Proofs

reflexive property of equality/congruence

a=a; sum equals itself (must be the same thing)

symmetric property of equality/congruence

b=a, a=b; flips the order after you establish 2 things are congruent (two things that are different but are congruent)

transitive property of equality/congruence

a=b, b=c, then a=c; each is used in two different equations, like a game of telephone (if one letter is used twice, you can use transitive)

definition of midpoint

a point that decides a segment into 2 congruent pieces

definition of segment bisector

point, ray, line, segment that intersects the segment at the midpoint, also makes something congruent

definition of complementary angles

two positive angles whose measures have a sum of 90, each angle is the complement of each other (right angle) (MUST DEFINE RIGHT ANGLES BEFORE MAKING CONGRUENT STATEMENT)

definition of supplementary angles

two positive angles with the sum of 180 (must be identified before linear pair postulate)

definition of angle bisector

a line, ray, or segment that intersect an angle creating two congruent angles

segment addition postulate

adding segments are possible

angle addition postulate

adding angles is possible

definition of linear pair

two adjacent angles that are supplementary

linear pair postulate

a pair of supplementary angles add up to 180 degrees

vertical angles congruence theorem

vertical angles are congruent

right angle congruence theorem

all right angles are congruent

definition of perpendicular

angles are right angles

congruent supplements theorem

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

congruent complements theorem

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

consecutive exterior angles theorem

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

corresponding angles theorem

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 alternative interior angles are congruent

alternate exterior angles theorem

if two parallel lines are cut by a transversal, then the pairs of alternate exterior 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 (diff from others)

Corresponding angles converse +alt interior and exterior and consecutive interior

if two parallel lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel

transitive property of parallel lines

if two lines are parallel to the same lines, then they are parallel to each other

linear pair perpendicular theorem

if 2 lines intersect to form a linear pair of congruent angles, then the lines are perpendicular

perpendicular transversal theorem

in a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line

lines perpendicular to a transversal theorem

In a plane, if two lines are perpendicular to the same line, they are perpendicular to each other

triangle sun theorem

the sum of the measures of the interior angles of a triangle is 180

exterior angle theorem

the measure of an exterior angle of a triangle is equal to the sum of the measure is the 2 nonadjacent interior angles

base angles theorem

if 2 sides of triangles are congruent, then the angles opposite to them are congruent

converse of base angles theorem

if 2 angles of a triangle are congruent, then the sides opposite are congruent

third angles theorem

in a triangle, if two angles of 1 triangle are congruent to 2 angles of another triangle, the 3rd angles are also congruent (NO THIRD SIDES THEOREM)

SAS congruence theorem

side angle side congruence theorem

SSS congruence theorem

side side side congruence theorem

ASA congruence theorem

angle side angle congruence theorem

AAS congruence theorem

angle angle side congruence theorem

HL congruence theorem

hypotenuse (longest side of a triangle) and a leg of the right triangle are congruent to the HL of the second triangle

CPCTC

corresponding parts of congruent triangles are congruent—> if we know a pair on triangles are congruent, their parts (angles and segments) must be too