Postulates, Theorems, Definitions

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Angle Addition Postulate

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

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