Postulates, Theorems, Definitions
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127 terms
Terms | Definitions |
|---|---|
 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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