Math

Concepts of geometry

Geometry -it is the branch of math that studies shapes, sizes, lines, positions, angles, and spaces



-it came from the latin word “geo” meaning earth, and “metron” meaning measurement



Undefined terms:



Point ( • )

-exact location, intersection, or a corner. It has no dimension , represented by a dot.

-named using a uppercase letter 



Line ( ↔ )

-set of points that extend in both directions endlessly.

-named by the two points or a lowercase letter



  • Line subsets/parts of a line



  • Line segment ( ━ )

- part of a line segment with two endpoints, it has a fixed length



  • Ray ( → )

-starts in one endpoint and continues endlessly in the other direction.



Plane

-two dimensional, meaning it has height and length.

-requires three noncollinear lines to make a plane



  • Noncollinear

-points that don't lie on the same line



  • Three ways to name a plane

  • 3 noncollinear points

  • 3 or more intersecting lines

  • A line & point not on the same line

Angle

-formed when two rays meet at a common endpoint



  • Vertex- common endpoint



Types of angles:



 Acute angle

-less than 90 

Right angle

-exactly 90

Obtuse angle

-more than 90, less than 180

Straight angle

-exactly 180

Reflex angle

-more than 180



Parallel lines cut by transversals

Parallel lines

-lines that dont meet and are equidistant



  • Equidistant

-is the same distant at a given point or line



Intersecting lines

-two lines having a common point



Transversal

-a line that cuts through two or more lines



Angles formed by transversals



Congruent angles:

Corresponding

-equal in measurement

Alternate interior

-inside the parallel lines, opposite sides of the transversal

Alternate exterior

-outside the parallel lines, opposite sides of the transversal

Vertical

-intersection of two lines



Supplementary angles: 

Same-side interior

-inside the parallel lines and on the same side of the transversal

Same-side exterior

-outside the parallel lines and on the same side of the transversal

Adjacent

-two angles that sit next to each other



Inductive and deductive reasoning



Inductive

-specific details of observation to a general conclusion



Deductive

-general details of observation to a specific conclusion

-uses facts



Triangle congruence

Congruent triangles

  • Two triangles that are the same shape and size.



Corresponding Parts - the matching sides and angles between two geometric figures that sit in the same relative positions.



  1. If two line segments are congruent, then their lengths are equal

  2. If two angles are congruent, then their measures are equal

  3. CPCTC - Corresponding Parts of Congruent Triangles are Congruent.



Included angle 

-angle located directly between two given sides of a triangle



Included side

 -It is the specific side located directly between two given angles

Postulates - Statements that are accepted as true; no need for a proof

Theorems - Statements that require a proof before accepted as true



Postulates: 



Side-Side-Side (SSS)

-three sides of a triangle are congruent to the corresponding sides of another triangle



Side-Angle-Side (SAS)

-two sides and an included angle of a triangle are congruent to the corresponding parts of another triangle



Angle-Side-Angle (ASA)

-two angles of a triangle and an included side of a triangle are congruent to the corresponding parts of another triangle



Angle-Angle-Side (AAS/SAA)

-two angles and a non-included side of a triangle are congruent to the corresponding parts of another triangle



Theorems:



Hypotenuse-Leg (HyL)

hypotenuse and one leg of a right triangle are congruent correspondingly



Leg-Leg (LL)

two legs of a right triangle are congruent correspondingly



Hypotenuse-Acute Angle (HyA)

hypotenuse and an acute angle of one right triangle are congruent correspondingly



Leg-Acute Angle (LA)

one leg and an acute angle of a right triangle are congruent correspondingly





Postulates

Postulate 1

  • A line is formed by at least two different points. A plane is formed by at least three noncollinear points. Space is formed by at least four noncoplanar points.


Postulate 2

  • Through any two different points, there is exactly one line.


Postulate 3

  • Through any three non collinear points, there is exactly one plane.



Postulate 4

  • two points lie in a plane, then the line containing these two points also lies in the same plane



Postulate 5

  • If two different planes intersect, then the intersection is a line



Postulate 6

  • There is a one-to-one correspondence between the points of a number line and the set of real numbers.



Postulate 7 (The Ruler Postulate)

  • There is a unique distance between any two different points on the number line



Segment Addition Postulate
-three points A, B, and C are collinear, and AB + AC = AC, then B is between A and C



Congruent Line Segments
-two line segments have equal lengths, then they are congruent.



Midpoint of a Line Segment
-a point divides a line segment into two congruent segments, then the point is called the midpoint of the line segment.

Bisector of a Line Segment
-a line, line segment, ray, or plane intersects a line segment at its midpoint, then it is called a bisector of the line segment.



Properties of equality

Reflexive

- a=a



Symmetric 

-  a=b, then b=a



Transitive 

- a = b and b = c, then a = c



Addition 

- a = b, then a + c = b + c



Subtraction 

-  a = b, then a - c = b - c



Multiplication 

-  a = b, then ac = bc



Division 

 - a = b, then a/c = b/c



Substitution 

- A quantity can be substituted or replaced by another quantity of equal or the same value in any expression or equation