#1 (permalink)

Postulate Definition: A statement that is taken to be true without proof

List of Postulates

• Any segment or angle is congruent to itself. (Reflexive Property)
• If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent. (SSS)
• If there exists a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. (SAS)
• If there exists a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. (ASA)
• Two points determine a line (or ray or segment).
• If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent. (HL)
• A line segment is the shortest path between two points.
• Through a point not on a line there is exactly one parallel to the given line. (Parallel Postulate)
• Three noncollinear points determine a plane.
• If a line intersects a plane not containing it, then the intersection is exactly one point.
• If two planes intersect, their intersection is exactly one line.
• If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar. (AAA)
• A tangent line is perpendicular to the radius drawn to the point of contact.
• If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle.
• Circumference of a circle = pi • diameter.
• The area of a rectangle is equal to the product of the base and the height for that base.
• Every closed region has an area.
• If two closed figures are congruent, then their areas are equal.
• If two closed regions intersect only along a common boundary, then the area of their union is equal to the sum of their individual areas.
• The volume of a right rectangular prism is equal to the product of its length, its width, and its height.
• For any two real numbers x and y, exactly one of the following statements is true x < y, x = y, or x > y. (Law of Trichotomy)
• If a > b and b > c, then Q > c. Similarly, if x < y and y < z, then x < z. (Transitive Property of Inequality)
• If a > b, then a + x > b + x. (Addition Property of Inequality)
• If x < y and a > 0, then a • x < a • y. (Positive Multiplication Property of Inequality)
• If x < y and a < 0, then a • x > a • y. (Negative Multiplication Property of Inequality)
• The sum of the measures of any two sides of a triangle is always greater than the measure of the third side.

Theorem Definition: A statement that can be demonstrated to be true by accepted mathematical operations and arguments.

List of Theorems

• If two angles are right angles, then they are congruent.
• If two angles are straight angles, then they are congruent.
• If a conditional statement is true, then the contrapositive of the statement is also true. (If p, then q ; If ~q, then ~p.)
• If angles are supplementary to the same angle, then they are congruent.
• If angles are supplementary to congruent angles, then they are congruent.
• If angles are complementary to the same angle, then they are congruent.
• If angles are complementary to congruent angles, then they are congruent.
• If a segment is added to two congruent segments, the sums are congruent. (Addition Property)
• If an angle is added to two congruent angles, the sums are congruent. (Addition Property)
• If congruent segments are added to congruent segments, the sums are congruent. (Addition Property)
• If congruent angles are added to congruent angles, the sums are congruent. (Addition Property)
• If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property)
• If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property)
• If segments (or angles) are congruent, their like multiples are congruent. (Multiplication Property)
• If segments (or angles) are congruent, their like divisions are congruent. (Division Property)
• If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. (Transitive Property)
• If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other. (Transitive Property)
• If two sides of a triangle are congruent, the angles opposite the sides are congruent.
• If two angles of a triangle are congruent, the sides opposite the angles are congruent.
• If two angles are both supplementary and congruent, then they are right angles.
• The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
• If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel.
• If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel.
• If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel.
• If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are parallel.
• If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are parallel.
• If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent.
• If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary.
• If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent.
• If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent.
• If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary.
• If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the transversal are supplementary.
• In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
• If two lines are parallel to a third line, they are parallel to each other. (Transitive Property of Parallel Lines)
• The sum of the measures of the three angles of a triangle is 180º.
• The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
• A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-half the length of the third side. (Midline Theorem)
• If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No-Choice Theorem)
• If there exists a correspondence between the vertices of two triangles such that two angles and a nonincluded side of one are congruent to the corresponding parts of the other, then the triangles are congruent. (AAS)
• In a proportion, the product of the means is equal to the product of the extremes. (Means-Extremes Products Theorem)
• If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion. (Means-Extremes Ratio Theorem)
• If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar. (AA~)
• If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of corresponding sides are equal, then the triangles are similar. (SSS~)
• If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar. (SAS~)
• If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. (Side-Splitter Theorem)
• If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem)
• The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs. (Pythagorean Theorem)
• In a triangle whose angles have the measures 30º, 60º, and 90º, the lengths of the sides opposite these angles can be represented by x, x , and 2x respectively. (30°-60°-90''-Triangle Theorem)
• In a triangle whose angles have the measures 45º, 45º, and 90º, the lengths of the sides opposite these angles can be represented by x, x, and x respectively. (45°-45°-90°-Triangle Theorem)

#2 (permalink)

OMG You are the secks. I wish I had this in Trig and Geometry

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#3 (permalink)

Yeah, Geometry sucked learning all of these. I hope people catch this and use it to their advantage.

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#4 (permalink)

Umm....WOW.

I distinctly remember saying, in THIS that me and Pollox would be working on a thread of this sort. I'm surprised that you would steal my idea that I said when I suggested to bring some activity to these sections.

Dissapointment....

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www.last.fm/user/shmads/

#5 (permalink)

Thats to bad Shmads.

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#6 (permalink)

I don't remember that thread saying a list of postulates and theorems. I remember it saying it was gonna be a math cheat sheet. That could be anything.

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#7 (permalink)

Haha. But I think at the back of the Geo textbooks is a list of these.

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#8 (permalink)

Thank you so much I will be using these.

#9 (permalink)

Looks really good to me man. Good job. Too bad I don't take Geometry anymore.

#10 (permalink)

um, the theorems are also named like

Theorem 3.1, Theorem 3.2 etc. Might want to go find those