Geometry Note-Sheet Overview

Transcription

1 Geometry Note-Sheet Overview 1. Logic a. A mathematical sentence is a sentence that states a fact or contains a complete idea. Open sentence it is blue x+3 Contains variables Cannot assign a truth variable Closed sentence the sky is blue 2+3 Can be either true or false No variables b. Negation: to cancel out. Symbol: ~ An odd number of negations is false, an even number (where no negations is true) is true c. Conjunction: Operation using and. Symbol: ^ Conjunctions are only true when both p and q are true d. Disjunction: Operation using or Symbol: v Disjunctions are only false when both p and q are false e. A tautology is a compound statement that is always completely true. f. A contradiction is a compound statement that is always completely false. g. Conditional: A premise and conclusion, using if, then Symbol: -> A conditional is only false when a true hypothesis leads to a false conclusion Hidden conditional: Instead of if then, it is flipped: you can assemble the bike if you follow the directions. Special/Related Conditional Statements Original: p -> q Converse: q -> p Inverse: ~p -> ~q Contrapositive: ~q -> ~p Logical Equivalence A conditional and its contrapositive are logically equivalent The inverse is the contrapositive of the converse, so they are logically equivalent Biconditionals A statement formed by the conjunction of the conditionals p -> q and q -> p if and only if iff (p->q) ^ (q->p) : p <-> q A biconditional is only true when both single statements have the same truth value. If two statements are logically equivalent, their biconditional is a tautology (and vice versa)

2 To prove that two statements are logically equivalent, make a truth table with their biconditional. h. Drawing Conclusions Draw a TRUE conclusion from a set of statements Therefore: Sometimes no conclusion can be drawn if there is not enough information If T or F is not given, assume statements are true i. Validity An argument is valid if and only if the premises are true and the conclusion is true, based on the premises. j. Logic Laws Law of contrapositive Conjuctive simplification DeMorgan s Law Law of Disjunctive Inference Law of Detachment Law of Modus Tollens Chain Law Negating a Conditional Law of Conjunction Disjunctive addition k. Direct proofs l. Indirect Proofs Proof by contradiction We assume opposite, and when we hit contradiction, then we say RAA 2. Intro to Geometry a. Point: a definite location in a plane or line, 0 dimensions Use capital letters:.a If two points are on the same line they are collinear If two points are on the same plane they are coplanar b. Line: A continuous extent of length, 1 dimension You can name line l or AB

3 Two lines intersect at a point Parallel lines are on the same plane but don t intersect Skew lines are not on the same plane and don t intersect The least number of points you need to define/create a line: 2 c. Plane: Shape/area, 2 dimensions You can name plane (capital letter) M or by using 3 points: plane ALC Two planes intersect at a line The least number of points you need to define/create a plane: 3 d. Euclidean Geometry e. Rays Parts of a line that contain an endpoint and continue forever in one direction. Note the ray with the endpoint first: JK Opposite rays are rays that share the same endpoint and continue in opposite directions f. Line segments Part of a line with two endpoints. g. Equality vs. Congruent statements h. Property of Betweeness (pg. 61,62) 3 collinear points, therefore one is between the others. If given AB=7, AC=12, BC=19, this can be true by either being collinear or not being collinear In order for a triangle to be true, sides with smaller lengths must add up to more than side with largest length i. Midpoint: divides line segment into 2 equal parts j. Bisector: Crosses line segment at midpoint k. Midpoint formula: x1 + x2, y1 + y2 2 2 l. Distance formula: D= (x2-x1) 2 + (y2-y1) 2 m. Angles Formed by two rays with the same endpoint Two sides, vertex Name angle <XYZ m<abt vs. <ABT Adjacent angles point must be on interior Angle bisector Acute Angle: below 90 degrees Obtuse angle: More than 90 less than 180 degrees Straight Angle: 180 degrees Right Angle: 90 degrees Reflex Angle: degrees n. Complementary Angles two angles add to 90 degrees o. Supplementary Angles Two angles add to 180 degrees

4 p. Vertical angles are equal q. Locus place that contains a property (pg. 89) 3. Geometry Proofs a. Postulates given information that doesn t need to be proven b. Theorem Proven statement; need to use postulates or previously proven theorems to prove c. Postulates of Equality- and some random reasons Reflexive Symmetric Transitive Addition POE Subtraction POE Multiplication POE Division POE Substitution Congruency vs. equality Bisector definition Midpoint definition Betweeness Angle addition postulate Definition of an angle bisector d. Parallel lines Don t intersect, same plane (symbol:, m n ) Segments and rays are parallel if they lie on parallel planes. A line is parallel to a plane if the line is in a plane parallel to the given plane. Through a point not on a line, there are infinite lines that go through that point. One of these lines is parallel to the given line. One is also perpendicular. Have the same slopes e. Transversal: a line that intersects two or more coplanar lines at different points. f. Different types of angles formed by two lines cut by a transversal. (if the two lines are, some of the angles are congruent. 1 2 a b Corresponding angles ( <1, <5) {if lines are parallel, <s are =} Alternate Interior angles (<4, <5) {if lines are parallel, <s are =} Alternate exterior angles (<2, <7) {if lines are parallel, <s are =} Same side interior angles (<3, <5) {if lines are parallel, <s are supplementary} Vertical angles (<1, <4) {always congruent} Linear Angles (<1, <2) {always supplementary} g. Conjecture: an unproven statement based on observations h. Angles supplementary to the same angle are equal

5 i. If two lines are parallel to the same line, the lines are parallel to eachother j. Converse postulates prove two lines are parallel. Converse Alt. Int, Alt. Ext, Corresponding k. Perpendicular Lines Form four right (90 ) angles Form two pairs of linear angles All right angles are equal If two exterior sides of two adjacent angles are perpendicular, then the angles are complementary If two lines are perpendicular to the same line, then the lines are parallel If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other. Have opposite reciprocal slopes Perpendicular Bisectors: Opposite reciprocal Goes through midpoint of given line. l. Slope formula: m = y2 y1 x2 x1 m. Different types of equations for lines Slope intercept form: y = mx + b Point slope form: y y1 = m(x x1) Standard form: ax + by = c n. Slopes of lines mean: m>0 : increasing m<0 : decreasing m = 0 : horizontal m = undefined : vertical o. y coordinate: ordinate; x coordinate: abscissa p. CONSTRUCTION: Copy of a line and angle Construction of bisectors and midpoints 4. Triangles a. A triangle is a polygon with three sides Made up of three vertexes and three sides Name triangle counter clockwise The sum of the measures of the interior angles of a triangle add to 180 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent Reflexive, Symmetric, Transitive Write the parts of the triangles in THE SAME ORDER b. Polygon closed figure composed of straight line segments c. Classify triangles by sides: Scalene: no congruent sides

6 Isosceles: At least two congruent sides Equilateral: 3 congruent sides d. Classify triangles by angles: Acute triangle: 3 acute angles Right triangle: 1 right angle Obtuse triangle: 1 obtuse angle Equiangular triangle: three equal angles e. Proving Triangle Congruency: All parts of one figure are congruent to the corresponding parts of another; same shape/same size Angles don t prove congruency Reasons to prove triangle congruency: SSS SAS ASA AAS RHL ASS/SSA doesn t work donkey rule unless it has a right angle, RHL f. Isosceles triangles Two congruent sides Angles opposite equal sides are equal (in an isosceles triangle) Sides opposite equal angles are equal (in an isosceles triange) g. Doubles of equals are equal h. Halves of equals are equal