Honors Geo God Doc Fall PDF

Written by Owen Ye ['27] Lesson 1 - Intro to Logic Definition: A proposition is a closed sentence that is true or false but not both. In logic, we can use letters (eg.: p, q, etc.) to represent propositions. Definition: An open sentence is a sentence whose truth cannot be determined, because it cont...

Written by Owen Ye ['27] Lesson 1 - Intro to Logic Definition: A proposition is a closed sentence that is true or false but not both. In logic, we can use letters (eg.: p, q, etc.) to represent propositions. Definition: An open sentence is a sentence whose truth cannot be determined, because it contains a variable element. -​ A question or command cannot be a proposition Basic Connectives: -​ Negation (~) -​ Conjunction (^) -​ Disjunction (v) -​ Conditional (->) Lesson 2 - Properties of Conjunctions/Disjunctions Associative Commutative Distributive DeMorgan’s Property Property Property Laws Conjunction (p^q)^r = p^q = q^p p^(qvr) = ~(p^q) = p^(q^r) (p^q)v(p^r) ~pv~q Disjunction (pvq)vr = pv(qvr) pvq = qvp pv(q^r) = ~(pvq) = (pvq)^(pvr) ~p ^ ~q Lesson 3 - Conditional Statements p = hypothesis/antecedent q = conclusion/consequent The inverse of a conditional statement negates the antecedent and consequent ~p->~q The converse reverses the order of the antecedent and consequent q->p The contrapositive negatives and reverses the order of the antecedent and the consequent. ~q->~p Lesson 4 - Biconditionals A biconditional is a conjunction of a condition and its converse (p->q) ^ (q->p) or p ↔ q Every definition is a biconditional A tautology is a statement that is always true A contradiction is a statement that is always false Lesson 5 - Laws of Logic Law of the excluded middle p or not p Law of Contradiction P and not p cannot both be true Law of double negation ~(~p) is the same as p Law of Simplification If p and q, then p Law of Disjunctive Addition If p, then p or q Law of Disjunctive Elimination If p or q and not q is true, then p must be true Law of Transposition If p then q is the same as if not q then not p (contrapositive) Law of modus ponens (detachment) If p then q and p, then q must be true Law of Modus Tollens If p then q and not q are true, then not p must be true Law of Syllogism If p then q and if q then r, then if p then r Lesson 6 - Writing Logic Proofs Two Column Proofs 1.​ Write the given info and what we want to prove 2.​ 2 columns: 1 for statements and 1 for reasons 3.​ Number each row 4.​ Introduce the given information when needed 5.​ Cite previous statements used to support new statements in parenthesis 6.​ After the final step, write a conclusion ∴ for “therefore” ________ Logic Proofs -​ Every statement in a logic proof should be true Lesson 7 - Introduction to Geometry Undefined Terms: A point has neither length nor width but indicates a position. We represent it with a dot. Points are denoted with capital letters. A line has no width and can be extended as far as desired in either direction. Lines are denoted by using two points on the line with a double-ended arrow above them or by using a script lower-case letter. A plane is a surface such that if any two points on the surface are connected by a line, all points of the line are also on the surface. Planes have infinite length and width but no depth. Planes are denoted by capital script letters. Defined Terms: If 3 or more points all belong to the same line, they are said to be collinear. If 3 or more lines contain the same point, they are said to be concurrent If point B is between A and C, then A, B, and C are distinct collinear points, and AB+BC=AC Points B and C are said to be on the same side of point A if A, B, C are collinear and A is not between B and C A line segment is a set of 2 points on a line and all the points between them. A ray is a set of points consisting of a fixed point of a line and all the points of that line on the same side of the fixed point. Opposite rays are two rays of the same line that have a common endpoint and no other point in common An angle is a set of points consisting of a union of two rays not lying on the same line that have a common endpoint Angles are denoted using their vertex, or if 2+ angles share the same vertex, then we use a point on one side, the vertex, and a point on the other side. One unit of measuring angles is the degree. One degree is defined to be 1/360th of a full circle. Degrees can be broken into smaller parts. 1 degree = 60’ (minutes) and 1’ = 60 ‘’ (seconds) A point P lies in the interior of an angle if there exist two points one on each side, neither at the vertex such that the point P is between said 2 points. Congruent line segments are segments that are equal in length. Congruent angles are angles that are equal in measure On a diagram, congruence is indicated with identical tick marks. The midpoint of a line segment is a point of the line segment such that the two segments formed are equal in length. The trisection points of a line segment are the 2 points of the line segment such that the 3 segments formed are congruent. A segment bisector is a line, line segment, or ray, that intersects the line segment at its midpoint. A right angle is an angle measuring 90 degrees. An acute angle is an angle measuring less than 90 degrees. An obtuse angle is an angle measuring more than 90 degrees. Complementary angles are 2 angles the sum of whose measures is 90 degrees. Supplementary angles are 2 angles the sum of whose measures is 180 degrees. Two lines are said to intersect if they have a point in common. Perpendicular lines are two lines that intersect and form a right angle, denoted by ⊥ If two lines in the same plane do not intersect, then they are parallel, denoted by || An angle bisector is the ray whose endpoint is the vertex of the angle and that divides it into two congruent angles. Angle trisectors are the 2 rays whose common endpoint is the vertex of the angle and that divide it into 3 congruent angles. The union (U) of 2 objects is the set of all points that are contained in at least 1 of the 2 objects. The intersection (∩) of 2 objects is the set of all points that the 2 objects have in common. Definitions Reversed 1.​ A point of a line segment that forms two congruent segments is the midpoint of the line segment 2.​ A line that intersects only the midpoint of a segment is a segment bisector 3.​ An angle whose measure is 90 degrees is a right angle 4.​ A ray whose endpoint is the vertex of an angle and that forms two congruent angles is an angle bisector 5.​ Angles that are equal in measure are congruent​ Line segments that are equal in length are congruent. Lesson 8 - First Postulates Writing a Geometry Proof -​ Copy diagrams using a straightedge and/or a compass -​ Copy Given Statements and statements to Prove -​ Write proof in a two-column format. -​ Finish with a conclusion. From a diagram, we may assume 1.​ Collinearity of points: Points drawn on one line really lie on one line 2.​ Betweenness of points: A point drawn on the same line as two other points that is in between them is actually between them. 3.​ Incidence of line: Two points shown as intersecting at a point do intersect at that point. 4.​ Coplanarity of the diagram: Everything shown in the diagram lies in the same plane. We may NOT assume: 1.​ That an angle is a right angle because it looks like one 2.​ That 2 segments are congruent because they appear to have the same length 3.​ That two angles are congruent because they appear to have the same measure 4.​ Anything regarding the relative sizes of segments and angles. Postulates: A postulate or axiom is a statement that is unproven, but we assume it to be true. The following are some basic postulates in geometry: 1.​ Line Axiom: -​ There are infinitely many points A, B, C, and infinitely many lines l, m, n, -​ Given any two distinct points, exactly one line contains them both. -​ Each line contains infinitely many points -​ Given a line, there exists a point not on the line 2.​ The Distance Assignment Postulate: To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points. The distance between 2 points is 0 if and only if the points are not distinct 3.​ The Segment Construction Postulate: Given ray XY and segment AB, there exists exactly one point P on ray XY such that segment XP is congruent to segment AB. 4.​ Line Segment Partition Postulate: If point B is between points A and C, then segment AB + segment BC is congruent to segment AC, and AB + BC = AC. 5.​ Angle Measure Assignment Postulate: To every angle there corresponds a unique real number between 0 and 180. This number is called its measure. 6.​ Angle Construction Postulate: Given ray XY and a real number k between 0 and 180, there exists exactly one ray XR on a given side of line XY such that the measure of RXY = k. 7.​ Angle Partition Postulate: If P lies in the interior of angle ABC, then angle ABP + angle PBC is congruent to angle ABC and same with measures. (The whole is congruent to the sum of its parts. 8.​ Any 2 points have a unique midpoint. 9.​ Each angle has a unique angle bisector 10.​Two (distinct) lines intersect at at most one point. Lesson 9 - Addition and Subtraction Postulate Properties of Equality Properties of Congruence Reflexive Property A value is equal to itself Segments/Angles are congruent to itself Transitive Property If a=b and b=c, then a=c If 2 segments/angles are congruent to the same segment/angle, they are congruent to each other. Symmetry Property a=b is equal to b=a Congruence is a biconditional  Substitution Property You can replace a quantity Objects can be replaced with an equal quantity with equivalent objects. The Addition Postulate of Algebra If equals are added to equals, then their sums are equal. Using symbols: If a = b and c = d, then a+c = b+d The Addition Postulate of Geometry If congruent segments or angles are added to congruent segments or angles, then their sums are congruent. The Subtraction Postulate of Algebra If equals are subtracted from equals, then their differences are equal. Using symbols: If a=b and c=d, then a-c=b-d The Subtraction Postulate of Geometry If congruent segments or angles are subtracted from congruent segments or angles, then their differences are congruent Lesson 10 - First Theorems Definitions: A theorem is a statement that has been proven by a chain of reasoning. A postulate is assumed to be true. Adjacent angles are 2 angles in the same plane that share a common vertex and a side in common but share no other points. Angles that form a linear pair are two adjacent angles such that their non-common rays are opposite rays. Lesson 11 - Multiplication and Division Theorems Multiplication Postulate for Algebra Given: a = b ​ ca = cb Division Postulate for Algebra Given: a = b and c ≠ 0 ​ a/c = b/c Division Theorem of Congruence If 2 segments (or angles) are congruent, then their like divisions are congruent. Special Case: Halves of congruent segments (or angles) are congruent. Multiplication Theorem of Congruence If two segments (or angles) are congruent, then their like multiples are congruent Special Case: Doubles of congruent segments (or angles) are congruent Lesson 12 - Practice Writing Proofs Theorem: Adjacent angles whose sum is a right angle are complementary Lesson 13 - Practice with Proofs Practice Questions Lesson 14 - SAS Postulate of Congruence There are 6 parts in a triangle (3 sides, 3 angles) If 2 triangles are congruent, all 6 parts are congruent In a simple diagram, we can name angles using the letter of their vertex. We can name sides using the lowercase letter of their opposite vertex. A construction in Euclidean Geometry is an accurate, precise drawing done using only a compass and a straightedge. Construction Supporting Reason Construct the line joining two points Two points determine a line (from Line Axiom) Extend a line segment or ray A line can be extended as far as desired Construct a desired length on a ray Segment construction postulate Given a ray, construct an angle of a desired Angle construction postulate size Construct the angle bisector of a given Every angle has a unique angle bisector angle Erect the line perpendicular to a given line Through a point on a line exactly one line at a point on the given line can be constructed perpendicular to the given line Drop the line perpendicular to a given line Through a not point on a line exactly one passing through a point not on the given line can be constructed perpendicular to line the given line  Construct the perpendicular bisector of a Every segment has a unique perpendicular given line segment bisector Definition: Two triangles are congruent if and only if there exists a correspondence between the vertices of one triangle and the vertices of the other triangle such that each pair of corresponding parts are congruent. Proving Triangles Congruent To prove two triangles congruent, we do not need to show all 6 pairs of parts are congruent. We only need to show 3 carefully chosen pairs of corresponding parts are congruent. Methods for Proving Triangles Congruent SAS - Two pairs of sides and the angles included by them ASA - Two pairs of angles and the sides included by them SSS - All 3 pairs of sides AAS - Two pairs of angles and a pair of sides that are not included by them HL - In a right triangle, one pair of legs and a pair of hypotenuses. Properties of Congruent Triangles By definition, corresponding parts of congruent triangles are congruent. (CPCTC) Once we have shown 2 triangles are congruent, we can conclude that the rest of their pairs of corresponding parts are also congruent Lesson 15 - Proving Congruent Triangles with SAS/ASA Practice problems Lesson 16 - Proving parts of Triangles Congruent Theorem: All radii of a circle are congruent Lesson 17 - Overlapping Triangles Practice Problems Lesson 18 - Medians and Altitudes of Triangles Classifying Triangles by Sides: A scalene triangle is a triangle in which no 2 sides are congruent An isosceles triangle is a triangle in which at least 2 sides are congruent An equilateral triangle is a triangle in which all 3 sides are congruent -​ A triangle is equilateral only if it is isosceles -​ A triangle is scalene if and only if it is not isosceles Classifying Triangles by Angles: An acute triangle is a triangle in which all 3 angles are acute A right triangle is a triangle in which one angle is a right angle and the other 2 angles are acute and supplementary The side opposite the right angle is called the hypothenuse and the sides including the right angle are called the legs An obtuse triangle is a triangle in which it has 1 obtuse angle and 2 acute angles. Angle Bisector of a Triangle An angle bisector of a triangle is a ray/line/line segment drawn from any vertex of a triangle to the opposite side in such a way that it creates 2 congruent angles. Median of a Triangle A median of a triangle is a line segment drawn from any vertex of a triangle to the midpoint of the opposite side Altitude of a Triangle An altitude of a triangle is a line segment drawn from any vertex of a triangle to the line containing the opposite side such that it is perpendicular to that line. Isosceles Triangles The legs of an isosceles triangle are the sides of the triangle that are congruent The vertex angle of an isosceles triangle is the angle included by the legs of the triangle The base of an isosceles triangle is the side of the triangle that is opposite the vertex angles The base angles of an isosceles triangle are the angles included by the base and each of the legs. An isosceles triangle is sometimes equilateral The base of an isosceles triangle is sometimes congruent to the legs A median of a triangle is always inside of the triangle An altitude of a triangle is sometimes inside of the triangle An angle bisector of an angle of a triangle is always inside of the triangle An angle bisector of a triangle is sometimes an altitude of the triangle A side of a triangle is sometimes an altitude of the triangle A side of a triangle is never a median of the triangle. Lesson 19 - Base Angles of an Isosceles Triangle Theorem: If 2 sides of a triangle are congruent, the angles opposite these sides are also congruent Lesson 20 - Triangles with Congruent Angles Theorem: If 2 angles of a triangle are congruent, then the sides opposite those angles are also congruent Lesson 21 - How do we prove the SSS Theorem of Congruence? SSS Theorem of Congruence If 3 sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent. Lesson 22 - How do we complete proofs requiring detours? Procedure for Detour Proofs 1.​ Determine which congruent triangles lead to the conclusion 2.​ If you are lacking info to prove them congruent, take a detour 3.​ Identify the missing parts you need 4.​ Find a pair of triangles that use the missing parts or a pair that is easy to prove congruent 5.​ Prove those triangles congruent 6.​ Use CPCTC to finish proving the original triangles from step #1 congruent Lesson 23 - Corollaries to the Theorem about Sides and Angles Theorem: If 2 lines intersect to form congruent supplementary angles, then the lines are perpendicular If 2 angles are congruent and supplementary, they are right angles A corollary is a theorem that comes as an easily provable result of another theorem that has been already proven Corollaries: The median to the base of an isosceles triangle bisects the vertex angle The median to the base of an isosceles triangle is also the altitude to the base The bisector of the vertex of an isosceles triangle is also the altitude to the base The bisector of the vertex of an isosceles triangle is also the median to the base In an equilateral triangle, all angles are congruent Lesson 24 - Equidistant Theorems The distance between two objects is the length of the shortest path joining them. The distance from a point to a line is the length of the perpendicular segment from the point to the line. The perpendicular bisector of a segment is the line perpendicular to the segment of its midpoint. Theorems of Perpendicular Lines 1.​ Through a point on the line, there exists exactly 1 line perpendicular to the given line 2.​ Through a point not on the line, there exists exactly 1 line perpendicular to the given line Equidistance Theorems 1.​ If 2 points are each equidistant from the endpoints of a segment, then the line joining them is the perpendicular bisector of segment 2.​ If a point is on the perpendicular bisector of a segment, it is equidistant from the endpoints. 3.​ If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. Lesson 25 - Proofs with Equidistance Theorems Practice Problems Lesson 26 - More practice with Equidistance Theorems Practice Problems Lesson 27 - Indirect Proofs p v ~p = Tautology p ^ ~p = Contradiction Law of the Excluded Middle -​ A statement is either true or false Law of Contradiction -​ A statement cannot be true AND false at the same time Law of Elimination -​ When all possibilities but one have been shown untrue, the remaining possibility must be true Indirect Proof Procedure 1.​ List all possibilities of the conclusion (Law of the Excluded Middle)  2.​ Assume the negation of the conclusion is true (Assumption) 3.​ Proceed with proof until a contradiction is reached. The statement may contradict a.​ Given Information b.​ Known Fact (definitions/theorems/postulates) 4.​ The remaining possibility must be true (Law of Elimination) Lesson 28 - Practice with Indirect Proofs Practice Problems Lesson 29 - Perpendicular Lines and Planes Undefined Term Plane - A surface such that if any 2 points in the surface are joined by a line, then the line lies entirely in the surface Definitions If points, lines, and segments lie in the same plane, we call them coplanar. If they do not lie in the same plane, they are not coplanar. A line is perpendicular to a plane if it is perpendicular to every line in the plane that passes through its foot. A line is oblique to a plane if it intersects the plane at exactly one point and is not perpendicular to the plane. Postulates: If a line passes through two points that lie in a plane, then the line lies entirely in the plane If a line intersects a plane not containing it, then they intersect at exactly one point. If 2 planes intersect, then their intersection is exactly one line, assuming they are distinct planes 3 non-collinear points determine a plane Given a plane and a point on the plane, there is exactly one line passing through the given point that is perpendicular to the plane Given a point and a point not on the plane, there is exactly one line passing through the given point that is perpendicular to the plane. Theorem: A line and a point not on the line determine a plane Two intersecting lines determine a plane If a line is perpendicular to at least 2 distinct lines that lie in a plane and pass through its foot, then it is perpendicular to the plane. Lesson 30 - Basic Inequality Postulates Law of Trichotomy For real numbers, exactly one of the following is true: ab A whole is greater than any of its parts If point X is between A and B, then AB>AX and AB>BX (same with angles) Transitive Property for Inequalities If a>b and b>c, then a>c Substitution Property for Inequalities A quantity may be substituted for its equal in any inequality If a>b and b=c, then a>c If ab+d b.​ If unequal quantities are added to quantities that are unequal in the same order, then their sums are unequal in the same order. i.​ If a>b and c>d, then a+c>b+d Subtraction Properties for Inequalities a.​ If equal quantities are subtracted from unequal quantities, then their difference are unequal in the same order i.​ If a>b and c=d, then a-c>b-d b.​ If unequal quantities are subtracted from equal quantities, then their difference are unequal in the opposite order. i.​ If a=b and c>d, then a-cb and c>0, then ac>bc and a/c>b/c d.​ If 2 quantities are unequal and they are multiplied/divided by a negative number, the resulting products are unequal in the opposite order  i.​ If a>b and c