Postulates, Formulas Theorems, Properties, Corollaries, and Congruence Statements
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ShannonKelly_ on January 13, 2012
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Super important things to know from the Holt McDougal Geometry book.
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72 terms
Terms | Definitions |
|---|---|
 1.1.1 | Through any two points there is exactly one line. |
| 1.1.2 | Through any three noncollinear points there is exactly one plane containing them. |
| 1.1.3 | If two points lie in a plane, then the line containing those points lies in the plane. |
| 1.1.4 | If two lines intersect, then they intersect in exactly one point. |
| 1.1.5 | If two planes intersect, then they intersect in exactly one line. |
| Ruler postulate | The points on a line can be put into a one-to-one correspondence with the real numbers. |
| Segment addition postulate | If B is between A and C, then AB + BC = AC. |
| Protractor postulate | Given line AB and a point O on AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180. |
| Angle addition postulate | If S is in the interior of angle PQR, then the measure of PQS + the measure of SQR = the measure of PQR. |
| Circumference of a circle | The circumference C of a circle is given by the formula C = 𝜋d or C = 2𝜋r. |
| Area of a circle | The area A of a circle is given by the formula A = 𝜋r2. |
| Midpoint formula | The midpoint M of segment AB with endpoints A and B is found by M = (x1 + x2 / 2, y1 + y2 / 2). |
| Distance formula | In a coordinate plane, the distance d between two points (x1, y1) and (x2, y2) is d = √(x2 - x1)squared + (x2 - y2)squared. |
| Pythagorean theorem | In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse (a2 + b2 = c2). |
| Law of detachment | If p → q is a true statement and p is true, then q is true. |
| Law of syllogism | If p → q and q → r are true statements, then p → r is a true statement. |
| Addition property of equality | If a = b, then a + c = b + c. |
| Subtraction property of equality | If a = b, then a - c = b - c. |
| Multiplication property of equality | If a = b, then ac = bc. |
| Division property of equality | If a =b and c ≠ 0, then a/c = b/c. |
| Reflexive property of equality | a = a |
| Symmetric property of equality | If a = b, then b +a. |
| Transitive property of equality | If a = b and b = c, then a = c. |
| Substitution property of equality | If a = b, then b can be substituted for a in any expression. |
| Reflexive property of congruence | figure A is congruent to figure A |
| Symmetric property of congruence | If figure A is congruent to figure B, then figure B is congruent to figure A. |
| Transitive property of congruence | If figure A is congruent to figure B and figure B is congruent to figure C, then figure A is congruent to figure C. |
| Linear pair theorem | If two angles form a linear pair, then they are supplementary. |
| Congruent supplements theorem | If two angles are supplementary to the same angle, then the two angles are congruent. |
| Right angle congruence theorem | All right angles are congruent. |
| Congruent complements theorem | If two angles are complementary to the same angle, then the two angles are congruent. |
| Common segments theorem | Given collinear points A, B, C, and D arranged in alphabetical order, if segment AB is congruent to segment CD, then segment AC is congruent to segment BD. |
| Vertical angles theorem | Vertical angles are congruent. |
| 2.7.3 | If two congruent angles are supplementary, then each angle is a right angle. |
| Corresponding angles postulate | If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. |
| Alternate interior angles theorem | If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. |
| Alternate exterior angles theorem | If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. |
| Same-side interior angles theorem | If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary |
| Converse of the corresponding angles postulate | If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. |
| Parallel postulate | Through a point P not on line l, there is exactly one line parallel to l. |
| Converse of the alternate interior angles theorem | If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. |
| Converse of the alternate exterior angles theorem | If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. |
| Converse of the same-side interior angles theorem | If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel. |
| 3.4.1 | If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. |
| Perpendicular transversal theorem | In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. |
| 3.4.3 | If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. |
| Parallel lines theorem | In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope Any two vertical lines are parallel. |
| Perpendicular lines theorem | In a coordinate plane, two nonvertical lines are perpendicular is and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. |
| Parallel lines | same slope, different y-intercept |
| Intersecting lines | different slopes |
| Coinciding lines | same slope, same y-intercept |
| Triangle sum theorem | The sum of the angle measures of a triangle is 180°. |
| 4.2.2 | The acute angles of a right triangle are complementary. |
| 4.2.3 | The measure of each angle of an equiangular triangle is 60°. |
| Exterior angle theorem | The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. |
| Third angles theorem | If two angles of one triangle are congruent to two angles of another triangle, then the third pair angles are congruent. |
| SSS | If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. |
| SAS | If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. |
| ASA | If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. |
| AAS | IF two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. |
| HL | If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. |
| Isosceles triangle theorem | If two sides of a triangle are congruent, then the angles opposite the sides are congruent. |
| Converse of isosceles triangle theorem | If two angles of a triangle are congruent, then the sides opposite those angles are congruent. |
| Equilateral triangle | If a triangle is equilateral, then it is equiangular. |
| Equiangular triangle | If a triangle is equiangular, then it is equilateral. |
| Perpendicular bisector theorem | If a point on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. |
| Converse of the perpendicular bisector theorem | If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. |
| Angle bisector theorem | If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. |
| Converse of the angle bisector theorem | If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. |
| Circumcentre theorem | The circumcentre of a triangle is equidistant from the vertices of the trangle (PA=PB =PC). |
| Incentre theorem | The incentre of a triangle is equidistant from the sides of the triangle (PX = PY = PZ). |
| Centroid theorem | The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side (AP = 2/3AY; BP = 2/3BZ; CP = 2/3 CX). |
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