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Transcript
Notes for Proofs: Definitions, Theorems, Properties, Postulates a = a, AB = AB, m A = m A (can also be used with ) 1 Reflexive Property 2 Symmetric Property 3 Transitive Property If a = b and b = c, then a = c (can also be used with ) 4 Addition Property If a = c, then a + b = c + b 5 Subtraction Property If a = c, then a – b = c – b 6 Multiplication Property If a = c, then ab = cb 7 Division Property If a = c, then a/b = c/b 8 Substitution Property If a = b, then a and b may be substituted for one another in an equation. 9 Distributive Property a(b + c) = ab + ac 10 Definition of Midpoint If B is the midpoint of AC , then AB BC . 11 Definition of Segment Bisector If BD bisects AC at D, then AD DC . 12 Segment Addition Postulate If B is between A and C, then AB+BC=AC. 13 Definition of Congruent Segments If AB CD , then AB = CD. 14 Definition of Congruent Angles If A B , then mA mB . 15 Definition of Perpendicular Lines 16 Definition of Right Angle 17 Right Angle Congruence Theorem 18 Angle Addition Postulate 19 Definition of Angle Bisector 20 Vertical Angles Theorem 21 Definition of Complementary Angles If 1 & 2 are complementary, then m 1 + m 2 = 90. 22 Definition of Supplementary Angles If 1 & 2 are supplementary, then m 1 + m 2 = 180. 23 Linear Pair Theorem If a = b, then b = a (can also be used with ) If two lines are perpendicular, then they intersect at 90° angles. If A is a right angle, then mA 90 . If A and B are right angles, then A B. If D is in the interior of ABC , then mABD mDBC mABC . If BD bisects ABC, then ABD DBC. If A and B are vertical angles, then A B . If two angles form a linear pair, then they are supplementary. Same Side Interior Angles Postulate Alternate Interior Angles Theorem Corresponding Angles Theorem Same Side Exterior Angles Theorem Alternate Exterior Angles Theorem Perpendicular Bisector Theorem If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary. If two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure. If two parallel lines are cut by a transversal, then the pairs of corresponding angles have the same measure. If two parallel lines are cut by a transversal, then the pairs of same-side exterior angles are supplementary. If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles have the same measure. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. 30 The Triangle Sum Theorem The sum of the angle measures of a triangle is 180°. 31 Exterior Angle Theorem 32 Isosceles Triangle Theorem 33 Equilateral Triangle Theorem 34 Triangle Inequality Theorem 35 ASA Triangle Congruence Theorem 36 SAS Triangle Congruence Theorem 37 SSS Triangle Congruence Theorem 38 AAS Triangle Congruence Theorem 39 HL Triangle Congruence Theorem 40 CPCTC 41 Angle Bisector Theorem 42 Triangle Midsegment Theorem AA Triangle Similarity Theorem If a point is on the bisector an of angle, then it is equidistant from the sides of the angle. The segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is half the length of that side. If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. 44 SSS Triangle Similarity Theorem If the three sides of one triangle are proportional to the corresponding sides of another triangle, then the triangles are similar. 45 SAS Triangle Similarity Theorem 46 Triangle Proportionality Theorem 24 25 26 27 28 29 43 The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. If two sides of a triangle are congruent, then the two angles opposite the sides are congruent. If a triangle is equilateral, then it is equiangular The sum of any two side lengths of a triangle is greater than the third side length. 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. 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. If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent. If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Corresponding parts of congruent triangles are congruent. If two sides of one triangle are proportional to the corresponding sides of another triangle and their included angles are congruent, then the triangles are similar. If a line that is parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally.
