Geometry Theorems, Definitions, and postulates

Definitions

Definitions

Complementary Angles

Two angles whose measures equal 90 degrees

Supplementary angles

two angles whose measures equal 180 degrees

theorem

a statement that can be proven

Vertical angles

4 angles formed by intersecting lines and facing in four different directions

transversal

a line that intersects two lines in the same plane at different points

corresponding angles

pairs of angles formed by two lines adn a transversal that make an F pattern

Same-sided interior angles

pairs of angles formed by two lines adn a transversal that make an c pattern

alternate interior angles

pairs of angles formed by two lines adn a transversal that make a Z pattern

congruent angles

triangles in which corresponding parts (sides and angles) are equal in measure

Similar triangles

Triangles in which corresponding angles are equal in measure and corresponding sides are in proportion (ratios equal)

angle bisector

a ray that begins at the vertex of an angle and divides the angle into two angles of equal measure.

segment bisector

a ray, line or segment that divides a segment into two parts of equal measure

legs of an isosceles triangle

the sides of equal measure in an isosceles triangle

Base of an isosceles triangle

The third side of an isosceles triangle

equiangular

having angles that are all equal in measure

perpendicular bisector

a line that bisects a segment and is perpendicular to it

altitude

a segment from a vertex of a triangle perpendicular to the line containing the opposite side

geometric mean

the value of x in proportion

a/x = x/b where a, b, and x are positive numbers (x is the geometric mean between a and b)

sine sin

for an acute angle of a right angle, the ratio of the side opposite the angle to the measure of the hypotenuse. (opp/hyp)

cosine, cos

for an acute angle of a right triangle the ratio of the side adjacent to the angle to the measure of the hypotenuse. (adj, hyp)

Tangent, tan

for an acute angle of a right triangle the ratio of the side opposite to the angle to the measure of the side adjacent.. (opp, adj)

Algebra Postulates

Algebra Postulates

addition prop of =

if the same number is added to equal numbers, then the sums are equal

Subtraction prop of =

if the same number is subtracted from equal numbers then the differences are equal.

multiplication prop of =

if the same number is multiplied from equal numbers then the products are equal.

division prop of =

if the same number is divided from equal numbers then the quotient are equal.

reflexive prop of =

a number is equal to itself

symmetric prop of =

if a=b, then b=a

substitution prop of =

if values are equal, then one value may be substituted for the other

transitive prop. of =

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

distributive prop. of =

a(b+c)=ab+bc

congruence postulates

Congruence postulates

reflexive prop of ≅

a≅a

symmetric prop of≅

if a=b, then b=a

transitive prop of ≅

if a≅b and b≅a, then a≅c

Angle Postulates and theorems

Angle postulates and theorems

Angle addition Postulate

for any angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts

Linear pair theorem

if two angles form a linear pair, then they are supplementary.

congruent supplements theorem

if two angles are supplements of the same angle, then they are congruent.

Congruent complements theorem

if two angles are complements of the same angle, then they are congruent.

right angle congruence theorem

all right angles are congruent

vertical angles theorem

vertical angles are congruent

Theorem

if two congruent angles are supplementary, then each is a right angle.

angle bisector theorem

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

converse of the angle bisector theorem

if a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

lines postulates and theorems

lines postulates and theorems

segment addition postulate

for any segment, the measure of the whole is equal to the sum of its non overlapping parts

postulate

through any two points there is exactly one line

postulate

if any two lines intersect, then they intersect at exactly one point

common segments theorem

given any colinear points, a b, c, and d arranged as shown, if line AB ≅ line CD, then line AC ≅ line BC

corresponding angles postulate

if two parallel lines are intersected by a transversal, then the corresponding angles are equal in measure

converse of corresponding angles postulate

if two lines are intersected by a transversal and corresponding angles are equal in measure, then the lines are parallel

postulate

through a point not on a given line, there is one and only one line parallel to the given line

alternate interior angles theorem

if two parallel lines are intersected by a transversal, then alternate interior angles are equal in measure

alternate exterior angles theorem

if two parallel lines are intersected by a transversal, then alternate exterior angles are equal in measure

same side exterior angles theorem

if two parallel lines are intersected by a transversal, then same side interior angles are supplementary

converse of alternate interior angles theorem

if two parallel lines are intersected by a transversal and alternate interior angles are equal in measure, then the lines are parallel.

converse of alternate exterior angles theorem

if two parallel lines are intersected by a transversal and alternate exterior angles are are equal in measure, then the lines are parallel.

converse of same side interior angles theorem

if two parallel lines are intersected by a transversal and same side interior angles are supplementary, then the lines are parallel.

theorem

if two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular

theorem

if two lines are perpendicular to one of two parallel lines, then ti is perpendicular to the other one

perpendicular transversal theorem

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

perpendicular bisector theorem

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

parallel lines theorem

in a coordinate plane, two nonvertical lines are parallel IF they have the same slope.

perpendicular lines theorem

in a coordinate plane, two nonvertical lines are perpendicular IF the product of their slopes is -1

two transversals proportionality corollary

if three or more parallel lines intersect two transversals, then they divide the traversals proportionally

triangle postulates and theorems

triangle postulates and theorems

angle angle similarity postulate

i two angles of one triangle are equal in measure to two angles of another triangle, then the two are similar

side side side similarity theorem

if the three side of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

third angles theorem

if two angles o one triangles are congruent to two angles of another triangle, then the third pair of angles are congruent

side angle side conguence postulate

if two sides and the included angles of one triangle are equal in measure, to the corresponding sides of another triangle, then the triangles are congruent