142 terms

# Geometry- All Definitions, Postulates, Theorems and Other properties

option x is ≈ option j is ∆ option 0 is º ∠ and ± and ² and ⊥ are under symbols option , is ≤ option . is ≥
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equidistant
equally distant to the same point
point
an undefined term; location in space; no size
line
an undefined term; extends in 2 directions never ending; no size
plane
an undefined term; extends without ending; no thickness; need at least 3 non collinear points to draw a plane
space
set of all points (everything)
collinear points
points on the same line
coplanar points
points on the same plane
coplanar
on the same plane
collinear
on the same line
intersection
set of pints in 2 figures where they meet/cross
line segment
2 points on a line and all the points in between
ray
starting at 1 point and going infinitely in one direction
parallel lines
2 lines that are coplanar and never touch
skew lines
not coplanar; not parallel; but never touch
length
distant between endpoints
congruent
same size and shape
congruent segments
segments that have the same length
midpoint
a point that divides the segment into two ≈ segments
bisector
line that crosses at the midpoint
opposite rays
(forms a line) must have connecting end point
postulate (axiom)
a statement accepted without proof
angle
a figure formed by two rays that have a common end point
acute ∠
less than 90º
obtuse ∠
greater than 90º
right ∠
equals 90º
strait ∠
equals 180º
congruent ∠s
have equal measures
2 coplanar ∠s with a common vertex and a common side but no common interior points (don't have to be congruent)
Bisector of an ∠
a ray that divides ∠ into 2 ≈ adjacent ∠s
protractor postulate
can use a protractor to measure
1st part of ∠ + 2nd part of ∠ = a whole ∠ ( ∠AOC+ ∠COD = ∠AOD)
unnamed postulate/theorem #1
a line contains at least 2 points
unnamed postulate/theorem #2
a plane contains at least 3 points that are not collinear
unnamed postulate/theorem #3
space contains at least 4 points that are non coplanar
unnamed postulate/theorem #4
through any 3 points there is at least one plane
unnamed postulate/theorem #5
if 2 points are on a plane then the line containing those points is also on the plane
unnamed postulate/theorem #6
if 2 planes intersect then their intersection forms a line
theorem
a statement that has to be proven
unnamed postulate/theorem #7
if 2 lines intersect they intersect at exactly 1 point
unnamed postulate/theorem #8
through a line and a point not in the line there is exactly 1 plane
unnamed postulate/theorem #9
if 2 lines intersect then exactly 1 plane contains both lines
conditional statement/ if then statement
if p then q
converse
if q then p
counter example
an example used to prove a statement is false (the hypothesis is true but the statement is false) (only takes 1 counter example to prove a statement false)
hypothesis of "if ellen studies then she will get an A"
ellen studies
conclusion of "if ellen studies then she will get an A"
she will get an A
Biconditional
if and only if (ex. a triangle is acute if and only if it has 3 acute∠s)
6 properties of Equality (algebra)
addition, subtraction, multiplication, division, substitution, distributive
reflexive property
a=a; a≈a
symmetric property
if a=b then b=a
transitive property
if a=b and b=c then a=c
midpoint theorem
parts = half the whole
( A_______M_______B) )if m is midpoint of AB then AM=1/2 AB and MB=1/2 AB)
Angle bisector theorem
part= half the whole
rule with vertical ∠s
vertical ∠s are ≈
perpendicular lines
2 lines that intersect to form right ∠s
unnamed postulate/theorem #10 (related to ⊥ lines)
2 lines are ⊥ if they form ≈ adjacent ∠s
converse of unnamed postulate/theorem #10 (related to ⊥ lines)
if 2 lines form ≈ adjacent ∠s then they are ⊥
unnamed postulate/theorem #11
if exterior sides of 2 adjacent acute ∠s are ⊥ then the ∠s are complementary
unnamed postulate/theorem #12
if 2 ∠s are ≈ then their supplements are also ≈
unnamed postulate/theorem #13
if 2 acute ∠s are ≈ then their complements are also ≈
unnamed postulate/theorem #14
if 2 ll planes are cut by a third plane then the lines of intersection are ll
transversal
a line that intersects 2 or more coplanar lines in different points
when dealing with Transversals and lines these 4 kinds of ∠s are formed
alt. int. ∠s
alt. ext. ∠s
same side int. ∠s
corresponding ∠s
what types of ∠s are ≈ when 2 ll lines are cut by a trans.
corresponding ∠s are ≈ and alt. int. ∠s are ≈
what type of ∠s are supp. when 2 ll lines are cut by a trans.
same side int. ∠s
unnamed postulate/theorem #15
if a trans. is ⊥ to 1 of 2 ll lines then it is ⊥ to the other line also
5 ways to prove lines are ll
if 2 lines are cut by a trans. and ...
alt. int. ∠s are ≈
corresponding∠s are ≈
same side int. ∠s are supplementary;
if in a plane 2 lines are ⊥ to the same line
if 2 lines are ll to the same line
unnamed postulate/theorem #16
though a point outside a line there is exactly 1 line ll to the given line
unnamed postulate/theorem #17
through a point outside a line there is exactly 1 line ⊥ to the given line
triangle
a figure formed by 3 segments joining 3 non collinear points
scalene ∆
no ≈ segments
isosceles ∆
at least 2 ≈ sides
acute ∆
all ∠s of the ∆ are ≈
equilateral
all sides are ≈
equiangular
all ∠s are ≈
auxiliary line
a line ray or segment added to a diagram to help in a proof
corollary
a statement that can be proved easily by applying the theorem
obtuse ∆
a ∆ with 1 obtuse ∠
right ∆
a ∆ with 1 right ∠
exterior ∠ theorem
the measure of the ext. ∠s of a ∆ equals the sum of the measures of the 2 remote interior ∠s
4 corollaries of ext. ∠ theorem
1. if 2 ∠s of 1 ∆ are ≈ of to 2 ∠s of another ∆ then the 3rd ∠s of both ∆s are ≈
2. Each ∠ of an equiangular has a measure of 60º
3. in a ∆ there can be at most 1 right or 1 obtuse ∠
4. the acute ∠s of a right ∆ are complementary
polygon
a figure formed by coplanar segments such that 1. each segment intersects exactly with 2 other segments one at each endpoint and 2. no 2 segments with a common endpoint are collinear
diagonal
a segment joining 2 non consecutive vertices
convex polygon
a polygon such that NO line containing a side of the polygon contains a point in the interior of the polygon OR a polygon is convex if NO diagonal contains points outside the polygon
concave polygon
if a diagonal contains points outside the polygon
regular polygon
a polygon that is equiangular and equilateral
equation to find the sum of the measures of of a convex polygon
(n-2)180
(n= # of sides in the polygon)
equation to find an ∠ of a regular polygon
[(n-2)180]÷n
unnamed postulate/theorem #18
the sum of the measure of the ext. ∠s of any convex polygon is 360
deductive reasoning
proving statements by reasoning from accepted postulate, definitions, theorems, and given info (must be true)
inductive reasoning
a kind of reasoning in which the conclusion is based on several past observations (conclusion is probably true but not necessarily true)
5 ways to prove ∆s are ≈
SSS postulate, SAS postulate, ASA postulate, HL theorem, and AAS theorem
SSS postulate (side, side, side)
if 3 sides of one ∆ are ≈ to 3 sides of another ∆ then the ∆s are ≈
SAS postulate (side, angle, side)
if 2 sides and the included ∠ of 1 ∆ are congruent to 2 sides and the included ∠ of the other ∆ then the ∆s are ≈
ASA postulate (angle side angle)
if 2 ∠s and the included side of 1 ∆ are ≈ to 2 sides and included ∠s of another ∆ then the ∆s are ≈
CPCTC (corresponding parts of ≈ ∆s are ≈)
by finding 2 ∆s ≈ then you can prove their corresponding parts are ≈
Isosceles ∆
∆ with at least 2 sides ≈
the isosceles ∆ theorem
if 2 sides are ≈ then the ∠s opposite those sides are ≈
converse of the isosceles ∆ theorem
if 2 ∠s of a ∆ are ≈ then the sides opposite those ∠s are ≈
unnamed corollary A (related to equilateral ∆s)
an equilateral ∆ is also equiangular
unnamed corollary B (related to isosceles ∆s)
the bisector of the vertex ∠ of an isosceles ∆ is ⊥ the base of the midpoint
AAS theorem (angle, angle side)
if 2 ∠s and a non included side of 1 ∆ are ≈ to the corresponding parts of the other ∆ then the ∆s are ≈
HL theorem
if the hypotenuse and a leg of 1 right ∆ are ≈ to the corresponding parts of another right ∆ then the ∆s are ≈
x= [-b ± √b²-4ac] ÷ 2a
median of a ∆
a segment from a vertex to the midpoint of the opposite side
altitude of a ∆
is the ⊥ segment from a vertex to the opposite side
perpendicular bisector of a segment
a ray or segment that is ⊥ to the segment at its midpoint
unnamed postulate/theorem #19
if a point lies on the ⊥ bisector or a segment then the point is equidistant from the endpoints of a segment
unnamed postulate/theorem #20
if a point is equidistant from the endpoints of a segment then the point is equidistant from the sides of the ∠
unnamed postulate/theorem #21
if a point lies on the bisector of an ∠ then the point is equidistant from the sides of the ∠
unnamed postulate/theorem #22
if a point is equidistant from the sides of the ∠ then the point lies on the bisector of the ∠
parallelogram
a quadrilateral with both pairs of opposite sides are ≈
3 theorems relating to parallelograms
opp. sides of a parallelogram are ≈; opp. ∠s of a parallelogram are ≈; diagonals of a parallelogram bisect each other
4 ways to prove the quad is a parallelogram
1. if both pairs of opp. sides of a quad. are ≈
2. if 1 pair of opp. sides of a quad are both ≈ and ll
3. if both pairs of opp. ∠s of a quad are ≈
4. if the diagonals of a quad bisect each other
2 unnamed postulates/theorems related to ll lines
if 2 lines are ll then all the points on 1 lines are equidistant to all the points on the other line; if 3 ll lines cut 1 transversal into ≈ segments then they cut every transversal into ≈ segments
2 unnamed postulates/theorems that relate to medians of a ∆
1. a line that contains the midpoint of 1 side of a ∆ and is ll to another side passes through the midpoint of the 3rd side; the segment that joins the midpoints of 2 sides of a ∆ is ll to the 3rd side and is 1/2 as long as the 3rd side
rectangle
a quad w/ 4 right ∠s
rhombus
a quad w/ 4 ≈ sides
square
a quad w/ 4 right ∠s and 4 ≈ sides
the diagonals of a rectangle are ≈
the diagonals of a rhombus are ⊥; each diagonal of a rhombus bisects 2 ∠s if the rhombus
unnamed postulate/theorem #23
the midpoint of the hypotenuse of a right ∆ is equidistant from the 3 vertices
unnamed postulate/theorem #24 (related to proving a parallelogram is a rectangle)
if an ∠of a parallelogram is a right ∠ then the parallelogram is a rectangle
unnamed postulate/theorem #25 (related to proving a parallelogram is a rhombus)
if 2 consecutive sides of a parallelogram are ≈ then the parallelogram is a rhombus
trapezoid
a quad with exactly 1 pair of ll sides
isosceles trapezoid
a trapezoid with 1 pair of ≈ sides
median of a trap.
1. is ll to the bases
2. has a length equal to the avg of the base length
unnamed postulate/theorem #26 (related to isosceles trap. and their base ∠s)
base ∠s of and isosceles trap are ≈
inequality
a mathematical sentance that contains < > ≤ ≥
properties of inequalities #1
if a>b and c≥d then a+c>b+d
properties of inequalities #2
if a>b and c>0 then ac>bc and a/c>b/c
properties of inequalities #3
if a>b and c<0 then ac<bc and a/c<b/c
properties of inequalities #4 (like transitive)
if a>b and b>c then a>c
properties of inequalities #5
if a=b+c and c>0 then a>b
ext. ∠ inequality theorem
the measure of an exterior ∠ of a ∆ is greater than the measure of either remote ∠s
inverse
if not p then not q
contrapositive
if not q then not p
venn diagram
a circle diagram that may be used to represent a conditional statement
indirect proof
a proof in which you assume temporarily that the conclusion is not true and then deduce a contradiction
3 theorems/postulates regarding inequalities of 1 ∆
1. the sum of 2 sides of a ∆ is greater than the 3rd side
2. if 1 side of a ∆ is longer than the 2nd side then ∠ opp. the 1st side is larger than the ∠ opp. the 2nd side
3. if 1 ∠ of a ∆ is larger than a 2nd ∠ then the side opp. the 1st ∠ is longer than the side opp. the 2nd ∠
SAS inequality
if 2 sides of a ∆ are ≈ to 2 sids of another ∆ but included 1st ∠ of the 1st ∆ is larger than the included ∠ of the 2nd ∆ then the 3rd side of the 1st ∆ is longer than the 3rd side of the 2nd ∆
SSS inequality
if 2 sides of 1 ∆ are ≈ to 2 sides of another ∆ but the 3rd side of the 1st ∆ is larger than the included ∠of the 2nd ∆