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Postulates: 1) Reflexive Property of Equality/ Congruence - a=a 2) Symmetric Property of Equality/ Congruence - If a = b, then b = a. 3) Transitive Property of Equality/ Congruence - If a = b, and b = c, then a = c. 4) Substitution Property of Equality/ Congruence - If ab = 2ac, and ad = ac, then ab = 2ad. 5) Addition Property (Theorem) of Equality (Congruence) - If a = b, and c = d, then a+c = b+d. 6) Subtraction Property (Theorem) of Equality (Congruence) - If a = b, and c = d, then a-c = b-d. 7) Multiplication Property (Theorem) of Equality (Congruence) - If a = b, and c = d, then ac = bd. 8) Division Property (Theorem) of Equality (Congruence) - If a = b, and c = d, then a/c = b/d, where c ≠ 0 and d ≠ 0. Inequality Postulates: 1) Addition/ Subtraction Property of Inequality - If a > b, then a+x > b+x. If a > b, then a-x > b-x. - If a < b, then a+x < b+x. If a < b, then a-x < b-x. 2) Positive Multiplication/ Division Property of Inequality - If x < y, and a > 0, then ax < ay. If x > y, and a > 0, then ax > ay. - If x > y, and a > 0, then x/a > y/a. If x < y, and a > 0, then x/a < y/a. 3) Negative Multiplication/ Division Property of Inequality - If x < y, and a < 0, then ax > ay. If x > y, and a < 0, then ax < ay. - If x > y, and a < 0, then x/a < y/a. If x < y, and a < 0, then x/a > y/a. 4) Substitution Property of Inequality - If a > b, and b = c, then a > c. - If a < b, and b = c, then a < c. 5) Transitive Property of Inequality - If a > b, and b > c, then a > c. - If a < b, and b < c, then a < c. 6) A whole is greater than any of its parts. 7) Law of Trichotomy: Either a < b, a = b, or a > b. 8) The shortest path (distance) between a point and a line is the perpendicular segment from the point to the line. * Most inequality proofs are numerical. Euclid’s 5th Postulate (The Parallel Postulate): Given a line and a point not on the line, there exists one and only one (unique) line through the given point that is parallel to the given line. Theorems: 1) If 2 angles are right angles, then they are congruent. 2) If 2 angles are complementary (supplementary) to congruent angles/ the same angle, then these angles are congruent. 3) If 2 adjacent angles form a right angle, then they are complementary. 4) If 2 angles form a linear pair, then they are supplementary. * Never say: If a line segment is a segment bisector, then it divides the segment into 2 congruent parts. - Prove that point __ is the midpoint of the segment. - Use the midpoint theorem to prove that it divides the segment into 2 congruent parts. 5) If 2 adjacent angles have noncommon sides that are opposite rays, then they are a linear pair of angles. 6) If 2 angles are congruent to congruent/ the same angle, then they are congruent. * Always use reflexive property to state what you are adding/subtracting: You need 2 pairs of congruent things. * Midpoints/ Bisectors: Think about utilising multiplication/ division theorem (property) of congruence (equality) * Multiplication (Division) Theorem of Congruence If segments or angles are congruent, then their multiples (divisions) are congruent. 7) If 2 angles’ sides form 2 pairs of opposite rays, then they are vertical angles. 8) If 2 sides of a ∆are congruent, then the angles opposite these sides are also congruent. 9) If 2 angles of a ∆are congruent, then the sides opposite these angles are also congruent. (Converse of 8) 10) If a segment is a median of a ∆, then it is drawn from any vertex of the ∆to the midpoint of the opposite side. 11) If a segment is drawn from any vertex of a ∆to the midpoint of the opposite side, then it is a median of the ∆. (Converse of 10) 12) If a segment is the altitude of a ∆, then it is drawn from any vertex of the ∆perpendicular to the line containing the opposite side. 13) If a segment is drawn from any vertex of a ∆perpendicular to the line containing the opposite side, then it is an altitude of the ∆. (Converse of 12) * Must say: the line containing the opposite side . An altitude could fall outside the ∆. * Essentially, a perpendicular bisector is both the median and the altitude. Isosceles ∆s: 14) If a ∆has at least 2 congruent sides, then it is isosceles. 15) The median to the base, the altitude to the base, and the bisector of the vertex angle in an isosceles ∆are all the same segment. 16) If a ∆is equilateral, then it is equiangular. 17) If a ∆is equiangular, then it is equilateral. (Converse of 16) * All equilateral ∆s are isosceles, but not all isosceles ∆s are equilateral. 3 Equidistance Theorems: 18) If 2 points are each equidistant from the endpoints of a segment, then they will determine the perpendicular bisector of that segment. * ^: used only to establish perpendicular bisector * Equidistant = congruent. No need to prove that segments are equal in measure in order to say that they are equidistant. 19) If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. 20) If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. (Converse of 19) Circles: 21) If a segment is a radius, then it is drawn from the centre of a circle to any point on the circle. 22) If a segment is drawn from the centre of a circle to any point on the circle, then it is a radius. (Converse of 21) 23) If 2 segments are of the same circle/ congruent circles, then they are congruent. Proof by Contradiction/ Indirect Proof: 24) Law of Excluded Middle: Either p or ~p is true. No other possibilities exist. 25) Law of Contradiction: Both p and ~p cannot be true at the same time. * Format: - Step 1: Negation of what you are trying to prove: Law of Excluded Middle - Step 2: Assumption of negation of given: Assumption - Last step: Law of contradiction in steps __ and __. Assumption in step 2 is false. Negation of assumption must be true. Inequality Proofs: 26) The measure of the exterior angles of a ∆is greater than the measure of either remote interior angle. 27) If 2 angles of a ∆are unequal, then the sides opposite them are unequal and the longer side is opposite the larger angle. 28) If 2 sides of a ∆are unequal, then the angles opposite them are unequal and the larger angle is opposite the longer side. (Converse of 28) 29) The sum of any 2 side lengths of a ∆is greater than the length of the third side. Parallel Lines: 30) If 2 parallel lines are cut by a transversal, then each pair of: - alternate interior angles/ alternate exterior angles/ corresponding angles is congruent - same side interior angles/ same side exterior angles is supplementary 31) If 2 coplanar lines are cut by a transversal such that a pair of: - alternate interior angles/ alternate exterior angles/ corresponding angles is congruent - same side interior angles/ same side exterior angles is supplementary , then the lines are parallel. (Converse of 30) 32) If 2 lines are parallel to a third line, then they are parallel to each other. 33) If 2 coplanar lines are perpendicular to the same line, then they are parallel. ∆Angle-Sum Theorems: 34) The sum of the measures of the interior angles of a ∆is 180°. 35) The acute angles of a right ∆are complementary. 36) No Choice Theorem: If 2 angles of one ∆are congruent to 2 angles of another ∆, then the third pair of angles is congruent. 37) The measure of an exterior angle of a ∆is equal to the sum of the measures of its remote interior angles. ∆Congruence: 38) If 2 ∆s are congruent, then all pairs of corresponding parts are congruent. * No SSA ∆congruence! 39) SAS Postulate : 2 sides + included angle of one ∆is congruent to corresponding sides and angle of another ∆. 40) SSS Theorem : All 3 sides of one ∆is congruent to corresponding sides of another ∆. 41) ASA Theorem : 2 angles + included side of one ∆is congruent to corresponding angles and side of another ∆. 42) AAA Theorem : All 3 angles of one ∆is congruent to corresponding angles of another ∆. 43) HL Theorem - needs 3 things: 1. right ∆(not just right angle, you have to state that it is a right ∆), 2. Hypotenuses (side opposite right angle) are congruent, 3. One pair of legs (sides adjacent to right angle) are congruent. * no need to state that right angles are congruent 44) If a ∆has a right angle, then it is a right ∆. * The Transitive/ Substitution Property can apply to figures (polygons). Polygons: 45) The sum of the measures of the interior angles of a n-gon is: (n-2) x 180°. 46) The sum of the measures of the exterior angles of a n-gon is: 360°. 47) The measure of each interior angle of an equiangular n-gon is: (n-2) x 180°/n. 48) The measure of each exterior angle of an equiangular n-gon is: 360°/n. Parallelograms: 49) Definition : A quadrilateral is a parallelogram if and only if both pairs of opposite sides are parallel. 50) Properties : If a quadrilateral is a parallelogram, then: - both pairs of opposite sides/ opposite angles are congruent, - any pair of consecutive angles are supplementary, - the diagonals bisect each other . 51) Proving a polygon is a parallelogram : If: - both pairs of opposite sides of a quadrilateral are parallel (converse definition), - one pair of opposite sides of a quadrilateral is both parallel and congruent, - both pairs of opposite sides/ opposite angles of a quadrilateral are congruent, - the diagonals of a quadrilateral bisect each other, - an angle of a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram. Rectangle: 52) Definition : A quadrilateral is a rectangle if and only if it is a parallelogram with at least one right angle. 53) Properties : If a quadrilateral is a rectangle, then: - it has all the properties of a parallelogram, - it is equiangular , - the diagonals are congruent . 54) Proving a polygon is a rectangle : If: - a quadrilateral is a parallelogram with at least one right angle (converse definition), - a quadrilateral is equiangular, - a quadrilateral is a parallelogram with congruent diagonals, then it is a rectangle. Rhombus: 55) Definition : A quadrilateral is a rectangle if and only if it is a parallelogram with at least 2 congruent consecutive sides. 56) Properties : If a quadrilateral is a rhombus, then: - it has all the properties of a parallelogram, - it is equilateral , - the diagonals are perpendicular , - each diagonal bisects a pair of opposite angles. 57) Proving a polygon is a rectangle : If: - a quadrilateral is a parallelogram with at least 2 congruent consecutive sides (converse definition), - a quadrilateral is a parallelogram whose diagonals are perpendicular, - a quadrilateral is a parallelogram whose diagonals bisect a pair of opposite angles, - a quadrilateral whose diagonals are perpendicular bisectors of each other, - a quadrilateral is equilateral, then it is a rhombus. Square: 58) Definition : A quadrilateral is a square if and only if it is both a rectangle and a rhombus. 59) Properties: If a quadrilateral is a square, then: - it has all the properties of a rectangle, - it has all the properties of a rhombus. (and thus has all the properties of a parallelogram) 60) Proving a polygon is a square : If a quadrilateral is both a rectangle and a rhombus, then it is a square (converse definition). Trapezoid: 61) Definition : A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides. 62) Prove a polygon is a trapezoid : If a quadrilateral has exactly one pair of opposite sides that is parallel, then it is a trapezoid (converse definition). * parallel sides = bases; non-parallel sides = legs. Isosceles Trapezoid: 63) Definition : A quadrilateral is an isosceles trapezoid if and only if it is a trapezoid in which the non-parallel sides/ legs are congruent. 64) Properties : If a quadrilateral is an isosceles trapezoid, then: - the lower base angles are congruent, - the upper base angles are congruent, - the diagonals are congruent . 65) Prove a polygon is an isosceles trapezoid : If: - a quadrilateral is a trapezoid with non-parallel sides/ legs that are congruent (converse definition), - a quadrilateral is a trapezoid with one pair of upper/lower base angles that is congruent, - a quadrilateral is a trapezoid with congruent diagonals, then it is an isosceles trapezoid. Kite: 66) Definition : A quadrilateral is a kite if and only if it has 2 disjoint pairs of consecutive sides that are congruent. 67) Properties : If a quadrilateral is a kite, then: - one of the diagonals is the perpendicular bisector of the other , - the diagonals are perpendicular , - one of the diagonals bisects a pair of opposite angles . 68) Prove a polygon is a kite : If: - a quadrilateral has 2 disjoint pairs of consecutive sides that are congruent (converse definition), - a quadrilateral has 1 diagonal that is the perpendicular bisector of the other diagonal, then it is a kite. Ratios and Proportions: Ratio = quotient of 2 numbers Proportion = an equation stating that 2 or more ratios are equal Mean proportion = proportion in which the means are equal 69) In a proportion, the product of the means is equal to the product of the extremes. (Cross Multiplication) 70) In a proportion, the means can be interchanged. 71) In a proportion, the extremes can be interchanged. 72) Midsegment (Midline) Theorem (of a ∆): A segment joining the midpoints of 2 sides of a ∆(called the midsegment) is parallel to the third side and is ½ the length of the 3rd side. 73) Midsegment (Midline) Theorem (of a trapezoid): The median (midsegment) of a trapezoid is parallel to the bases, and its length is ½ the sum of the lengths of the 2 bases. * Format of 72: A segment joining the midpoints of 2 sides of a ∆is - parallel to the 3rd side, OR - ½ the length of the 3rd side. * Format of 73: - The median of a trapezoid is parallel to the bases, OR - The length of the median of a trapezoid is ½ the sum of the lengths of the 2 bases. 74) Side-Splitter Theorem: If a line is parallel to 1 side of a ∆and intersects the other 2 sides, then it divides those 2 sides proportionally. 75) If 3 or more parallel lines are intersected by 2 transversals, then the parallel lines divide the transversals proportionally. (no need for similar ∆s) 76) Angle-Bisector Theorem: If a ray bisects an angle of a ∆, then it divides the opposite side into segments that are proportional to the adjacent sides. Right ∆s: 77) Right ∆s, Hypotenuse, Altitude: If an altitude is drawn to the hypotenuse of a right ∆, then: a) the 2 ∆s formed are similar to the given right ∆and to each other, b) either leg of the given ∆is the mean proportional between the hypotenuse of the given ∆and the segment of the hypotenuse adjacent to that leg, c) the altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse. * b) hypotenuse/leg 1 = leg 1/ adj. seg. 1 OR hypotenuse/leg 2 = leg 2/ adj. seg. 2 c) adj. seg. 1/altitude = altitude/adj. seg. 2 78) Pythagorean Theorem: The square of the measure of the hypotenuse of a right ∆is = to the sum of the squares of the measures of the legs. Similar Polygons: 79) Ratio of Similitude/ Constant of Proportionality: The ratio of the perimeters of 2 similar polygons is equal to the ratio of any pair of corresponding sides. 80) If 2 polygons are similar, then: - all pairs of corresponding angles are congruent, - the corresponding sides are proportional. Proving ∆s Similar: 81) AA~ Theorem: 2 angles of one ∆are congruent to corresponding angles of another ∆. * All congruent ∆s are similar, but not all similar ∆s are congruent 82) SAS~ Theorem: 2 sides of one ∆is proportional to the corresponding sides of another ∆+ included angle of 1st ∆is congruent to 2nd ∆. 83) SSS ~ Theorem: All 3 sides of one ∆is proportional to corresponding sides of another ∆. * Drawings are always dotted Miscellaneous: 84) If 2 segments intersect each other at their midpoints, then they bisect each other. 85) If 2 segments bisect each other, then they intersect each other at their midpoints. (Converse of 83)