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Transcript
REASONS for your Proofs gathered in one convenient location through Section 2.8 Reasons for Properties of Equality ONLY Addition Subtraction Multiplication Division Distributive Segment Addition Postulate Angle Addition Postulate If a = b, then a + c = b + c If AB = XY and BC = YZ then AB + BC = XY + YZ If ๐โก๐จ=๐โก๐ฟ and ๐โก๐ฉ=๐โก๐, ๐โก๐จ + ๐โก๐ฉ=๐โก๐ฟ + ๐โก๐ If a = b, then a - c = b - c If AB = XY and BC = YZ then AB BC = XY - YZ If ๐โก๐จ=๐โก๐ฟ and ๐โก๐ฉ=๐โก๐, ๐โก๐จ โ ๐โก๐ฉ=๐โก๐ฟ โ ๐โก๐ If a = b, then ac = bc If x = y, then 12x = 12y If a = b, the a/c = b/c If a = b and c = d, then a/c = b/d If m = n, then m/3 = n/3 a(b+c) = ab + ac w(x+y+z) = wx + wy + wz 5(3x+2y-6z) = 15x + 10y - 30z If B is between A and C, then AC = AB + BC If D is in the interior of โก๐จ๐ฉ๐ช, then ๐โก๐จ๐ฉ๐ช = ๐โก๐จ๐ฉ๐ซ + ๐โก๐ฉ๐ซ๐ช You can add the same value to both sides! (it could be the exact same, or two different (segment, angles, variables) that are equal! Exactly the same in nature as addition, but with a subtraction sign! You can multiply the same value to both sides of an equation! You can divide the same value into both sides of an equation! If a value is in front of a set of parenthesis, you can โdistributeโ or multiply that value to EVERY term inside the parenthesis! A whole segment will always equal the sum of its two parts! An angle will always equal the sum of its two parts! Reasons that work for BOTH properties of Equality AND Congruence Definition of Congruence ฬ ฬ ฬ ฬ , then AB = CD. ฬ ฬ ฬ ฬ โ ๐ช๐ซ If ๐จ๐ฉ If ๐โก๐จ = ๐โก๐ฉ, then โก๐จ โ โก๐ฉ Reflexive a = a, AB = AB, ฬ ฬ ฬ ฬ ๐จ๐ฉ โ ฬ ฬ ฬ ฬ ๐จ๐ฉ , โก๐จ โ โก๐จ, ๐โก๐จ = ๐โก๐จ, Symmetric If a = b, then b = a If ๐โก๐จ๐ฉ๐ช=๐โก๐ฟ๐๐, then ๐โก๐ฟ๐๐=๐โก๐จ๐ฉ๐ช ฬ ฬ ฬ ฬ , then ๐ช๐ซ ฬ ฬ ฬ ฬ โ ๐จ๐ฉ ฬ ฬ ฬ ฬ โ ๐ช๐ซ ฬ ฬ ฬ ฬ If ๐จ๐ฉ 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 โproveโ statement look exactly the same. Transitive Substitution If a = b and b = c, then a = c If AB = BC and BC = CD, then AB = CD If โก๐จ โ โก๐ฉ and โก๐ช โ โก๐ซ, then โก๐จ โ โก๐ซ 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 ๐โก๐จ๐ฉ๐ช = ๐โก๐จ๐ฉ๐ซ + ๐โก๐ฉ๐ซ๐ช terms OR to combine like terms and ๐โก๐ฟ๐๐ = ๐โก๐จ๐ฉ๐ซ, then on one side of the equality. ๐โก๐จ๐ฉ๐ช = ๐โก๐ฟ๐๐ + ๐โก๐ฉ๐ซ๐ช If AC = AB + BC and AB = BC, then AC = 2AB If 2x โ x = 5 + y + 2, then x = 7 + y Reasons for Angle Proofs Definition of Supplementary โกโs Definition of Complementary โกโs Supplement Theorem Complement Theorem Theorem 2.6 Theorem 2.7 Definition of Angle Bisector Angle Bisector Theorem Angle Addition Postulate ๐โก๐ + ๐โก๐ = ๐๐๐ ๐โก๐ + ๐โก๐ = ๐๐ โก๐ ๐๐๐ โก๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐ Supp. Angles add to 180 Comp. Angles add to 90 If angles 1 and 2 form a linear pair in a picture, then 1 and 2 are supplementary! โก๐ ๐๐๐ โก๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐ If angles 1 and 2 are adjacent angles whose noncommon sides are perpendicular, then 1 and 2 are complementary! โฒ โก ๐ supplementary to โ โกโฒ๐ are โ If a set of angles is supplementary to the same angle or two congruent angles, then those angles must be congruent to each other! โกโฒ ๐ complementary to โ โกโฒ๐ are โ If a set of angles is complementary to the same angle or two congruent angles, then those angles must be congruent to each other! โโโโโโ bisects โก๐จ๐ฉ๐ช, then An angle bisector cuts an angle If ๐ฉ๐ซ into two angles of equal measure ๐โก๐จ๐ฉ๐ซ = ๐โก๐ซ๐ฉ๐ช If โโโโโโ ๐ฉ๐ซ bisects โก๐จ๐ฉ๐ช, then โก๐จ๐ฉ๐ซ โ An angle bisector cuts an angle into two congruent angles! โก๐ซ๐ฉ๐ช If D is in the interior of โก๐จ๐ฉ๐ช, then An angle will always equal the ๐โก๐จ๐ฉ๐ช = ๐โก๐จ๐ฉ๐ซ + ๐โก๐ฉ๐ซ๐ช sum of its two parts! Reasons about Perpendicular Lines 2.9 Perpendicular lines intersect to form four right angles 2.10 All right angles are congruent 2.11 Perpendicular lines form congruent adjacent angles 2.12 If two angles are congruent and supplementary, then each angle is a right angle 2.13 If two congruent angles form a linear pair, then they are right angles If ฬ ฬ ฬ ฬ ๐จ๐ฉ โฅ ฬ ฬ ฬ ฬ ๐ช๐ซ and they intersect at point E, then โก๐จ๐ฌ๐ช, โก๐ช๐ฌ๐ฉ, โก๐ฉ๐ฌ๐ซ, โก๐ซ๐ฌ๐จ are all right angles! If โก๐จ๐ฉ๐ช ๐๐๐ โก๐ฟ๐๐ are both right, then โก๐จ๐ฉ๐ช โ โก๐ฟ๐๐! 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! A C E D B 90 + 90 = 180 X + Y = 180 2X = 180 X = 90 Some Basic Point, Line, Plane Postulates 2.1 Through any two points, there is exactly one line AB = BA 2.2 Through any three points not on the same line, there is exactly one plane. 2.3 A line contains at least two points. 2.4 A plane contains at least three points not on the same line. 2.5 If two points lie in a plane, then the entire line containing those points lies in that plane. Any 3 noncollinear points define a plane! Used to switch order of points on a line Used for adding points onto a line if necessary Used for adding points onto a plane if necessary If โกโโโโ ๐ญ๐ฎ is on plane P, then points H, I, and J are on plane P as well! J G I H F P 2.6 If two lines intersect, then their intersection is exactly one point. 2.7 If two planes intersect, then their intersection is a line. โกโโโโ then โกโโโโ intersects ๐ฉ๐ช If ๐จ๐ฉ they MUST intersect at point B. ALWAYS list a LINE as an intersection of two planes!