Geometry units 1 and 2

acute angle

an angle that measures less than 90 degrees

adjacent angles

two angles that share a common vertex and side, but have no common interior points

angle

two rays with a common endpoint

angle bisector

a ray, segment, line, or plane that divides an angle into two congruent angles

collinear

points that lie on the same line

complementary angles

two angles whose measures have a sum of 90 degrees

congruent

having the same measure

coplanar

points that lie on the same plane

distance formula on a number line

absolute value of the difference of the two points

distance formula on a coordinate plane

pythagorean theorem solved for the hypotenuse

intersection

the point or set of points at which two or more figures intersect

linear pair

two adjacent angles whose noncommon sides are opposite rays

line segment

a part of a line that has two endpoints

midpoint

a point that divides a segment into two congruent segments

midpoint formula on a number line

average of the two numbers

midpoint formula on a coordinate plane

average of the x's and average of the y's

obtuse angle

an angle that measures more than 90 degrees but less than 180 degrees

opposite rays

two rays that have a common endpoint and form a line

perpendicular

intersecting at or forming right angles

ray

a part of a line that starts at an endpoint and extends forever in one direction

right angle

an angle that measures 90 degrees

segment bisector

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

supplementary angles

two angles whose measures have a sum of 180 degrees

vertex

the common endpoint of an angle

vertical angles

two nonadjacent angles formed by two intersecting lines

pythagorean theorem

the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse

compound statement

two or more statements joined by the word "and" or "or"

conclusion

the part of a conditional statement following the word then

conditional statement

a statement that can be written in if-then form; it has a hypothesis and a conclusion

conjecture

an educated guess formed on the basis of given information

conjunction

a compound statement using the word and

contrapositive of "if p then q"

If not q, then not p

converse of "if p then q"

If q, then p

counterexample

an example that proves that a conjecture or statement is false

deductive reasoning

based on facts, applies general facts to specific cases

disjunction

a compound statement that uses the word or

hypothesis

The part of a conditional statement following the word if

inductive reasoning

reasoning from detailed facts to general principles

inverse of "if p then q"

If not p, then not q

negation

the opposite of the original statement

statement

any sentence that is either true or false, but not both

truth value

whether a statement is true or false

through any two points there is...

exactly one line

through any three non collinear points, there is...

exactly one plane

a line contains...

at least two points

a plane contains...

at least three non collinear points

if two points lie in a plane, then...

the entire line containing those points lies in that plane

if two lines intersect, then their intersection is...

exactly one point

if two planes intersect, then their intersection is..

a line

Midpoint Theorem

If M is the midpoint of seg. AB, then seg. AM is congruent to seg. MB

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

Division Property of Equality

if a = b and c is not equal to 0, then a/c = b/c

Reflexive Property of Equality

a = a

Symmetric Property of Equality

If a = b, then b = a

Transitive Property of Equality

If a = b and b = c, then a = c

Substitution Property of Equality

If a = b, then a may be replaced by b in any equation or expression

Distributive Property

a(b + c) = ab + ac

Segment Addition Postulate

If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC

Angle Addition Postulate

B is in the interior of

Supplement Theorem (Linear Pair Theorem)

If two angles form a linear pair, then they are supplementary angles

Complement Theorem

If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles), then they are congruent.

Vertical Angles Theorem (VAT)

If two angles are vertical angles, then they are congruent.

Perpendicular lines intersect to form...

four right angles

All right angles are...

congruent

Perpendicular lines form congruent...

adjacent angles

If two angles are congruent and supplementary, then...

each angle is a right angle

If two congruent angles form a linear pair, then ...

they are right angles

Parallel lines

coplanar lines that do not intersect

Skew lines

non-coplanar lines that do not intersect and are not parallel

Transversal

a line that intersects two or more coplanar lines at different points

Consecutive interior angles

interior angles that lie on the same side of the transversal

Alternate interior angles

nonadjacent interior angles that lie on opposite sides of the transversal

Alternate exterior angles

nonadjacent exterior angles that lie on opposite sides of the transversal

Corresponding angles

lie on the same side of the transversal and in the same positions

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent

Perpendicular Transversal Theorem

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Slope

The steepness of a line on a graph

Slope Formula

rise/run or change in y over change in x

Positive slope

line rises from left to right

Negative slope

Line falls from left to right

Zero slope

The slope of a horizontal line (the change in y-values is 0).

Undefined slope

The slope of a vertical line (cannot be found because it would force a division by 0, undefined in math)

Slopes of Parallel Lines Postulate

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

Slopes of Perpendicular Lines Postulate

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. The slopes are opposite reciprocals.

slope-intercept form

y= mx + b, where m is the slope and b is the y-intercept of the line.

point slope form

y-y1 = m(x-x1), where m is the slope and (x1,y1) is the point the line is passing through.

horizontal line equation

y = b, where b is the y-intercept of the line

vertical line equation

x = a, where a is the x-intercept of the line

Converse of the Corresponding Angles Postulate

If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel.

Alternate Exterior Angles Converse Theorem

If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

Consecutive Interior Angles Converse Theorem

If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel

Alternate Interior Angles Converse Theorem

If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel

Perpendicular Transversal Converse

In a plane, if two lines are perpendicular to the same line, then they are parallel.