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Transcript
Proof Tool Box - SEGMENTS Midpoint: A point bisects a segment into two congruent segments iff it is the midpoint. Angle Addition Postulate: mABD mDBC mABC Segment Addition Postulate: AB + BC = AC iff B is between A and C. Angle Bisector: A ray that divides an angle into two congruent angles. Segment Bisector: A segment, ray, line, or plane that intersects a plane at its midpoint. Congruent Angles: Two angles are congruent iff their measures are equal. - ALGEBRA PROPERTIES - Complimentary Angles: Two angles are complimentary iff the sum of their measures is 90 degrees. Addition Property: If a = b, then a + c = b + c Subtraction Property: If a = b, then a – c = b – c Multiplication Property: If a = b, then ac = bc Division Property: If a = b, then a/c = b/c Reflexive Propery: a=a Symmetric Property: If a = b, then b = a - ANGLES - Congruent Compliments Theorem: If two angles are complimentary to the same angle, then they are congruent. Ex) If 1 and 2 are complementary and 2 and 3 are complementary, then 1 3 . Supplementary Angles: Two angles are supplementary iff the sum of their measures is 180 degrees. Congruent Supplements Theorem: If two angles are supplementary to the same angle, then they are congruent. Acute Angle: An angle is acute iff its measure is less than 90 degrees. Right Angle: An angle is a right angle iff its measure is 90 degrees. Transitive Property: If a = b and b = c, then a = c. Obtuse Angle: An angle is obtuse iff its measure is greater than 90 degrees. Straight Angle: An angle is straight iff its measure is 180 degrees. Substitution Property: If a = b, then you can substitute a in for b. - INTERSECTING LINES Vertical Angles Theorem: Vertical angles are congruent. Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. MORE ON THE NEXT SIDE… Proof Tool Box - POINTS / LINES / PLANES Through any two points there exists exactly one line. A line contains at least 2 points If two lines intersect, then their intersection is exactly one point Through any 3 non-collinear points, there exists 1 plane A plane contains at least 3 non-collinear points If 2 points lie on the same line, them lies in the same plane. If 2 planes intersect, then their intersection is a line. - PERPENDICULAR Perpendicular Lines: Two lines are perpendicular iff they intersect to form a right angle. Perpendicular Lines Cut By Transversal Perpendicular Transversal Thm: If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. **If 2 lines are perpendicular to the same line, then they are parallel to each other. - PARALLEL Parallel Lines: Two lines are parallel iff they do not intersect and are coplanar. Parallel Lines Cut By Transversal Corresponding Angles Postulate: If parallel lines are cut by a transversal, then corresponding angles are congruent. Alternate Interior Angles Thm: If parallel lines are cut by a transversal, then alternate interior angles are congruent. Consecutive Interior Angles Thm: If parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Alternate Exterior Angles Thm: If parallel lines are cut by a transversal, then alternate exterior angles are congruent. **The converse of each is also true. **Do not remember these by their name…. remember them by their definition. **If 2 lines are parallel to the same line, then they are parallel to each other. **If 2 lines are perpendicular to the same line, then they are parallel to each other.