Math exam

collinear

points that lie on the same line

Coplanar

points that lie on the same plane

perpendicular lines

Lines that intersect to form right angles

skew lines

non parallel, non coplanar (they do not intersect but go in different directions)

midpoint

A point that divides a segment into two congruent segments

segment bisector

either a point, line, ray or other segment that intersects a segment at its midpoint

Midpoint formula on a coordinate plane

(x₁+x₂)/2, (y₁+y₂)/2

midpoint formula on a number line

(a + b)/2

angle

formed by two rays with the same endpoint

rays

a part of a line that has a fixed starting point but no endpoint

vertex

the point shared by both rays

What unit is used to measure angles?

degrees

congruent angles

angles with the same measure

angle bisector

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

acute angle

less than 90 degrees

rigth angle

90 degrees

obtuse angle

An angle between 90 and 180 degrees

straight angle

an angle that measures 180 degrees

adjacent angles

two coplanar angles with a common side, a common vertex, and no common interior points (no overlap)

vertical angles

two angles whose sides are opposite rays (vertical angles are congruent)

complementary angles

Two angles whose sum is 90 degrees

supplementary angles

Two angles whose sum is 180 degrees

linear pair

a pair of adjacent angles whose non-common sides are opposite rays. the angles of a linear pair form a straight angle (180 degrees) *angles are always supplementary

the distance formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

perpendicular bisector of a segment

a line segment, or ray that is perpendicular to a segment at it's midpoint

perimeter of a square

4s

area of a square

s²

perimeter of a triangle

a+b+c

Area of a triangle

1/2bh

Perimeter of a rectangle

2(b+h)

Area of a rectangle

bh

circmference of a circle

2πr

area of a circle

πr²

inductive reasoning

reasoning based on patterns you observe

conjecture

a conclusion you reach using inductive reasoning

Counter example

a specific example that shows a conjecture is incorrect

conditional

an if-then statement

What is the "if" part of a conditional statement called?

hypothesis

What is the "then" part of a conditional statement called?

conclusion

negation

the negative or opposite of that statment

Inverse

negate the hypothesis and the conclusion

converse

switch the hypothesis and conclusion

Contrapositive

switch and negate the hypothesis and conclusion

biconditional statement

A statement that contains the phrase "if and only if"

how can you determine if a biconditional is true?

1. re-write the biconditional as 2 statements.

2. determine whether the biconditional is true

3. if not, provide a counterexample

What is a good definition?

uses clearly understood terms, is precise, and is reversible

deductive reasoning

reasoning using logic, facts, and definitions to draw a conclusion

Law of Detachment

1. p--\>q

2. p is true

conclusion: q is true

law of sylllogism

1. p--\>q

2. q--\> r

conclusion: p--\>r

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

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 can be substituted for b in any equation or expression

proof

a convincing argument that uses deductive reasoning. logically shows why a conjecture is true

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.

parallel lines

coplanar lines that do not intersect

parallel planes

planes that do not intersect

Transversal

a line that intersects two or more lines

alternate interior angles

angles between 2 lines and on opposite sides of a transversal

alternate exterior angles

Angles that lie outside a pair of lines and on opposite sides of a transversal.

same side interior angles

two interior angles on the same side of the transversal

same side exterior angles

two exterior angles on the same side of the transversal

corresponding angles

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

same side interior angle postulate

if two lines are parallel, then same side interior angles are supplementary

Triangle Angle Sum Theorem

The sum of the measures of the angles of a triangle is 180.

exterior angle

An angle formed by one side of a polygon and the extension of an adjacent side

remote interior angles

the two nonadjacent interior angles of a triangle

Triangle Exterior Angle Theorem

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

slope

the ratio of the vertical change to the horizontal change between any two points

how do you find slope?

y2-y1/x2-x1

what type of slope is this?

slope is positive

what type of slope is this?

slope is negative

what type of slope is this?

zero

what type of slope is this?

undefined

slope-intercept form

y=mx+b

point slope form

y-y1=m(x-x1)

horizontal lines

have a slope of 0

vertical line

have a undefined slope

Third Angles Theorem

if two angles of one triangle are congruent to two angles of another triangle, then the third angle are congruent

Included Angles and Sides

in the same triangle and between both angles or sides

non included angles or sides

in the same triangle but is not between both angles or sides

SSS

side side side; 3 sides of one triangle are congruent to 3 sides of another triangle

AAA

angle angle angle; 3 angles of one triangle are congruent to 3 angles of another triangle

SAS

side angle side; two sides and an included angle are congruent to that of another triangle

ASA

angle side angle; two angles and an included side are congruent to that of another triangle

AAS

angle angle side; two angles and a non-included side are congruent to that of another triangle

SSA

side side angle; two sides and a non-included angle are congruent to that of another triangle

5 theorems used to prove triangle congurence

sss, sas, asa, aas, hl

2 theorems not used to prove triangle congruence

aaa, ssa