* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Angles - www .alexandria .k12 .mn .us
Survey
Document related concepts
Duality (projective geometry) wikipedia , lookup
Integer triangle wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Noether's theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Multilateration wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Pythagorean theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Rational trigonometry wikipedia , lookup
Line (geometry) wikipedia , lookup
Transcript
Postulates, Theorems, Corollaries & Definitions (Things that can be used in proofs) Items are organized by type of figure. Please note that addition, subtraction, division, multiplication and substitution properties can be used in any proof. Segments Segment Addition Postulate (part + part = whole) PQ + QR = PR Midpoint Theorem if M is the midpoint of AB, then AM = MB Reflexive Property AB = AB Symmetric Property if AM = MB, then MB =AM Transitive Property if AB = CD and CD = EF, then AB = EF Definition of Congruence if AB = CD, then AB = CD Definition of Bisector if line l bisects MO at N, then MN = NO or MN=NO Angles Angle Addition Postulate (part + part = whole) mPQR + mRQS = mPQS Supplement Theorem if two angles form a linear pair, then they are supplementary. Ex: 1 and 2 form a linear pair, so m1 + m2 = 180 Reflexive Property ABC = ABC Symmetric Property if ABC = DEF, then DEF = ABC Transitive Property if ABC = DEF and DEF = GHI, then ABC = GHI Congruent Supplements Theorem angles supp. to the same angle or to = angles are =. Ex: 1 and 2 are supp. and 2 and 3 are supp., therefore 1 = 3 Congruent Complements Theorem angles comp. to the same angle or to = angles are =. Ex: 4 and 5 are comp. and 5 and 6 are comp., therefore 4 = 6 Right Angles Theorem All right angles are =. Vertical Angles Theorem Vertical angles are =. Ex: 7 = 9 and 8 = 10 Perpendicular Lines intersect to form 4 right angles. Ex: if a b, then 11, 12, 13 and 14 are all right angles Definition of Congruent Angles if A = B, then mA = mB Definition of Angle Bisector Ex: if UV bisects TUW, then mTUV = mVUW or TUV = VUW Use the following picture for all theorems involving parallel lines: Corresponding Angles Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. Ex: in the picture 1 = 5, 2 = 6, 3 = 7, 4 = 8 Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Ex: in the picture 3 = 6 and 4 = 5 Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Ex: in the picture m3 + m5 = 180 and m4 + m6 =180 Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Ex: in the picture 1 = 8 and 2 = 7 Perpendicular Transversal Theorem If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. Ex: if a b and b c, then a c The following theorems are all converses of those on the other side of this page. The same picture is used. Corresponding Angles Converse If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. Ex: in the picture if 1 = 5, or 2 = 6, or 3 = 7, or 4 = 8, then m n Alternate Interior Angles Converse If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. Ex: in the picture if 3 = 6 or 4 = 5, then m n Consecutive Interior Angles Converse If two lines are cut by a transversal and consecutive interior angles are supplementary, then the lines are parallel. Ex: if m3 + m5 = 180 and m4 + m6 =180, then m n Alternate Exterior Angles Converse If two lines are cut by a transversal and alternate exterior angles are congruent, then the lines are parallel. Ex: in the picture if 1 = 8 or 2 = 7, then m n Perpendicular Transversal Converse In a plane, if two lines are perpendicular to the same line, then they are parallel. Ex: if a b and a c, then b c