Plane and Solid Geometry PDF

Plane and Solid Geometry 2SEDM-A Week 1: The Modern Axiomatic System two right angles, then the two lines, if extended...

Plane and Solid Geometry 2SEDM-A Week 1: The Modern Axiomatic System two right angles, then the two lines, if extended indefinitely, meet on that side. Modern Axiomatic System - is a logical framework that Illustration: Two lines that are not parallel will serves as the foundation for constructing mathematical eventually intersect if the interior angles add up to theories. less than 180 degrees. 1. Undefined Terms - do not require a definition but can describe. The three primary undefined terms are: 3. Theorems Derived from Postulates Theorems are propositions that can be logically derived - a location in space and has no dimension from the postulates. no length, width, or height. Theorem 1: The sum of the interior angles of a triangle is 180 degrees. - one-dimensional figure that extends infinitely in both directions. It is made up Illustration: In any triangle ABC, the angles ∠A, ∠B, of an infinite number of points and has no and ∠C add up to 180°. thickness. Theorem 2: The base angles of an isosceles triangle are - a flat, two-dimensional surface that congruent. extends infinitely in all directions. Illustration: In an isosceles triangle, where sides AB and AC are equal, the base angles ∠B and ∠C are also equal. - Defined Terms - can defined using undefined terms. Line segment Ray Theorem 3: The Pythagorean Theorem. Angle Illustration: In a right triangle ABC, with the right Space angle at C, the square of the length of the Collinear points hypotenuse AB is equal to the sum of the squares of Non-collinear points the lengths of the other two sides (AC and BC). Coplanar points Non-conplanar points 2. Initial Postulates (Axioms) - the foundational assumptions about the undefined terms. Postulate 1: A straight line can be drawn from any Week 2: Introduction to Proving point to any other point. Illustration: If you have two points, A and B, you Proving - the process of demonstrating the truth or can always draw a straight line connecting them. falsehood of a statement using logical reasoning, based on previously established axioms, definitions, and theorems. Postulate 2: A finite straight line can be extended Proofs - the backbone of mathematics, ensuring that indefinitely in both directions. mathematical concepts and relationships are rigorously and Illustration: A line segment AB can be extended to universally valid. form a full line. Components of a Proof Postulate 3: A circle can be drawn with any center and 1. Axioms/Postulates - fundamental assumptions accepted any radius. without proof. They form the foundational building blocks Illustration: A circle can be drawn with a center at from which other statements are derived. point O and a radius r. 2. Definitions - explanations of mathematical concepts. Definitions are used to establish a common understanding Postulate 4: All right angles are congruent. (equal) of terms within the context of a proof. Illustration: Right angles are all equal in measure 3. Theorems - Statements that have been proven to be true (90 degrees). based on axioms, definitions, and previously proven theorems. The process of proving a theorem is called proof. Postulate 5: If a line intersects two other lines such that the sum of the interior angles on one side is less than 4. Lemmas and Corollaries: K.D.A. Plane and Solid Geometry 2SEDM-A - lemma is a subsidiary theorem used as a stepping stone in the proof of a larger theorem. Structure of a Proof - corollary is a statement that follows directly from a 1. Statement of the Theorem - state the proposition or theorem with little or no additional proof. theorem you intend to prove. 2. Assumptions (Hypotheses) -List the assumptions or given Types of Proofs conditions. These are the starting points of your proof. Direct Proof - start with known facts (axioms, 3. Proof - Provide a logical sequence of steps, using definitions, previously proven theorems) and use definitions, axioms, and previously proven theorems, to logical steps to arrive at the statement to be proven. arrive at the conclusion. This method is straightforward and commonly used. 4. Conclusion - Summarize the result, often with a Example: To prove that the sum of two even numbers is statement like "Thus, the theorem is proven" or "Therefore, even, you start by expressing even numbers as 2a and 2b, the statement is true." where a and b are integers. Adding them gives 2a+2b=2(a+b), which is even. Properties of Equality Indirect Proof (Proof by Contradiction) - assume the Equality means that two expressions represent the same opposite of what you want to prove and show that this value. assumption leads to a contradiction. Once a 1. Reflexive Property - Any quantity is equal to itself. contradiction is found, you conclude that the original a=a statement must be true. 2. Symmetric Property - The order of equality does not Example: To prove that there is no largest prime number, matter; if one quantity equals another, you can switch them. assume the opposite, that there is a largest prime. By If a=b, then b=a. constructing a new prime number that is larger than this, a 3. Transitive Property - If two quantities are both equal to a contradiction is reached, proving the original statement. third quantity, they are equal to each other. Proof by Contrapositive - Instead of proving P→Q If a=b and b=c, then a=c. directly, you prove the contrapositive ¬Q→¬P. Since a 4. Substitution Property - Equal quantities can replace each statement and its contrapositive are logically other in any expression or equation. equivalent, proving the contrapositive proves the If a=b, then a can be substituted for b in any expression. original statement. 5. Addition Property - Adding the same quantity to both Example: To prove "If is even, then n is even," you can sides of an equation does not change the equality. prove its contrapositive: "If n is odd, then is odd." If a=b, then a+c=b+c for any c. Proof by Induction - used to prove statements 6. Subtraction Property - Subtracting the same quantity about natural numbers. It involves two steps: from both sides of an equation does not change the - Base Case: Prove the statement is true for the first value equality. (usually n=1). If a=b, then a−c=b−c for any c. - Inductive Step: Assume the statement is true for some 7. Multiplication Property - Multiplying both sides of an n=k, and then n=k+1. equation by the same non-zero quantity does not change - If both steps are successful, the statement is true for all the equality. natural numbers. If a=b, then ac=bc for any c. Example: To prove that the sum of the first n natural 8. Division Property - Dividing both sides of an equation by the same non-zero quantity does not change the equality. numbers is , you would use induction. If a=b and c≠0 then Proof by Exhaustion - involves checking every possible case individually. It is feasible when there are a limited number of cases. Example: Proving a property for all triangles by considering Properties of Inequality cases such as equilateral, isosceles, and scalene. Inequalities describe the relationship between two Constructive Proof - not only demonstrates that expressions where one is greater than or less than the other. something exists but also provides a method to construct the example. Example: Proving there is an even prime number by explicitly showing that 2 is even and prime. Non-constructive Proof - shows that something exists without necessarily providing a specific example. 1. Reflexive Property - A quantity is always less than or Example: Using the pigeonhole principle to prove that at equal to, or greater than or equal to, itself. least two people in a room must have the same birthday. K.D.A. Plane and Solid Geometry 2SEDM-A For any number a, a ≤ a and a ≥ a. o Divide both sides by 4 to solve for xxx. 2. Transitive Property - If one quantity is less than or equal Since we are dividing by a positive number, to (or strictly less than) a second quantity, and the second is the direction of the inequality remains the less than or equal to (or strictly less than) a third, then the same. first is less than or equal to (or strictly less than) the third. If a ≤ b and b ≤ c, then a ≤ c. If a < b and b < c, then a < c 3. Addition Property - Adding the same quantity to both sides of an inequality preserves the inequality. Example 3: Applying the Properties of Real Numbers If a ≤ b, then a + c ≤ b + c for any c. Problem: Simplify and solve the expression 2(3x−4)+5x=19. If a < b, then a + c < b + c for any c. Solution: 4. Subtraction Property - Subtracting the same quantity o Distributive Property: from both sides of an inequality preserves the inequality. o Apply the distributive property to eliminate If a ≤ b, then a – c ≤ b − c for any c. the parentheses. If a < b, then a − c < b – c for any c. 5. Multiplication Property - Multiplying both sides of an inequality by a positive number preserves the direction of o Combine Like Terms: the inequality. Multiplying by a negative number reverses o Combine the terms involving x. the direction. If a ≤ b and c > 0, then ac ≤ bc. If a ≤ b and c < 0, then ac ≥ bc. 6. Division Property - Dividing both sides of an inequality by o Addition Property of Equality: a positive number preserves the direction of the inequality. o Add 8 to both sides to isolate the term with Dividing by a negative number reverses the direction. the variable. If a ≤ b and c > 0, then If a ≤ b and c < 0, then o Division Property of Equality: Examples o Divide both sides by 11 to solve for xxx. Example 1: Solving an Equation Using Properties of Equality Problem: Solve the equation. Solution: o Subtraction Property of Equality: o Subtract 7 from both sides to isolate the term with the variable. Example 4: Using the Properties of Inequality in Real-World Problems Problem: A company pays its employees 15 dollars per hour o Division Property of Equality: and gives a bonus of 50 dollars if the employee works more o Divide both sides by 3 to solve for x. than 40 hours in a week. If an employee earns more than 700 dollars in a week, how many hours did the employee work? Solution: o Set Up the Inequality: o Let x be the number of hours worked. The Example 2: Solving an Inequality Using Properties of total pay can be expressed as: Inequality Problem: Solve the inequality. (Note: The 50 dollar bonus is added only if the Solution: employee worked more than 40 hours.) o Addition Property of Inequality: o Subtract the Bonus: o Add 5 to both sides to isolate the term with o Subtract 505050 from both sides of the the variable. inequality. o Division Property of Inequality: K.D.A. Plane and Solid Geometry 2SEDM-A o Division Property of Inequality: o For any negative number a, the absolute o Divide both sides by 151515 to solve for value ∣a∣ is −a, which is positive. xxx. o Example: ∣−5∣=5, which is the distance from −5 to 0 on the number line. Zero: o The absolute value of zero is 0, because it is at zero distance from itself. Conclusion: The employee worked more than 43.33 hours. o Example: ∣0∣=0. Since the number of hours must be a whole number, the employee worked at least 44 hours. 2. Geometric Interpretation with an Example Example: Consider the expression ∣x−3∣. Steps of Formal Proof Interpretation: Formal proof is a logical argument that establishes the truth o The expression ∣x−3∣ represents the of a statement in a structured manner. It is used extensively distance between the point x and the point in mathematics, particularly in geometry, algebra, and logic. 3 on the number line. It consists of a sequence of statements, each supported by Geometric Meaning: axioms, definitions, or previously proven theorems. o If x = 7, then ∣7−3∣ = ∣4∣ = 4. This means that Here are the key steps in constructing a formal proof: the distance between 7 and 3 on the 1. State the Theorem or Proposition - Begin by clearly number line is 4 units. stating what you want to prove. o If x = 1, then ∣1−3∣ = ∣−2∣ = 2. This means 2. List the Given Information (Hypotheses) - Identify and list that the distance between 1 and 3 on the all the information that is assumed to be true, known as the number line is 2 units. hypotheses or given information. Visual Representation: 3. State the Conclusion - Clearly state what needs to be o Imagine the number line where the point 3 proven, often referred to as the conclusion. is fixed. The absolute value ∣x−3∣ measures 4. Draw a Diagram (if applicable) - In geometry, it's helpful how far any point x is from 3, regardless of to draw a diagram that represents the given information. whether x is to the left (negative side) or This visual aid can help in understanding the relationships right (positive side) of 3. between the elements involved. 5. Write the Proof - Construct the proof by writing a logical 3. Absolute Value as a Distance Function in Geometry sequence of steps that lead from the given information to In a geometric context, the absolute value can be used to the conclusion. Each step must be justified by a definition, describe the distance between points on a plane or in space. postulate, axiom, or previously proven theorem. Example in Geometry: 6. Use Clear and Precise Language - Ensure that each o If you have two points on a coordinate statement is clear and unambiguous. Avoid unnecessary assumptions and make sure every claim is supported by a plane, and B , the valid reason. distance between them on the x-axis can 7. Conclude the Proof - After the final step, state that the be represented by ∣x2−x1∣|. This is the theorem has been proven. This is often done with a absolute value of the difference in their x- statement like "Thus, the theorem is proven" or using the coordinates, which geometrically is the Latin phrase "Q.E.D." (quod erat demonstrandum), which distance between these two points along means "which was to be demonstrated." the x-axis. Geometric Interpretation of Absolute Value Introduction to Angles and Perpendicular Lines - The absolute value of a number represents its distance Angle - formed when two rays (or line segments) share a from zero on the number line, regardless of direction. common endpoint called the vertex. The amount of "turn" Geometrically, the absolute value of a number a, denoted by between the two rays is measured in degrees (°). ∣a∣, is interpreted as the distance from a to 0 on the number line. Types of Angle 1. Absolute Value on the Number Line Positive Numbers: o For any positive number a, the absolute value ∣a∣ is simply a. o Example: ∣5∣=5, which is the distance from 5 to 0 on the number line. Negative Numbers: K.D.A. Plane and Solid Geometry 2SEDM-A Pair of Lines In coordinate geometry, if two lines are perpendicular, the product of their slopes is -1 (i.e., ).∆ Basic Theorems Concerning Angles and Angle Pairs 1. Vertical Angles Theorem - When two lines intersect, the pairs of opposite (or vertical) angles are equal. 2. Linear Pair Theorem - If two angles form a linear pair (they are adjacent and their non-common sides form a straight line), then the angles are supplementary. 3. Corresponding Angles Postulate (Parallel Lines) - If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. > Name the lines through point H that appear 4. Alternate Interior Angles Theorem - If two parallel skew to (CD). answer: AH lines are cut by a transversal, then each pair of alternate > Name the lines containing point H that interior angles is congruent. appear parallel to (CD). answer: EH 5. Alternate Exterior Angles Theorem - If two parallel > Name a plane that is parallel to plane CDE lines are cut by a transversal, then each pair of alternate and contains point H. answer: GH exterior angles is congruent. 6. Same-Side (Consecutive) Interior Angles Theorem - If two parallel lines are cut by a transversal, then each pair In a plane, two lines are either of same-side interior angles is supplementary. ◦Parallel 7. Exterior Angle Theorem (Triangle) - The measure of an ◦Intersect exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. Perpendicularity for Lines ✓ Transversal - line that intersects two coplanar lines. Two lines are said to be perpendicular if they intersect to ✓ Interior - angles that are between the lines. form four right angles (each measuring 90°). The symbol ∠2,∠3∠,5,∠6 used to denote that two lines are perpendicular is ⊥. For ✓ Exterior - angles that are outside the lines. example, if line AB is perpendicular to line CD, we write it ∠1,∠4,∠7,∠8 as AB ⊥ CD. ✓ Alternate interior angles - interior angles on opposite Properties of Perpendicular Lines: sides of the transversal. ∠2&∠5, ∠3&∠6 1. Right Angles - When two lines are perpendicular, they ✓ Alternate exterior angles - exterior angles on opposite create four right angles at the point of intersection. Each sides of the transversal. ∠1&∠8, ∠4&∠7 of these angles measures 90°. ✓ Consecutive interior angles - interior angles on the 2. Congruent Angles - The four angles formed by the same side of the transversal. ∠2&∠6, ∠3&∠5 intersection of two perpendicular lines are congruent. ✓ Corresponding Angles - angles on the same location This is because they all measure 90°. relative to the transversal. ∠1&∠6, ∠2&∠7, ∠3&∠8, 3. Slopes in Coordinate Geometry - In a Cartesian plane, ∠4&∠5 the slopes of two perpendicular lines have a special relationship. If the slope of one line is m1 and the slope of Perpendicular Lines - When two lines intersect at a right the other is m2, then m1×m2 = −1. angle (90°) - For example, if one line has a slope of 2, the Properties of Perpendicular Lines: perpendicular line will have a slope of −1/2. When two lines are perpendicular, they meet or 4. Equidistant Points - Any point on one of the cross each other at a 90° angle. perpendicular lines is equidistant from the line to which it The symbol for perpendicularity is ⊥. For is perpendicular. This is particularly useful in determining example, if line AB is perpendicular to line CD, we distances and constructing perpendicular bisectors. write it as AB ⊥ CD. Constructing Perpendicular Lines: 1. Using a Compass and Straightedge 2. In Coordinate Geometry K.D.A. Plane and Solid Geometry 2SEDM-A Theorems Involving Perpendicular Lines: 6. Perpendicular Transversal: Two lines perpendicular to 1. The Perpendicular Bisector Theorem - A point on the the same transversal are parallel. perpendicular bisector of a line segment is equidistant 7. Same Slopes: In coordinate geometry, two lines with from the endpoints of the segment. the same slope are parallel. 2. The Converse of the Perpendicular Bisector Theorem - 8. Parallel Postulate: There is exactly one line parallel to If a point is equidistant from the endpoints of a segment, a given line through a point not on the line. then it lies on the perpendicular bisector of the segment. 3. Theorems in Right Triangles - In a right triangle, the Ways of showing that lines are parallel altitude drawn from the right angle to the hypotenuse 1. Corresponding Angles: Show that corresponding creates two smaller right triangles, each similar to the angles are congruent. original triangle. 2. Alternate Interior Angles: Show that alternate interior 4. Perpendicularity in Polygon - There are two sides are angles are congruent. perpendicular therefore a right triangle are only one side 3. Alternate Exterior Angles: Show that alternate exterior angle exactly of 90 degree and the form by 2 sides are angles are congruent. perpendicular sides of triangle. Close geometry figures. 4. Consecutive Interior Angles: Show that consecutive interior angles are supplementary. Parallel lines - two or more lines that remain the same 5. Consecutive Exterior Angles: Show that consecutive distance apart along their entire length and never exterior angles are supplementary. intersect. 6. Perpendicular Transversals: Show that both lines are Key Characteristics of Parallel Lines: perpendicular to the same transversal. 1. Equidistant: Parallel lines are always the same distance 7. Equal Slopes (Coordinate Geometry): Show that both apart. This distance never changes, making the lines lines have the same slope. "perfectly parallel" throughout their entire length. 8. Parallel Postulate: Use the Parallel Postulate to 2. No Intersection: Parallel lines do not intersect at any construct a parallel line through a given point. point. This is a defining feature—unlike intersecting lines, which meet at a point, parallel lines maintain their Angles Formed/Relationship by Parallel Lines and a separation indefinitely. Transversal: 3. Coplanar: Parallel lines lie in the same plane. If two 1. Corresponding Angles: Congruent. lines are not in the same plane, they may be skew lines, 2. Alternate Interior Angles: Congruent. which are not parallel and do not intersect, but also do 3. Alternate Exterior Angles: Congruent. not lie on the same plane. 4. Consecutive Interior Angles (Same-Side Interior 4. Same Direction: While parallel lines may have different Angles): Supplementary. starting points, they always point in the same direction. 5. Consecutive Exterior Angles (Same-Side Exterior Their angles relative to other lines or axes are the same, Angles): Supplementary. ensuring they never converge. 6. Vertical Angles: Always congruent (for any intersection, not just with parallel lines). Parallel Postulate, states that given a line and a point not on the line, there is exactly one line parallel to the given These angle relationships are used to: line that passes through the point. This postulate helps to 1. Prove lines are parallel: By showing that define the behavior of parallel lines in flat, two- corresponding, alternate interior, or alternate exterior dimensional spaces. angles are congruent, or that consecutive angles are supplementary, we can prove two lines are parallel. Conditions Guaranteeing Parallelism: 2. Solve for unknown angles: Given some angles, you can 1. Corresponding Angles: Congruent corresponding use these relationships to calculate other angles when angles imply parallel lines. parallel lines are intersected by a transversal. 2. Alternate Interior Angles: Congruent alternate interior 3. Verify parallelism: In constructions and practical angles imply parallel lines. applications, these relationships are used to verify that 3. Alternate Exterior Angles: Congruent alternate structures or elements like walls or roads are parallel. exterior angles imply parallel lines. 4. Consecutive Interior Angles: Supplementary consecutive interior angles imply parallel lines. 5. Consecutive Exterior Angles: Supplementary consecutive exterior angles imply parallel lines. K.D.A. Plane and Solid Geometry 2SEDM-A Angle Pairs Linear Pair - a pair of adjacent angles where noncommon sides are opposite rays, then the angles are supplementary. Angle Bisector - a ray that divides an angle into two congruent angles. It’s end point is at the angle vertex. Square and Rectangle in perpendicularity - all adjacent side are perpendicular - all side are congruent CTTO: LES K.D.A.