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Transcript
Date _________ Period_________ A Word Bank of Possible Reasons in Proofs Definitions, Postulates, Properties and Theorems – and the Pictures Statement in Proof (Example) A AB XY ABX & ABY are right angles Definition of perpendicular lines X mABX = 90° mABY = 90° Y B BD bisects ABC Definition of right angles A D ABD CBD or B Reason Given in Proof 1 2 C Definition of an angle bisector 1 2 M is the midpoint of AB M B A AM MB Definition of a midpoint m1 m2 180 Angle addition (postulate) (just the picture) 1 2 (just the picture) 1 3 1 or 2 4 3 2 4 Vertical angles theorem (VAC) m1 m2 m3 and m2 m3 m4 so m1 m4 1 2 2 3 so… 1 3 Any time you “plug in” angle measures or side lengths for other angles or side lengths you are using Substitution. This only works with =, not . The Transitive property is like substitution but only when it fits the pattern like the one shown (can use with .) Any time 2 triangles share a side. C (both white and grey triangles share side CD) CD CD Substitution Transitive Property Reflexive Property D AB YZ so… YZ AB DE = LM so… DE + XY = LM + XY You can flip sides of an equation.. You can add a length to an equation just like you can add a number to an equation in algebra Symmetric Property Addition Property (added XY to both sides) You can also subtract a length from an equation the same way… DE + XY = LM + XY so… DE = LM Subtraction Property (subtracted XY from both sides) 6x 30 so… x5 There are also multiplication and division properties, but they don’t work the same way for lengths. Use them for algebra. Division Property (just the picture) A B C AB BC AC (the two pieces of the segment make the entire segment) A (just the picture) m1 m2 mABC D 1 B Segment Addition (postulate) or mABD mDBC mABC 2 C Angle Addition Postulate Y B A C SSS Postulate ABC XYZ SAS Postulate ABC XYZ ASA Postulate ABC XYZ AAS Theorem ABC XYZ HL Theorem (Hypotenuse-Leg) Z X B ABC XYZ Y A X C Z B Y A C X Z B Y A C X B Z Y C A Z X Once you say that ABC XYZ by any of the 5 methods, then… AB XY A X B Y A C X Z B Y & BC YZ C Z AC XZ CPCTC Corresponding Parts of Congruent Triangles are Congruent 2 m1 m2 m3 180 1 Triangle Sum Theorem (3 angles add up to 180°) 3 2 1 1 2 If two angles of one triangle are congruent to two angles of another, then the 3rd angles are congruent too. C A B Isosceles Triangle Theorem XZ YZ Converse of the Isosceles Triangle Theorem 1 2 Corresponding Angles Postulate 3 4 Alternate Interior Angles Theorem 5 6 Alternate Exterior Angles Theorem B A Z X Y (lines are parallel in) this and the next 4) 1 2 3 4 5 6 7 m7 m8 180 8 Same Side Interior Angles Theorem 9 m9 m10 180 10 Same Side Exterior Angles Theorem l m and l n l m n mn If two lines are perpendicular to the same line, then those two lines are parallel