0, then a > b

• THEOREM 2-2- If a=b+c and c > 0, then a > b

• THEOREM 2-3- Let A,B, and C be points of a line, with coordinates x, y, and z

respectively. If x < y < z, then A-B-C

• THEOREM 2-4- If A, B, and C are three different points of the same line, then exactly

one of them is between the other two

• THEOREM 2-5: The Point-Plotting Theorem- Let ray AB be a ray, and let x be a positive number. Then there is

exactly one point P of AB such that AP =x

• THEOREM 2-6: The Midpoint Theorem- Every segment has exactly one midpoint

• THEOREM 3-1- If two different lines intersect, their intersection contains only one

point

• THEOREM 3-2- If a line intersects a plane not containing it, the intersection contains

only one point.

• THEOREM 3-3- Given a line and a point not on the line, there is exactly one plane

containing both



• THEOREM 3-4- Given two intersecting lines, there is exactly one plane containing

both

• THEOREM 4-1- Congruence between angles is an equivalence relation

• THEOREM 4--2

- If the angles in a linear pair are congruent, then each of them is aright angle

• THEOREM 4-3- If two angles are complementary, then both are acute

• THEOREM 4-4- Any two right angles are congruent

• THEOREM 4-5- If two angles are both congruent and supplementary, then each is a

right angle

• THEOREM 4-6: The Supplement Theorem- Supplement of congruent angles are congruent

• THEOREM 4-8: The Vertical Angle Theorem

- Vertical angles are congruent• THEOREM 4-9

- If two lines are perpendicular, they form four right angles

• THEOREM 5-1- Congruence for segments is an equivalence relation

• THEOREM 5-2- Congruence for triangles is an equivalence relation

TERMS DEFINITION

•Absolute Value - the distance of a number from 0 in the number line

- denoted by |x|; x being the number•Distance - the number given by the Distance Postulate. If the points are P and Q,

then the distance is denoted by PQ.

•Coordinate System - A correspondence the sort described in the ruler postulate

•Coordinate of a Point - The number corresponding to a given point

•Distance Formula - MN = |x-y| if the coordinate of M is x and the coordinate of Nis y

•Betweenness - B is between A and C if (a) A, B, and C are different points of the same line(b) AB + BC = AC. When B is between A and C, we write A-B-C or C-B-A

•Midpoint - A point B is called a midpoint of a segment AC if B is between A and Cand AB = AC

•Convex - A set M is called convex if for every two points P and Q of the set, theentire segment PQ lies in M

•Edge - divides planes into half-planes

•Angle - the union of two rays having the same endpoint but do not lie on thesame line.

•Sides - the two rays of an angles

•Vertex - the common endpoint of the sides

•Interior of angles - the interior of angles BAC is the set of all points P in the plane



of angle BAC such that(a) P and B are on the same side of line AC(b) P and C are on the same side of line AB

•Exterior of angles - The exterior of angle BAC is the set of all points of the planeof angle BAC that lie neither on the angle nor in its interior

• Triangle - the union of the segments AB, AC and AB if A, B, and C are any three

noncollinear points and is denoted by triangle ABC. Every triangle determines3 angles.

•Vertices - The point A, B, and C in the triangle ABC

•Sides - The segments AB, AC, and BC in the triangle ABC

•Perimeter - The sum of the lengths of the sides

•Interior of triangles - A point lies in the interior of a triangle if it lies in the interiorof the angles of the triangle.

•Exterior of triangles - A point lies in the extterior of a triangle if it lies in the planeof the triangle but does not lie on the triangle or the interior

•Degree - The unit of measure of angles

•Measure of Angles - The number of degrees in an angle

•Linear Pair - Angles BAC and CAD form a linear pair if ray AB and ray AD areopposite rays, and ray AC is any other ray

•Supplementary Angles - If the sum of the measure of two angles is 180 and eachis called a supplement of the other

•Right Angle - an angle having measure 90

•Acute Angle - an angle with measure less than 90

•Obtuse Angle - an angle with measure greater than 90 but less than 180

•Complementary - two angles whose sum of the measures is 90, and each of them is a complement of the other

•Congruent - two angles with the same measure

•Perpendicular Rays - two rays are perpendicular if they are the sides of a right

angle•Perpendicular Lines - two lines are perpendicular if they contain a pair of

perpendicular rays

•Perpendicular Sets - two sets are perpendicular if:(a) each of them is a line, a ray, or a segment(b) they intersect(c) the lines containing tem are perpendicular

•Vertical Angles - two pairs of opposite rays

•Congruent Angles - angles are congruent if they have the same measure

•Congruent Segments - segments are congruent if they have the same length.

*SUBSETS OF A LINE

•Segment - For any two points A and B, the segment AB is the union of A and Band all points that are between A and B. The points A and B are called theendpoints of ray AB. The distance between A and B, AB is called the length of segment AB.

•Ray - Lat A and B be point of a line, the ray AB is the set which is the uniopn of segment AB and the set of all point C for which A-B-C. The point A is called theendpoint of ray AC or ray AB.

•Opposite Rays - If A is between B and C, then ray AB and ray AC are called



opposite ray

*EQUIVALENCE RELATIONS

•Reflexive Propertya = a for every a

•Symmetric Property

If a = b, then b = a• Transitive Property

If a = b, and b = c, then a = c

geometry quater 1

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