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