| A | B |
|---|
| ReflexiveProperty | For every number a, a = a. |
| SymmetricProperty | For all numbers a & b, if a = b, then b = a.(ex. the segment GH = segment HG) |
| TransitiveProperty | For all numbers a, b & c, if a = b & b = c, then a = c. (A bit like the law of syllogism) |
| Add/Subtract Property | For all numbers a, b, & c, if a = b, then a + c = b + c and a - c = b - c.(ex. 1 ft = 12 inches, 1 ft + 3 inches = 12 in ches+ 3 inches) |
| Mult/Division Property | For all numbers a, b, and c, if a = b, then a * c = b * c, and if c not equal to zero, a ÷ c = b ÷ c.(ex. 1 m = 1000 mm, 1 m * 5 = 1000 mm * 5, 5 m = 5000 mm) |
| SubstitutionProperty | For all numbers a & b, if a = b, then a may be replaced by b in any equation or expression. |
| DistributiveProperty | For all numbers a, b, & c, a(b + c) = ab + ac. |
| THEOREM 2-1 Segment Properties | Congruence of segments is reflexive, symmetric, and transitive. |
| Theorem 2-2 Supplement Theorem | If two angles form a linear pair,then they are supplementary angles. |
| Theorem 2-3 Angle Properties | Congruence of angles is reflexive, symmetric, and transitive. |
| Theorem 2-4 supplementary congruent | Angles supplementary to the same angle or to congruent angles are congruent. |
| Theorem 2-5complementary congruent | Angles complementary to the same angle or to congruent angles are congruent. |
| Theorem 2-6 right congruent | All right angles are congruent. |
| Theorem 2-7 vertical angles | Vertical angles are congruent. |
| Theorem 2-8 perpendicular lines form | Perpendicular lines intersect to form four right angles.. |
| Postulate 3-1 Corresponding Angles | If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent., |
| Theorem 3-1 Alternate Interior | If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent, |
| Theorem 3-2 Consecutive Interior Angle | If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary |
| Theorem 3-3 Alternate Exterior Angle | If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent, |
| Theorem 3-4 Perpendicular Transversal | In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other., |
| Postulate 3-5 Euclidean Parallel Postulate | In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. |
| Theorem 3-5 transversal alt int angles | If there is a line and a point not on the line, then there exists exactly one line though the point that is parallel to the given line., |
| Theorem 3-5 transversal alt int angles | If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the two lines are parallel., |
| Theorem 3-6 | If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel., |
| Theorem 3-8 | In a plane, if two lines are perpendicular to the same line, then they are parallel., |
| Theorem 3-7 | to be added |
| Postulate 3-2 | Two nonvertical lines have the same slope if and only if they are parallel., |
| Postulate 3-3 | Two nonvertical lines are perpendicular if and only if the product of their slopes is -1., |
| Postulate 3-4 | If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel., |
| Theorem 4-2 Third Angle Theorem | If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent., |
| Theorem 4-1 Angle Sum Theorem | The sum of the measures of the angles of a triangle is 180., |
| Theorem 4-3 Exterior Angle Theorem | The measure of an exterior angle of a trianlge is equal to, |
| Corollary 4-1 | The acute angles of a right triangle are complementary., |
| Postulate 4-1 SSS | (Side - Side - Side) - If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent., |
| Postulate 4-2 SAS | Side - Included Angle - Side) - 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., |
| Postulate 4-3 ASA | (Angle - Included Side - Angle) - 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. |
| Postulate 4-3 AAS | (Angle - Angle - Side) - If two angles and a NON-INCLUDED side of one triangle are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent. |
| Theorem 4-6 Isosceles Triangle Theorem (ITT) | If two sides of a triangle are congruent, then the angles opposite those sides are congruent., |
| Theorem 4-7 Converse of the ITT | If two angles of a triangle are congruent, then the sides opposite those angles are congruent. |
| Corollary 4-3 | A triangle is equilateral if and only if it is equiangular. |
| Corollary 4-4 | Each angle of an equilateral triangle measures 60 degrees. |
| Theorem 5-5 LL (Leg - Leg) | If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent., |
| Theorem 5-6 HA (Hypotenuse - Angle) | If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent., |
| Theorem 5-7 LA (Leg - Angle) | If the leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent., |
| Postulate 5-1 HL (Hypotenuse -Leg) | If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. |
| Theorem 5-8 Exterior Angle Inequality Theorem | If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles., |
| Theorem 5-9 | If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite |
| Theorem 5-10 | If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle, |
| Theorem 5-11 | The perpendicular segment from a point to a line is the shortest segment from the point to the line., |
| Theorem 5-12 | The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. |
| Theorem 5-12 Triangle Inequality Theorem | The sum of the lengths of any two sides of a triangle is greater than the length of the third side. |
| Theorem 6-5 | If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| Theorem 6-6 | If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| Theorem 6-7 | If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. |
| Theorem 6-8 | If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. |
| Parallelogram Property Summary | 1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are congruent. 3. Both pairs of opposite angles are congruent. 4.Diagonals bisect each other. 5.A pair of opposite sides is both parallel and congruent. |
| Theorem 6-9 | If a parallelogram is a rectangle, then its diagonals are congruent |
| Theorem 6-10 (Converse of Thm 6-9) | If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. |
| Rectangle Property Summary | 1. Opposite sides are congruent and parallel. (All parallelograms) 2. Opposite angles are congruent. (All parallelograms) 3. Consecutive angles are supplementary. (All parallelograms) 4. Diagonals bisect each other. (All parallelograms) 5.Diagonals are congruent. (Rectangle) 6.All four angles are right angles. (Rectangle) |
| Theorem 6-11 | The diagonals of a rhombs are perpendicular |
| Theorem 6-12 (Converse of Thm 6-11) | If the diagonals of a parallelogram are perpendicular, |
| Theorem 6-13 | Each diagonal of a rhombus bisects a pair of opposite angles. |
| Trapezoids | Definition #1: A trapezoid is a quadrilateral with exactly one pair of parallel sides.Definition #2:A trapezoid is a quadrilateral with at least one pair of parallel sides. |
