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127 terms

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