S: SALKE | PERIOD 5 | JULISSA PARANZINO
BC: JULISSA PARANZINO SALKE PERIOD 5 MARCH 2011
FC: JULISSA PARANZINO SALKE PERIOD 5 MY GEOMETRIC PROJECT
1: TABLE OF CONTENTS! | 2 - LOGIC 4 - GEOMETRY BASICS 6 - TRIANGLE CONGRUENCE PROOFS 8 - PARALLEL LINES & TRIANGLES 10 - QUADRILATERALS 12 - COORDINATE GEOMETRY 14 - GRAPHING SYSTEMS & CIRCLES 3 - LOGIC : Theorems, Pictures, Definitions & Postulates 5 - GEOMETRIC BASICS : Theorems & Pictures 7 - TRIANGLE CONGRUENCE : Theorems & Postulates 9 - PARALLEL LINES & TRIANGLES : Pictures 11- QUADRILATERALS: Pictures & Explanations 13- COORDINATE GEOMETRY: Theorems & Explanations 15- GRAPHING SYSTEMS & CIRCLES: Pictures & Tables
2: LOGIC | Logic is used in everyday life, as almost anyone needs logic to ponder everyday decisions such as "It's raining outside, I should wear a jacket." In Logic, or mostly a statement, there are only two truth values, which are true or false. The Statement is either true, or it is false.
3: Statement: The basis of logic. Can be deemed as true or false, NEVER both. Negation: A statement that denies another statement. Contrapositive: Negation of a statement. Inverse: By switching the statement around. ex : the bird is flying today. inverse: today, the bird is flying. converse: Combination of contrapositive & inverse. ex: the bird is flying today. converse: today, the bird is NOT flying.
4: Geometry Basics All of geometry is based off of the basics, such as lines, rays, angles,vertex's, bases, legs, and sides. They're what makes up triangles & other shapes. Line- also known as a straight or 180 degree angle, it has no start or end. Ray-Similar to a line, has a starting point, but no end. Similar to the 'rays' of the sun. Angle-compiled of two rays and a vertex. Vertex-The starting point of two rays. | Every single building ever constructed has geometry in it because geometry is based on equality and congruence. If a building isn't congruent, it won't be stable and it will eventually collapse.
6: TRIANGLE CONGRUENCE PROOFS | Now that we know the basics of triangles, and the logic behind them, we can prove them. To prove a triangle, you have to prove all sides OR angles of it to be congruent. To be congruent, the sides must be the same length, and the angles must be the same measure. To prove triangles congruent, they have to be the same size & shape. There are several postulates to proving triangles congruent, lets go over some of them. (: | ASA: ANGLE, SIDE, ANGLE. Two angles and one included side of one triangle, are congruent to the other. an included side is the side connecting or between the two angles.
7: SSS:SIDE, SIDE, SIDE. Three sides of the triangle(which make up the triangle are proved congruent to three sides of another triangle. essentially, these triangles are identical. SAS:SIDE,ANGLE,SIDE Two sides and the angle that they create are proved congruent to those of another triangle. AAS:AGLE, ANGLE, SIDE. Two angles and a non-included or non-connective side are proved congruent to those of another triangle. | Triangle proofs are essential to those in the air- plane industry because one side of the plane MUST be absolutely identical to the other side, or flight could be interrupted and peoples lives at risk.
8: Parallel Lines & Triangles | Parallel lines are lines that NEVER meet and are always the same distance apart. Their home is a plane, which is a surface that has no thickness but width&height. They are DIRECTLY related to triangles, which are a shape made up of three vertices, three sides, and three angles. HENCE the name TRIangle. Depending on the type of triangle(scalene,equilateral, right) it may contain parallel lines in it. Included with parallel lines are transversals.
9: Transversals are straight lines that run through parallel lines & create exterior/interior corresponding angles. | In This transversal, 1=8, they are congruent exterior angles. 3=6, they are congruent interior angles. EF is a transversal that cuts the corresponding parallel lines AB & CD. | Rhombus, squares, rectangles, and parallelograms are all perfect examples of parallel lines WITHIN common everyday shapes. railroad tracks are also another great example of parallel lines as they never touch.
10: QUADRILATERALS | Quadrilaterals are shapes in geometry that contain 4 sides.. A TRIANGLE is NOT a quadrilateral. Squares, rectangles, rhombus, and trapezoids are all examples of quadrilaterals. | Squares are quadrilaterals that are congruent all around.. They have 90degree angles, and all 4 sides are congruent to eachother. | Trapezoids do not have 90degree angles, however they do contain one set of parallel lines.
11: Rectangles, like squares, have 4 90degree angles, however they have 2 sets of congruent sides, and 2 sets of parallel lines. | Rhombus's have 2 sets of parallel lines . | Basically every single building structure you EVER come across will have some type of quadrilateral or geometric shape within in. Buildings, cars, phones, computers, paper even, all have quadrilaterals in them. If you look closely at everyday&household items you will be surprised how many shapes you can find.
12: COORDINATE GEOMETRY | Coordinate geometry uses possibility tables, relative to logic and takes place on a plane, the coordinate plane. The most important variables of coordinate geometry are x and y. Slope: The measure of a line going up or downward when measured from left to right. Positive slope: The line goes up from left to right. Negative slope: The lines goes down from left to right Negative reciprocal: Slopes that are negated and turned upsidedown.
13: Perpendicular lines have negative reciprocal slopes, as they have to intersect at a right angle. Parallel lines share the same slope, because they never intersect. Coordinate geometry can be related to real life because when roads are built, they require coordinate plans. Same goes for building fences around property and mapping out houses.
14: GRAPHING SYSTEMS & CIRCLES | When solving a table, or system, you solve it graphically. A graphing calculator is your bestfriend here because it does all the work for you, all YOU need to do is plug in the values for x & y. (: A table is set up as such... x | y ------ 1 | 2 3 | 6 4 | 8 5 | 10 16| 32 If you go a little further in tables, then there are two lines given. y= mx + b m=slope b=y intercept
15: y=5x + 2 When these two lines are graphed, they y=2x + 1 will intersect, unless they are parallel but they are not, since they DO NOT share the same slope. The point at which they intersect(x,y) is the solution of this system. | Circles have a diameter, circumference, and radius. Diameter:The length of 1 side of the circle to another radius:half of the diameter, or from 1 side of the triangle to the midpoint(center) circumference:diameter of the circle x pi(3.14)