# Geometry : Corrolaries, Theorems, Postulates

## 87 terms

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

### Congruent Segments

segments that have equal lengths

### Midpoint

he point that divides a segment into two congruent segments; must be between two other points

### Segment Bisector

line, segment, ray, or plane that intersects a segment at its midpoint

### Angle

figure formed by two rays with a common endpoint; rays are the sides of the angle, the endpoint, the vertex of the angle
*name an angle by: vertex, number inside angle, all three letters (points) [vertex in the middle]

### Acute Angle

measure is between 0º and 90º

90º measure

### Obtuse Angle

measure is between 90º and 180º

### Straight Angles

180º measure

something accepted without proof; if point B lies in the interior of angle AOC, then angle AOB + angle BOC = angle AOC

### Congruent Angles

angles that have the same measure

coplanar with common vertex, share side, no common interior points (can't overlap)

### Angle Bisector

ray that divides an angle into two congruent parts (similar to midpoint)

### Postulate

a statement accepted without proof

### Theorem

a statement that must be proven

### 1

a line contains at least two points; a plane contains at least three non-linear points; space contains at least four non-coplanar points

### 2

through any two points, there is exactly one line

### 3

through any three points, there is at least one plane, and through any three non-collinear points, there is exactly one plane

### 4

if two points are in a plane, then the line that contains the points is in that plane

### 5

if two planes intersect, then their intersection is a line

### 6

the intersection of two lines is a point

### 7

through a line and a point not in a line, there is exactly one plane

### 8

if two lines intersect, then exactly one plane contains the lines

### Deductive Reasoning

proving statements by reasoning from accepted postulates, theorems, and given information (logic)

### Conditionals

Formula: If p (hypothesis), then q (conclusion); p implies q; p only if q; q if p.

If q then p.

### Biconditional Statement

if a conditional and its converse are both true, they can be combined into a single statement using "if and only if" (IFF)
eg: definition
Formula: p IFF q

### Counter Examples

example which disproves a conditional

if a = b and c = d, then a + c = b + d

### Subtraction Property

if a = b and c = d, then a - c = b - d

### Multiplication Property

if a = b, then ca = cb

### Division Property

if a = b and c = 0, then a/c = b/c

### Substitution Property

if a = b, then either a or be may be substituted for each other in any equation or inequality

### Distributive Property

a (b+c) = ab + ac

a = a

### Symmetric Property

a = b, then b = a

### Transitive Property

AKA Substitution Property
if a = b and b = c, then a = c

### Midpoint Theorem

if M is the midpoint of line AB, then AM = 1/2 AB and MB = 1/2 AB

### Proofs

Reasons: given information, definitions, postulates, theorems that have already been proved

### Complementary Angles

two angles whose measures have the sum of 90º; each angle is the complement of the other

### Supplementary Angles

two angles whose measures have the sum of 180º; each angle is the supplement of the other

### Vertical Angles

angles opposite from each other when two lines cross; share the same vertex

### Vertical Angles Theorem

vertical angles are congruent

### Perpendicular Lines Theorem(s)

If two lines are perpendicular, then they form congruent, adjacent angles
If two lines meet to form congruent, adjacent angles, the lines are perpendicular

### Exterior Sides Theorem

If the exterior sides of two acute, adjacent angles are perpendicular, the angles are complementary

### Supplementary Angles Theorem

supplements of congruent angles are congruent

### Complementary Angle Theorem

complements of congruent angles are congruent

### Parallel Planes Theorem

If two parallel planes are cut by a third plane, then the lines of intersection are parallel

### Parallel Lines Form...

Parallel -> corresponding angles congruent
Parallel -> alternate interior angles congruent
Parallel -> same-side interior angles supplementary
If a line is perpendicular to one of two parallel lines, then it's perpendicular to the other
Two lines parallel to a third line are parallel to each other
Through a point outside a line, there is exactly one line parallel to the given line

### Proving Lines Parallel

1) Corresponding angles congruent -> parallel
2) Alternate interior angles congruent -> parallel
3) Same-side interior angles supplementary -> parallel
4) In a plane, two lines perpendicular to the same plane are parallel
5) Two lines parallel to a third line are parallel to each other

### Point -> Perpendicular Lines

Through a point outside a line, there is exactly one line perpendicular to the given line

### Triangle

The sum of the angles of a triangle is 180º

### Proving Parts of Triangles Congruent

If two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent

### Equiangular Triangle

Each angle of an equiangular triangle measures 60º

### Triangle Angle(s) Exceptions

In a triangle, there can be at most one right or obtuse angle

### Right Triangle Angles

The acute angles of a triangle are complementary

### Exterior Angles

The exterior angle of a triangle is equal to the sum of the two remote interior angles

180 (n - 2)

180 (n - 2) / n

360º

360/n

### Inductive Reasoning

A conclusion based on several past observations, is probably true; uses patterns

### Deductive Reasoning

A conclusion based on statements

### Congruent Figures

two figures that have the same size and shape

### CPCTC

corresponding parts of congruent triangles are congruent

### SSS

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent

### SAS

If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent

### ASA

If two angles and an included side of one triangle are congruent to two angles and an included side of another triangle, then the triangles are congruent

### False Triangle Congruence Postulates

(in all triangles) AAA, ASS

### Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite of those sides are also congruent; Base angles of an isosceles triangles are congruent
(Converse) If the angles opposite of the sides are congruent, then the two sides are congruent

### Vertex Angle Bisector

The bisector of the vertex angle of an isosceles triangles is perpendicular to the base at its midpoint

### Equiangular ->Equilateral

An equiangular triangle is also equilateral

### Equilateral -> Equiangular

An equilateral triangle is also equiangular

### Equilateral Triangle

An equilateral triangle has three 60º angles

### HL

(Works with right triangles) Hypotenuse and leg of a right triangle

### Proving Triangles Congruent

ASA, SAS, SSS, AAS, (right triangles only) HL

### Median

segment from vertex to midpoint of opposite side; every triangle has three midpoints

### Altitude

perpendicular segment from vertex to line that contains the opposite side; every triangle has three altitudes
right triangle: two altitudes are legs, third is inside
obtuse triangle: two altitudes are outside, third is inside

### Perpendicular Bisector

line, ray, or segment that is perpendicular to the segment at its midpoint

### Perpendicular Bisector Theorem

points on a perpendicular bisector are equidistant from the endpoints

### Converse of Perpendicular Bisector Theorem

if a point is equidistant from the endpoint of a segment, then the point lies on the perpendicular bisector of the segment

### Length of Perpendicular Segment

distance from point to line (or plane)

### Equidistant Theorems

if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle
if a point is equidistant from the sides of an angle, then the point lies on the angle bisector

### Parallelogram

a quadrilateral with both pairs of opposite sides parallel
parts: opposite sides, opposite angles, diagonals, consecutive sides, consecutive angles

### Opposite Sides Theorems

opposite sides of a parallelogram are congruent

### Opposite Angles Theorems

opposite angles of a parallelogram are congruent

### Diagonals Theorems

diagonals of a parallelogram bisect each other