Theorems and Definitions

right angle

If an angle is a right angle, then it’s measure is 90 degrees

straight angle

If an angle is a straight angle, then it’s measure is 180 degrees

congruent angles

If two angles have the same measure, then they are congruent

congruent right angles

If two angles are right angles, then they are congruent

congruent straight angles

If two angles are straight angles, then they are congruent

congruent segments

If two segments have the same measure, then they are congruent

acute angle

If an angle is an acute angle, then it’s measure is greater than 0 and less than 90

obtuse angle

If an angle is an obtuse angle, then it’s measure is greater than 90 and less than 180

angle bisector

If a ray bisects an angle, then it divides the angle into two congruent angles 

angle trisector

If two rays trisect an angle, then it divides the angle into three congruent angles

segment bisector

If a (point, line, ray) bisects a segment, then it divides the segment into two congruent segments

segment trisector

If a (point, line, ray) trisects a segment, then it divides the segments into three congruent segments

midpoint

If a point is the midpoint of a segment, then it divides the segment into two congruent segments.

supplementary angles

If two angles are supplementary, then the sum of the angles is 180

complementary angles

if two angles are complementary, then the sum of the angles is 90

perpendicular

If two (segments, lines, rays) are perpendicular, then they intersect to form a right angle

Linear pairs being supplementary

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

non common sides being complementary

If the non common sides of two adjacent angles are perpendicular, then the angles are complementary

adjacent angles being complementary

If two adjacent angles are complementary, then their non common sides are perpendicular 

right adjacent angles

If two adjacent angles are right, then they form perpendicular lines

conclusions made when angles are supplementary to the same angle

If angles are supplementary to the same angle, then they are congruent

conclusions made when angles are supplementary to congruent angles

If angles are supplementary to congruent angles, then they are congruent

conclusions made when angles are complementary to the same angle

If angles are complementary to the same angle, then they are congruent

conclusions made when angles are complementary to congruent angles

If angles are complementary to congruent angles, then they are congruent

Assumed from diagram (so far)

Straight angles, linear pairs, vertical angles

addition theorem 

if congruent segments (or angles) are added to congruent segments (or angles), then their sums are congruent

subtraction theorem

if congruent segments (or angles) are subtracted from congruent segments (or angles), then their differences are congruent

reflexive property

every number, segment, angle, etc. is congruent to itself

addition theorem part two

if congruent segments (or angles) are added to the same segment (or angle), then their sums are congruent

subtraction theorem part two

if congruent segments (or angles) are subtracted from the same segment (or angle), then their differences are congruent

reflexive property

every number, segment, or angle is congruent to itself. used before saying two angles are congruent to the same angle

subtraction theorem part three

if the same segment (or angle) is subtracted from congruent segments (or angles), then their differences are congruent

division theorem

if wholes are congruent, then halves are congruent

multiplication theorem

if halves are congruent, then wholes are congruent

congruent vertical angles

if two angles are vertical angles, then they are congruent

substitution property

if A is congruent to B, and A is supplementary to C, then B is supplementary to C. used when converting congruency to something else

transitive property

property of congruent segments or angles. used when converting congruency to congruency, with the chain rule.

SSS

side-side-side

SAS

side-angle-side

ASA

angle-side-angle

AAS

angle-angle-side

congruent radii

if segments are radii of the same circle, then they are congruent

congruent triangles and corresponding parts

if triangles are congruent, then their corresponding parts are congruent

the reason you have to say when you draw a new line on the diagram

any two points determine a line

median divider

if a segment is a median of a triangle, then it divides the opposite side into two congruent segments

median bisector

if a segment is a median of a triangle, then it bisects the side to which it is drawn

altitude

if a segment is the altitude of a triangle, then it is perpendicular to the side to which it is drawn

isosceles triangle

if a triangle is isosceles, then at least two sides are congruent

congruent bases

if a triangle is isosceles, then the base angles are congruent

congruent legs

if a triangle is isosceles, then the legs are congruent

perpendicular bisector

if two points are equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of the segment

equidistant endpoints 

if a point of the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

right angle theorem

if two angles are supplementary and congruent, then they are right angles

corresponding angles

if corresponding angles are congruent, then the lines are parallel

alternate interior angles

if alternate interior angles are congruent, then the lines are parallel

alternate exterior angles

if alternate exterior angles are congruent, then the lines are parallel

same side interior angles

if same side interior angles are supp, then the lines are parallel

same side exterior angles

if same side exterior angles are supp, then the lines are parallel

perpendicular to the same line

if 2 lines are perpendicular to the same line, then they are parallel

parallel to the same line

if 2 lines are parallel to the same line, then they are parallel

parallelogram

if a quad has both pairs of opposite sides parallel, then it is a parallelogram

parallelogram

if a quad has both pairs of opposite sides congruent, then it is a parallelogram

parallelogram

if a quad has one pair of opposite sides both parallel and congruent, then it is a parallelogram

parallelogram

if a quad has diagnols which bisect each other, then it is a parallelogram

parallelogram

if a quad has both pairs of opposite angles congruent, then it is a parallelogram

parallelogram

if a quad has one angle supp to both of its consecutive angles, then it is a parallelogram

rectangle

if a parallelogram has at least one right angle, then it is a rectangle

rectangle

if a parallelogram has congruent diagonals, then it is a rectangle

rectangle

if a quad has four right angles, then it is a rectangle

rectangle

if a quad has four congruent angles, then it is a rectangle

rhombus

if a parallelogram has one pair of consecutive sides that are congruent, then it is a rhombus

rhombus

if a parallelogram has either diagonal bisect opposite angles, then it is a rhombus

rhombus

if a quad has diagonals that are the perp bisector of each other, then it is a rhombus

rhombus

if a quad has four congruent sides, then it is a rhombus

square

if a quad is both a rectangle and a rhombus, then it is a sqaure

trapezoid

if a quad has exactly one pair of opposite sides parallel, then it is a trapezoid

isosceles trapezoid

if a trap has the nonparallel sides congruent, then it is an isosceles trapezoid

isosceles trapezoid

if a trap has the lower or the upper base angles congruent, then it is an isosceles trapezoid

isosceles trapezoid

if a trap has congruent diagonals, then it is an isosceles trapezoid

kite

if a quad has two disjoint pairs of consecutive sides congruent, then it is a kite

kite

if a quad has one diagonal which the perp bisector of the other diagonal, then it is a kite