Geometry- All Definitions, Postulates, Theorems and Other properties

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option x is ≈ option j is ∆ option 0 is º ∠ and ± and ² and ⊥ are under symbols option , is ≤ option . is ≥

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

adjacent ∠s

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

∠ addition postulate

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 ≈

quadratic formula

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

theorems/rules about rectangles (1)

the diagonals of a rectangle are ≈

theorems/rules about rhombuses (2)

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 ∆

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