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

### 142 terms by zoe11

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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

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

less than 90º

greater than 90º

equals 90º

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

if p then q

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)

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

a=a; a≈a

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

no ≈ segments

### isosceles ∆

at least 2 ≈ sides

### acute ∆

all ∠s of the ∆ are ≈

all sides are ≈

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)

[(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 ∆

Example: