Geometry FINAL EXAM- Chapters 1, 2, and 3 Combined!!

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perimeter of a rectangle

P= 2L + 2w

perimeter of a triangle

P=a+b+c

Ruler Postulate

Points on a line can be paired one to one with real numbers. The real number that corresponds to a point is its coordinate. (Simpler Definition: Distance is ALWAYS positive!!)

endpoints

Points that represent the ends of a line segment or ray

Two Point Postulate

Postulate 2.1

Line-Point Postulate

Postulate 2.2

Line-Intersection Postulate

Postulate 2.3

Three Points Postulate

Postulate 2.4

Plane-Point Postulate

Postulate 2.5

Plane-Line Postulate

Postulate 2.6

Plane-Intersection Postulate

Postulate 2.7

Ruler Postulate

The points on a line can be put into a one-to-one correspondence with the real numbers. (Distance is always positive!!)

horizontal lines

They are "y= lines". They ALWAYS cross the y-axis. The slope is 0!

vertical lines

They are ALWAYS "x= lines". They ALWAYS cross the x-axis. They have no slope!

Vertical Angles Congruence Theorem

Vertical angles are congruent

no slope vs. 0 slope

What do skiers say when they are skiing down a flat hill? "This is a real 0 of a slope!"

Take complicated problems and break them up into easy steps.

What do truth tables do?

the slope and y-intercept

What do you need to know in order to write an equation?

the T/F pattern doubles (2, 4, 8, 16, etc.)

What happens to the T/F pattern as more letters are added?

the letter pattern doubles (1, 2, 4, 8, etc.)

What happens to the letter pattern as more letters are added?

the shortest distance

What is implied with distance?

p = T q = F

What is the only p → q combo that is false?

equal slope

What kind of slope do parallel lines always have?

opposite slopes

What kind of slope do perpendicular lines have?

p and q

What letters do you use to start the table?

the distributive property

What property does NOT apply to truth tables?

reduce to simpler terms

What should you do to fractions?

with words!

When CAN'T you use substitution?

Conditional statements are false when the first part is TRUE and the second part is FALSE.

When are conditional statements false?

in between 90 degree angles and right angles

When do you use "definition of a right angle"?

in between right angles and perpendicular lines

When do you use "definition of perpendicular lines"?

only 90 degree angles

Which angles only will work with the linear perpendicular theorem?

whole equation

Which equation should you have when you are done solving the table.

Be prepared for proofs!

Why do we learn about truth tables?

line segment (segment)

a part of a line that has two endpoints

midpoint

a point that divides a segment into two congruent or equal parts

decagon

a polygon with 10 sides

Icosagon

a polygon with 20 sides

pentagon

a polygon with 5 sides

hexagon

a polygon with 6 sides

heptagon

a polygon with 7 sides

octogon

a polygon with 8 sides

nonagon

a polygon with 9 sides

quadrilateral

a polygon with four sides

"n"-gon

a polygon with n sides

postulate (axiom)

a statement assumed to be true (like a theory)

triangle

a three-sided polygon

Distributive Property

a(b + c) = ab + ac

Reflexive Property

a=a

linear pair

adjacent angles whose non common sides are opposite rays (two rays that form a line)

equiangular

all angles are congruent/equal

rule for alternate exterior angles

alternate exterior angles are congruent

rule for alternate interior angles

alternate interior angles are congruent

Right Angle

an angle that measures 90 degrees

counterexample

an example that disproves a conjecture

segment bisector

anything that divides a segment into two congruent or equal parts

Pythagorean Theorem

a²+b²=c²

contructions

compass and straight edge

rule for consecutive exterior angles

consecutive exterior angles are supplementary

rule for consecutive interior angles

consecutive interior angles are supplementary

parallel lines

coplanar lines that do not intersect

rule for corresponding angles

corresponding angles are congruent

concave

curves/caves inward

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

distance formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

perimeter

distance around the outside (units)

regular

equilateral, equiangular, and convex

line

extends infinitely in two directions; contains infinite points; no width; one dimension; represented by a line with two arrow heads; exactly one through any two points; you can use any two points to name it

consecutive exterior angles

exterior angles that lie on the same side of the transversal

"you are given"

factual information you can use

tautology

final column is true Symbol: ⊥

equilateral

having all sides equal/congruent

Postulate 2.6 - Plane-Line Postulate

if 2 points lie in a plane, then the entire line containing them lies in the plane

Division Property of Equality

if a = b and c is not equal to 0, then a/c = b/c

Symmetric Property

if a=b, then b=a

interior of an angle

inside of an angle

consecutive interior angles

interior angles that lie on the same side of the transversal

coresponding angles

lie on the same side of the transversal and in corresponding positions

If two lines are cut by a transversal, and alternate exterior angles are congruent, then ....?

lines are parallel

If two lines are cut by a transversal, and alternate interior angles are congruent, then....?

lines are parallel

If two lines are cut by a transversal, and consecutive exterior angles are supplementary, then....?

lines are parallel

If two lines are cut by a transversal, and consecutive interior angles are supplementary, then ...?

lines are parallel

If two lines are cut by a transversal, and corresponding angles are congruent, then .....?

lines are parallel

Angle Addition Postulate

little angle + little angle = big angle

~

negation

undefined terms

no formal definitions; building blocks for EVERY geometric shape (points, lines, and planes)

point

no width; no length; represented by a dot

skew lines

non-coplanar lines that do not intersect

alternate exterior angles

nonadjacent exterior angles that lie on opposite sides of the transversal

alternate interior angles

nonadjacent interior angles that lie on opposite sides of the transversal

circle

not a polygon; it is a closed shape with curved lines

convex

not concave

negation

opposite of something "not"

o.c.

original conditional

exterior of an angle

outside of an angle

T, F, T, F

pattern for p

T, T, F, F

pattern for q

parallel planes

planes that do not intersect

intersection

point/multiple points that lie on two/more geometric shapes

between

points must be collinear to be "between"

Non-collinear points

points that do not lie on the same line

non-coplanar points

points that do not lie on the same plane

collinear points

points that lie on the same line

coplanar points

points that lie on the same plane

straight edge

ruler without tick marks

congruent

same size and shape

rays

sides of an angle

Definition of Angle Bisector

something that divides an angle into two congruent angles

area

surface inside (units squared)

defined terms

terms that can be described using known words such as point or line

hypothesis

the part that happens 1st

perpendicular distance

the shortest distance from a point to a line

conclusion

the thing that happens as a result of the hypothesis

supplementary angles

two angles whose measures have a sum of 180 degrees

complementary angles

two angles whose measures have a sum of 90 degrees

vertical angles

two non adjacent angles formed by two intersecting lines (vertical angles are congruent)

opposite rays

two rays that extend in opposite directions and share the same endpoint (collinear)

angle

two rays that share a common endpoint

conjecture

unproven statement based on observations

deductive reasoning

using a general rule and applying it to a specific situation (stereotyping) (BIG to SMALL!) (two types)

inductive reasoning

using specific examples to make a general conclusion

common endpoint

vertex (vertices)

special cases=

vertical and horizontal lines

logically equivalent

when two truth tables' final column is identical Symbol: ≡

y-intercept

where the line crosses the y axis

Quadratic Formula

x = -b ± √(b² - 4ac)/2a

slope-intercept form

y = mx + b

"and"

compass

"circle-maker"

ray

"half-line"; AB is a ray if it consists of the endpoint A and all of the points on AB that lie on the same side of A as B

(p -> q)

"if p, then q"

conditional statement

"if-then" sentence; hypothesis and conclusion

"or"

(~p ∧ q)

"p before q"

biconditional statement

( p <-> q)

law of detachment

(1st type of deductive reasoning) "If p →q is true and a specific p is true, then the specific q is also true."

incorrect example of the law of detachment

(goes backwards) shows the end result first! Don't start with q!

hypothesis

(p)

converse

(q -> p)

conclusion

(q)

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

negation

(~)

inverse

(~p -> ~q)

contrapositive

(~q -> ~p)

slope

- "rise over run" - y = mx +b - slope intercept form

Four Constructions

1) Copy/Double a Segment 2) Bisecting a Segment 3) Bisect an Angle 4) Copy an Angle

convex or concave?

1- Choose two points inside the shape. 2- Connect those points. 3- If the segment went outside, it's concave. 4- If the segment did not, it's convex.

Steps To Solving Problems With the Quadratic Formula:

1- Combine like terms. 2- Get everything on one side and make the other side equal zero. 3- Try Factoring 1st! 4- Then use the quadratic formula. 5- Use which ever answer makes the most sense.

Steps for Calculating Distance From a Point to a Line:

1- plot the point and the line 2- figure out the slope of the line (that will give you the opposite reciprocal, which is the slope of the perpendicular line) 3- From the point, use the perpendicular line to find where it crosses the given line-What are it's coordinates? 4- use the distance formula to find that distance

Protractor Postulate

Angle measures (degrees) are always positive!

Protractor Postulate

Angle measures are always positive!

Congruent Complements Theorem

Angles complementary to the same angle (or to congruent angles) are congruent.

adjacent angles

Angles in the same plane that have a common vertex and common side, but no common interior points.

Congruent Supplements Theorem

Angles supplementary to the same angle (or to congruent angles) are congruent.

standard form

Ax + By = C

Postulate 2.2 - Line-Point Postulate

A line consists of at least 2 points

linear pair

A pair of adjacent angles whose noncommon sides are opposite rays.

vertical angles

A pair of opposite congruent angles formed by intersecting lines

Postulate 2.5 - Plane-Point Postulate

A plane contains at least 3 noncollinear points

perpendicular postulate

If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary

Converse of the Perpendicular Transversal Theorem

If two lines are perpendicular to the same line, then they are perpendicular to each other.

The Linear Pair Perpendicular Theorem

If two lines intersect to form a linear pair (which will always happen) of congruent angles, then lines are perpendicular.

inductive reasoning vs. deductive reasoning

Inductive= small to big Deductive= big to small

yes!

Is it ok to leave slope as an improper fraction?

polygon

It is a closed figure (with at least three sides) made up of line segments and sides that only intersect at their endpoints.

Q: What is the intersection of two different lines called?

A: a point

Q: When do two or more geometric features intersect?

A: when they have one or more points in common

Q: When are segments and rays collinear?

A: when they lie on the same line

Q: When are lines, segments, and planes coplanar?

A: when they lie on the same plane

area of a triangle

A=1/2bh

area of a rectangle

A=lw

area of a square

A=s²

Right Angles Congruence Theorem

All right angles are congruent

transversal

a line that intersects two or more lines

truth table

a listing of the possible truth values for a set of one or more propositions

Construct a perpendicular bisector

1. open the compass more than half of the segment 2. draw an arc above and below the line 3. move the compass to the other endpoint 4. use the same compass setting and make an arc above and below so that the arcs intersect 5. connect the intersection points

Construct a line perpendicular to another line and go through a specific point

1. put needle on given point and put pencil on endpoint 2. make arc 3. place a point where lines intersect 4. place needle on new point 5. make compass more than halfway 6. flip compass 7. make arc 8. place needle on endpoint 9. make arc 10. draw line where arcs intersect

perimeter of a square

4s (where s = length of a side)

You cannot add a symbol unless it refers to something on the chart.

How do you know when to add a symbol?

alphabetical order

How should you place the letters on a truth table?

Postulate 2.3 - Line Intersection Postulate

If 2 lines intersect, then their intersection is exactly 1 point

Postulate 2.7 - Plane Intersection Postulate

If 2 planes intersect, then their intersection is a line

Segment Addition Postulate

If B is between A and C, then AB + BC = AC

Segment Addition Postulate

If Q is between P and R, then PQ + QR = PR.

Perpendicular Transversal Theorem

If a line is perpendicular to one parallel line, then is perpendicular to the second parallel line.

Addition Property of Equality

If a=b, then a+c=b+c

Subtraction Property of Equality

If a=b, then a-c=b-c

Multiplication Property of Equality

If a=b, then ac=bc

Substitution Property of Equality

If something has the same value as something else, then you can replace it.

parallel postulate

If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

measure (distance)

Symbol: AB (spoken as the measure of segment AB) The distance/length from point A to point B.

mx

The "slope part" of slope intercept form.

b

The "y-intercept" part of slope intercept form.

right angle

This is what length and width must form.

Postulate 2.1 - Two Point Postulate

Through any 2 points there exists exactly 1 line

Postulate 2.4 - Three Point Postulate

Through any 3 non-collinear points, there exists exactly 1 plane

Linear Pair

Two angles that are adjacent and supplementary (two angles that form a line)

Supplementary Angles

Two angles whose sum is 180 degrees

Complementary Angles

Two angles whose sum is 90 degrees

examples

Used to prove things FALSE...not used to prove things TRUE!

do-decagon

a 12 sided polygon

inverse

a condition where the hypothesis and the conclusion are both negated

converse

a conditional statement where the hypothesis and conclusion are switched

biconditional statement

a conditional statement where the o.c. and the converse are both true; "if and only if"

contrapositive

a conditional where the hypothesis and conclusion are both negated and switched

plane

a flat surface that extends infinitely in all directions; defined by three "non-collinear" points; has two dimensions; represented by a shape that looks like a floor/wall; through any same points not on the same line, there is exactly one plane

construction

a geometric drawing in which a set of tools are used (usually a compass or straightedge)


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