Comprehensive Study Guide: Geometry Fundamentals, Economics, and Christian Studies, and Values

Geometry Fundamentals

• Angle Pair: Consists of two angles that have a specific relationship.

• Adjacent Angles: Must meet three specific conditions:

  • They share a common vertex.

  • They share a common side.

  • They have no common interior points.

  • Note: Adjacent angles may or may not have equal measures.

• Complementary Angles:

  • Definition: Two positive angles whose measures add up to $90^\circ$.

  • Complement: The name of the other angle in a complementary pair.

  • Calculation: To find a missing complement, subtract the known angle from $90^\circ$.

• Supplementary Angles:

  • Definition: Two positive angles whose measures add up to $180^\circ$.

  • Supplement: The name of the other angle in a supplementary pair.

  • Calculation: To find a missing supplement, subtract the known angle from $180^\circ$.

  • Note: Adjacent angles are not always supplementary.

• Congruent Angles: Angles that have the same measure. They do not necessarily have to be adjacent.

• Linear Pair: Two adjacent angles whose noncommon sides are opposite rays. Every linear pair is supplementary, but not every supplementary pair is a linear pair.

• Vertical Angles: Two opposite angles formed when two lines intersect.

  • Vertical angles are always congruent.

  • Vertical angles are never adjacent.

Lines and Rays

• Opposite Rays: Two rays that form a straight line.

• Perpendicular Lines ($\perp$): Two lines that intersect to form a right angle ($90^\circ$). An example is the corner of a notebook.

• Parallel Lines ($\parallel$): Coplanar lines that never intersect and remain the same distance apart forever. Railroad tracks are a common example.

• Transversal: A line that intersects two or more lines at different points.

  • When a transversal crosses two parallel lines, 8 angles are formed.

• Angles Formed by a Transversal:

  • Alternate Interior Angles: Two interior angles on opposite sides of the transversal. If lines are parallel, these are congruent.

  • Alternate Exterior Angles: Two exterior angles on opposite sides of the transversal. If lines are parallel, these are congruent.

  • Corresponding Angles: Angles in the same relative position at each intersection. If lines are parallel, these are congruent.

  • Same-Side Interior Angles: Two interior angles on the same side of the transversal. These are supplementary ($\Sigma = 180^\circ$).

  • Same-Side Exterior Angles: Two exterior angles on the same side of the transversal. These are supplementary ($\Sigma = 180^\circ$).

Basic Geometric Concepts and Undefined Terms

• Geometry: The branch of mathematics studying size, shape, and position. It measures objects and describes shapes.

• The Three Undefined Terms:

  1. Point: Represented by a dot; named using a capital letter. It has no dimensions. Example: the tip of a pencil.

  2. Line: Extends infinitely in two directions; named using any two points on the line. Represented by a double-pointed arrow ($\leftrightarrow$). Example: a tight string.

  3. Plane: A flat surface with no thickness that extends infinitely. It is uniquely determined and named by three non-collinear points. Example: a floor.

• Collinear Points: Points that lie on the same line.

• Coplanar Points: Points that lie on the same plane.

• Skew Lines: Lines that exist only in three-dimensional figures and are not on the same plane.

Quadrilaterals: Properties and Classifications

• General Definition: A 2-dimensional flat shape (polygon) with 4 straight sides, 4 corners (vertices), and 4 interior angles.

• Interior Angle Sum: The sum of all interior angles in any quadrilateral is exactly $360^\circ$.

• Types of Quadrilaterals:

  1. Parallelogram: Has two pairs of parallel sides. Opposite sides are equal in length. Example: a slanted window.

  2. Rectangle: A parallelogram with 4 right angles ($90^\circ$). Opposite sides are equal and parallel. Example: a door.

  3. Rhombus: A parallelogram with 4 equal sides. Opposite sides are parallel and opposite angles are equal. Example: diamond shape on playing cards.

  4. Square: A special quadrilateral that is both a rectangle (4 right angles) and a rhombus (4 equal sides).

  5. Trapezoid (Trapezium): Has at least one pair of parallel sides.

  6. Kite: Has two pairs of equal adjacent sides. A "dart" is a concave kite with an inward corner.

• Geometric Terms for Shapes:

  • Adjacent: Next to each other; adjacent sides share a common corner.

  • Bisect: To divide into two equal parts.

  • Diagonal: A line connecting two opposite corners.

  • Convex: All corners point outward.

  • Concave: Has at least one inward corner.

Introduction to Economics (Ekonomiks)

• Definition: An "agham panlipunan" (social science) studying how people use limited resources ("pinagkukunang-yaman") to satisfy infinite needs ("pangangailangan") and wants ("kagustuhan").

• Etymology: Derived from Greek "Oikonomia" (‘Oiko’ meaning household; ‘Nomos’ meaning management).

• Core Concepts:

  • Kakapusan (Scarcity): Long-term limitation of resources (natural, human labor, capital).

  • Kakulangan (Shortage): Short-term imbalance where supply does not meet demand.

  • Choice: Individual/Personal vs. Social choice.

  • Opportunity Cost: The value of the alternative sacrificed to choose a specific option.

  • Trade-off: Compromising by sacrificing one thing for another.

History of Economic Thought and Famous Economists

• Xenophon (430-354 BC): Wrote Oeconomicus regarding household and agricultural management.

• Plato (380-360 BC): Wrote The Republic emphasizing specialization.

• Aristotle (384-322 BC): Wrote Politics focusing on private property.

• Mercantilists (16th-18th Century): Believed national wealth depends on accumulating gold and silver.

• Physiocrats (18th Century): Believed in the "Rule of Nature" where agriculture is the main source of wealth.

• Francois Quesnay: Father of Physiocracy; wrote Tableau Economique.

• Adam Smith: Father of Modern Economics; wrote An Inquiry into the Nature and Causes of the Wealth of Nations. Promoted "Laissez-Faire" (minimal government intervention) and specialization.

• Thomas Robert Malthus: Wrote An Essay on the Principle of Population. Proposed that population grows geometrically ($2, 4, 8, 16, \dots$) while resources grow arithmetically ($1, 2, 3, 4, \dots$).

• David Ricardo: Known for the Law of Comparative Advantage, Labor Theory of Value, and Law of Diminishing Marginal Returns. Wrote The Principles of Political Economy and Taxation.

• John Maynard Keynes: Father of Modern Macroeconomics; wrote General Theory of Employment, Interest and Money.

• Karl Marx: Father of Communism; defended the proletariat (workers) against capitalism. Wrote Das Kapital.

Sustainable Development (Likas-Kayang Kaunlaran)

• Definition: Development meeting present needs without sacrificing the ability of future generations to meet theirs.

• The Three Pillars: Environment (Kapaligiran), Economy (Ekonomiya), and Society (Lipunan).

• The Four Factors of Sustainability: Human (Tao), Environment (Kapaligiran), Economy (Ekonomiya), and Social (Panlipunan).

• Historical Milestones:

  • 1972: UN Conference on the Human Environment (Stockholm).

  • 1987: Publication of Our Common Future (World Commission on Environment and Development).

  • 1992: UN Conference on Environment and Development (Rio de Janeiro); Agenda 21.

  • 2015: Establishment of the 17 Sustainable Development Goals (SDGs) for 2030.

• Philippine Strategy for Sustainable Development (PSSD): A national framework integrating economy, environment, social equity, people empowerment, and good governance.

Production Factors and Business Costs

• Factors of Production (Inputs):

  1. Lupa (Land): Natural resources.

  2. Paggawa (Labor): Physical and mental effort.

  3. Kapital (Capital): Man-made materials (machinery, buildings).

  4. Entreprenyur: The "Brain of the Business" who combines other factors. Traits: risk-taker, creative, stress-tolerant, humble.

• Types of Inputs: Variable Input (changes with quantity) and Fixed Input (does not change).

• Economic Cost Formulas:

  • Total Fixed Cost (TFC): $TFC = TC - TVC$

  • Total Variable Cost (TVC): $TVC = TC - TFC$

  • Total Cost (TC): $TC = TFC + TVC$

  • Average Fixed Cost (AFC): $AFC = \frac{TFC}{TP}$

  • Average Variable Cost (AVC): $AVC = \frac{TVC}{TP}$

  • Average Total Cost (ATC): $ATC = AFC + AVC$

  • Marginal Cost (MC): $MC = \frac{\Delta TC}{\Delta Q}$

• Business Organizations:

  • Sole Proprietorship: Managed by one person; low capital, personal liability.

  • Partnership: Two or more owners; shared talent, potential conflict.

  • Corporation: Legal entity separate from owners; limited liability, double taxation.

  • Cooperative: Shared ownership to help members; small starting capital, low interest on loans.

Christian Living: Prayer and the Church

• Pentecost: The event when the Holy Spirit descended on the Apostles and Mary, marking the Birth of the Church.

• Note: The early Church devoted itself to four things: Apostles' teaching, fellowship, breaking of bread, and prayers (Acts 2:42).

• Christian Prayer Defined: A loving, conscious, personal relationship and conversation with God. St. Teresa of Avila described it as an "intimate conversation with God who we know loves us."

• Nature of Prayer: It is Trinitarian, offered in the name of the Father, Son, and Holy Spirit. The Holy Spirit is the "Teacher of Prayer."

• The ACTS Prayer Model:

  1. Adoration: Praising God's greatness and holiness. Acknowledgement of creaturely status before the Creator.

  2. Contrition: Expressing sorrow for sins and promising to amend life. "Perfect Contrition" is motivated by love for God rather than fear.

  3. Thanksgiving: Gratitude for blessings (life, family, health). Psalm 138:1-2 is a biblical example.

  4. Supplication (Petition): Making requests for personal needs, wisdom, or the needs of others (intercession).

Saints and Virtues

• Virtue: A good habit or regular performance of good deeds.

• Saint Roque:

  • Born in Montpellier, France. Joined the Franciscan Third Order.

  • Patron saint of protection against the plague.

  • Iconography: Often depicted with a dog that brought him bread.

  • Primary Virtue: Compassion.

• St. Martha:

  • Lived in Bethany with siblings Lazarus and Mary.

  • Declared Jesus "The Resurrection and the Life."

  • Patron saint of cooks.

  • Primary Virtue: Hospitality.

• Pondo ng Pinoy:

  • Founded by Cardinal Gaudencio B. Rosales.

  • Key concept: Small acts (crumbs) of 25 centavos.

  • Total collected (2004 - June 2019): PhP 400 million.

  • Impact: Helped approximately 250,000 individuals/communities.

  • Pillars: Small Acts, Regularity, Rooted in Love, Empowerment.