# Postulates, Theorems, Definitions

## 127 terms

the sum of the measures of adjacent angles equals the measure of the larger angle they form together

### Two points determine a line

Any two points are collinear

### Substitution

if two things are congruent, then they are interchangeable

### Reflexive Property

anything is equal/congruent to itself

### The Parallel Postulate

Through any point not on a given line, there exists exactly one line parallel to the given line. (Given a line and a point not on that line, there exists exactly one line that can be drawn through that point, parallel to the given line).

### Definition of Betweenness of Points

if B is between A and C, then AB + BC = AC

### If an angle is a right angle

then its measure is

### If an angle is a straight angle

then its measure is

### If two angles (segments) are congruent

then they are equal in measure

### If a line

ray or segment divides an angle into two congruent angles, then it bisects the angle.

### If a line

ray or segment divides a segment into two congruent segments, then it bisects the segment.

### If a point divides a segment into two congruent segments

then it is the midpoint of that segment.

### If two lines rays or segments divide an angle into 3 congruent angles

then they trisect the angle.

### If two points divide a segment into 3 congruent segments

then they trisect the segment.

### If two lines (rays or segments) are perpendicular

then they intersect to form right angles.

### If two angles are complementary

then the sum of their measures is 90 degrees.

### If two angles are supplementary

then the sum of their measures is 180 degrees.

### If two triangles are congruent

then all pairs of corresponding sides and corresponding angles are congruent.

### If a segment is a median of a triangle

then it drawn from a vertex to the midpoint of the opposite side.

### If a segment is drawn from a vertex of a triangle to the midpoint of the opposite side

then it is a median.

### If a segment is a median of a triangle

then it divides the side of the triangle that it intersects into two congruent segments.

### If a segment drawn from a triangle's vertex divides the opposite side into 2 congruent segments

then it is a median.

### If a segment is an altitude of a triangle

then it is drawn from a vertex perpendicular to the opposite side (the side of the triangle which it intersects)

### If a segment is drawn from a vertex of a triangle perpendicular to the opposite side

then it is an altitude.

### If a segment is an altitude of a triangle

then it forms right angles with the side it intersects.

### If a segment is drawn from a vertex forming right angles with the opposite side

then it is an altitude.

### If a triangle has no sides congruent to one another

then it is scalene.

### If at least two sides of a triangle are congruent

then the triangle is isosceles.

### If all sides of a triangle are congruent

then it is equilateral.

### If all angles of a triangle are congruent

then it is equiangular.

### If a triangle has all acute angles

then it is an acute triangle.

### If a triangle has one right angle

then it is a right triangle.

### If a triangle has an obtuse angle

then it is an obtuse triangle.

### If a quadrilateral is a parallelogram

then both pairs of opposite sides are parallel

### If both pairs of opposite sides of a quad are parallel

then it is a parallelogram

### If a quad is a rhombus

then it is a parallelogram in which at least one pair of consecutive sides is congruent

### If a quad is a rectangle

then it is a parallelogram in which there is at least one right angle

### If a quad is a square

then it is a parallelogram that is both a rhombus and a rectangle

### If a quad is a trapezoid

then it has exactly one pair of parallel sides

### If a quad is an isosceles trapezoid

then it is a trapezoid with congruent legs (it has exactly one pair of parallel bases and congruent legs)

### If two angles are straight angles

then they are congruent.

### If two angles are right angles

then they are congruent.

### If two adjacent angles form a right angle

then they are complementary.

### If two adjacent angles are complementary

then they form a right angle.

### If two adjacent angles form a straight angle

then they are supplementary.

### If two angles are complementary to the same angle

then they are congruent to each other.

### If two angles are complementary to congruent angles

then they are congruent to each other.

### If two angles are supplementary to the same angle

then they are congruent to each other.

### If two angles are supplementary to congruent angles

then they are congruent to each other.

### If a segment is added to two congruent segments

then the resulting segments are congruent

### If a segment is subtracted from two congruent segments

then the resulting segments are congruent

### If an angle is added to two congruent angles

then the resulting angles are congruent

### If an angle is subtracted from two congruent angles

then the resulting angles are congruent

### If congruent segments are added to congruent segments

then the resulting segments are congruent

### If congruent segments are subtracted from congruent segments

then the resulting segments are congruent

### If congruent angles are added to congruent angles

then the resulting angles are congruent

### If congruent angles are subtracted from congruent angles

then the resulting angles are congruent

### If 2 angles are vertical angles

then they are congruent. (Vertical angles are congruent)

### Congruent

All radii of a circle are

### If 2 congruent segments are bisected (by midpoints)

then all the resulting segments are congruent.

### If 2 congruent angles are bisected

then all the resulting angles are congruent.

### If congruent segments are doubled (or tripled)

then the resulting segments are congruent.

### If congruent angles are doubled (or tripled)

then the resulting angles are congruent.

### If two sides of a triangle are congruent

then the base angles are congruent

### If a triangle is isosceles

then the base angles are congruent

### If the base angles of a triangle are congruent

then the legs opposite them are congruent

### If at least two angles of a triangle are congruent

then the triangle is isosceles

### If a segment is an altitude to the base of an isosceles triangle

then it is also a median

### then they are right angles

If two angles are supplementary AND congruent

### If two points are each equidistant from the endpoints of a segment

then they determine the perpendicular bisector of that segment

### If a point is on the perpendicular bisector of a segment

then it is equidistant from the endpoints of that segment

### If a point is equidistant from the endpoints of a segment

then it LIES ON the perpendicular bisector of that segment

### the measure of either of the remote interior angles

The measure of an exterior angle of a triangle is greater than

### If two lines are cut by a transversal such that a pair of alternate interior angles are congruent

then the lines are parallel

### If two lines are cut by a transversal such that a pair of alternate exterior angles are congruent

then the lines are parallel

### If two lines are cut by a transversal such that a pair of corresponding angles are congruent

then the lines are parallel

### If two lines are cut by a transversal such that a pair of same side interior angles are supplementary

then the lines are parallel

### If two lines are cut by a transversal such that a pair of same side exterior angles are supplementary

then the lines are parallel

### In a plane if a line is perpendicular to one of 2 parallel lines,

then it is perpendicular to the other

### If 2 lines are parallel to a third line

then they are parallel to each other

### If 2 parallel lines are cut by a transversal

then alternate interior angles are congruent

### If 2 parallel lines are cut by a transversal

then alternate exterior angles are congruent

### If 2 parallel lines are cut by a transversal

then corresponding angles are congruent

### If 2 parallel lines are cut by a transversal

then same side exterior angles are supplementary

### If 2 parallel lines are cut by a transversal

then same side interior angles are supplementary

### In a plane if a line is perpendicular to one of 2 parallel lines

then it is perpendicular to the other

### If 2 lines are parallel to a third line

then they are parallel to each other

### If a quad is a parallelogram

then both pairs of opposite sides are congruent

### If a quad is a parallelogram

then opposite angles are congruent

### If a quad is a parallelogram

then the diagonals bisect each other

### If a quad is a parallelogram

then consecutive angles are supplementary

### If a quad is a rectangle

then both pairs of opposite sides are congruent

### If a quad is a rectangle

then opposite angles are congruent

### If a quad is a rectangle

then the diagonals bisect each other

### If a quad is a rectangle

then consecutive angles are supplementary

### If a parallelogram is a rectangle

then all angles are right angles

### If a parallelogram is a rectangle

then the diagonals are congruent

### If a parallelogram is a rhombus

then all sides are congruent

### If a quad is a rhombus

then both pairs of opposite sides are congruent

### If a quad is a rhombus

then opposite angles are congruent

### If a quad is a rhombus

then the diagonals bisect each other

### If a quad is a rhombus

then consecutive angles are supplementary

### If a parallelogram is a rhombus

then the diagonals bisect the angles of the rhombus

### If a parallelogram is a rhombus

then the diagonals are perpendicular bisectors of each other

### If a parallelogram is a rhombus

then the diagonals form 4 congruent right triangles

### If a quadrilateral is a square

then...(all properties of rectangle and rhombus)

### If a parallelogram is a square

then the diagonals form four isosceles right triangles

### If a quad is a kite

then one of the diagonals is the perpendicular bisector of the other diagonal

### If a quad is a kite

then one pair of opposite angles is congruent

### If a quad is a kite

then one diagonal bisects a pair of opposite angles

### If a quad is an isosceles trapezoid

then lower base angles are congruent

### If a quad is an isosceles trapezoid

then upper base angles are congruent

### If a quad is an isosceles trapezoid

then the diagonals are congruent

### If a quad is an isosceles trapezoid

then any lower base angle is supp to any upper base angle

### If both pairs of opposite sides of a quad are congruent

then it is a parallelogram

### If two sides of a quad are both parallel and congruent

then the quad is a parallelogram

### If the diagonals of a quad bisect each other

then the quad is a parallelogram

### If both pairs of opposite angles of a quad are congruent

then it is a parallelogram

### If a parallelogram contains at least one right angle

then it is a rectangle

### If the diagonals of a parallelogram are congruent

then the parallelogram is a rectangle

### If a quad has 4 right angles

then it is a rectangle

### If a parallelogram contains a pair of consecutive congruent sides

then it is a rhombus

### If the diagonals of a quad are perpendicular bisectors of each other

then the quad is a rhombus

### If a quad is both a rectangle and a rhombus

then it is a square

### If the legs of a trapezoid are congruent

then it is isosceles

### If the lower or upper base angles of a trapezoid are congruent

then it is isosceles

### If the diagonals of a trapezoid are congruent

then it is isosceles