MIDTERM DEFINITIONS THEOREM ETC
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CHAPTER ONE: CHAPTER TWO Properties of Equality: Segment Addition Postulate Angle Addition Postulate Definition of Midpoint Definition of Angle Bisector If B is between A and C, then AC = AB + BC If D is in the interior of β‘π¨π©πͺ, then πβ‘π¨π©πͺ = πβ‘π¨π©π« + πβ‘π©π«πͺ If B is the midpoint of AC, then AB = BC ββββββ bisects β‘π¨π©πͺ, then If π©π« πβ‘π¨π©π« = πβ‘π«π©πͺ Definition of Segment Bisector D B C A E A whole segment will always equal the sum of its two parts! An angle will always equal the sum of its two parts! If there is a point in the middle of a segment, it cuts it into two equal parts! An angle bisector cuts an angle into two angles of equal measure If a ray, segment, or line bisects a segment, it cuts through the midpoint, thus creating two equal parts! Μ Μ Μ Μ , then AB = BC β‘ββββ bisects π¨πͺ If π«π¬ Reasons that work for BOTH properties of Equality AND Congruence Definition of Congruence If Μ Μ Μ Μ π¨π© ≅ Μ Μ Μ Μ πͺπ«, then AB = CD. If πβ‘π¨ = πβ‘π©, then β‘π¨ ≅ β‘π© Reflexive a = a, AB = AB, Μ Μ Μ Μ π¨π© ≅ Μ Μ Μ Μ π¨π© , β‘π¨ ≅ β‘π¨, πβ‘π¨ = πβ‘π¨, Symmetric If a = b, then b = a If πβ‘π¨π©πͺ=πβ‘πΏππ, then πβ‘πΏππ=πβ‘π¨π©πͺ If the segments are congruent, then their measures are equal! (and vice versa!) Can be applied to angles as well! A value will always equal itself! On overlapping segments and angles when we need to add the same thing to both sides we use this reason first. Basically, this is just for switching sides of an equation if necessary to make the “prove” statement Transitive Substitution Midpoint Theorem If Μ Μ Μ Μ π¨π© ≅ Μ Μ Μ Μ πͺπ«, then Μ Μ Μ Μ πͺπ« ≅ Μ Μ Μ Μ π¨π© If a = b and b = c, then a = c If AB = BC and BC = CD, then AB = CD If β‘π¨ ≅ β‘π© and β‘πͺ ≅ β‘π«, then β‘π¨ ≅ β‘π« look exactly the same. Whenever you have a connecting piece that is congruent or equal to two other things, then those two things are congruent/equal to each other! If you are the same height as your friend Joe, and Joe is the same height as his friend Sue, then YOU and SUE are the same height! (Joe is the connecting piece) If a = b and b=2, then a = 2 Substitution is used to replace like If πβ‘π¨π©πͺ = πβ‘π¨π©π« + πβ‘π©π«πͺ terms OR to combine like terms and πβ‘πΏππ = πβ‘π¨π©π«, then on one side of the equality. πβ‘π¨π©πͺ = πβ‘πΏππ + πβ‘π©π«πͺ If AC = AB + BC and AB = BC, then AC = 2AB If 2x – x = 5 + y + 2, then x = 7 + y A midpoint cuts a segment into If B is the midpoint of Μ Μ Μ Μ π¨πͺ, then Μ Μ Μ Μ Μ Μ Μ Μ two congruent segments! π¨π© ≅ π©πͺ Reasons for Angle Proofs Definition of Supplementary β‘’s Definition of Complementary β‘’s Supplement Theorem Complement Theorem Congruent Supplements Theorem Congruent Complements Theorem If πβ‘π + πβ‘π = πππ , then β‘π πππ β‘π are supplementary πβ‘π + πβ‘π = ππ, then β‘π πππ β‘π are complementary β‘π πππ β‘π πππ πππππππππππππ Suppl. Angles add to 180 Compl. Angles add to 90 If angles 1 and 2 form a linear pair in a picture, then 1 and 2 are supplementary! β‘π πππ β‘π πππ πππππππππππππ If angles 1 and 2 are adjacent angles whose noncommon sides are perpendicular, then 1 and 2 are complementary! ′ β‘ π supplementary to ≅ β‘′π are ≅ If a set of angles is supplementary to the same angle or two ≅ πππππ. πππ congruent angles, then those angles must be congruent to each other! β‘′ π complementary to ≅ β‘′ π are ≅ If a set of angles is complementary to the same ≅ πππππ. πππ angle or two congruent angles, then those angles must be congruent to each other! Definition of Angle Bisector Angle Bisector Theorem Angle Addition Postulate Vertical Angle Theorem If ββββββ π©π« bisects β‘π¨π©πͺ, then πβ‘π¨π©π« = πβ‘π«π©πͺ If ββββββ π©π« bisects β‘π¨π©πͺ, then β‘π¨π©π« ≅ β‘π«π©πͺ If D is in the interior of β‘π¨π©πͺ, then πβ‘π¨π©πͺ = πβ‘π¨π©π« + πβ‘π©π«πͺ β‘π¨π¬πͺ ≅ β‘π«π¬π© A β‘π¨π¬π« ≅ β‘πͺπ¬π© Definition of Linear Pairs C E D B Adjacent angles whose noncommon sides are opposite rays. An angle bisector cuts an angle into two angles of equal measure An angle bisector cuts an angle into two congruent angles! An angle will always equal the sum of its two parts! Vertical angles are always congruent! They are your “bowtie angles” We take linear pairs right off pictures and use the Supplement Theorem to say they are supplementary! In the picture above, β‘π¨π¬πͺ πππ β‘πͺπ¬π© are linear pairs Reasons about Perpendicular Lines and Right Angles Definition of a Right Angle Definition of Perpendicular Lines Perpendicular lines intersect to form four right angles All right angles are congruent Perpendicular lines form congruent adjacent angles If two angles are congruent and supplementary, then each angle is a right angle If two congruent angles form a linear pair, then they are right angles If β‘π¨π©πͺ is a right angle, then πβ‘π¨π©πͺ = ππ Lines that intersect to form right angles If Μ Μ Μ Μ π¨π© ⊥ Μ Μ Μ Μ πͺπ« and they intersect at point E, then β‘π¨π¬πͺ, β‘πͺπ¬π©, β‘π©π¬π«, β‘π«π¬π¨ are all right angles! If β‘π¨π©πͺ πππ β‘πΏππ are both right, then β‘π¨π©πͺ ≅ β‘πΏππ! Of course they do!! They are all right angles! The only combination of two congruent, supplementary angles are 90 and 90. Linear pairs are supplementary. So if both angles are congruent as well, they have to be 90 each! Right angles measure 90 degrees Use this, or the reason below to talk about the right angles formed from the perp. Lines. A C E D B 90 + 90 = 180 X + Y = 180 2X = 180 X = 90 CHAPTER THREE SLOPE INTERCEPT FORM CHAPTER FOUR CHAPTER FIVE
