Geometry Unit 1: Measurements and Proof Flashcards

Unit Objectives for Measurements and Proof

• Foundational Knowledge: Learn key geometric vocabulary and the proper representation of points, lines, angles, and shapes using accepted notation.

• Measurement and Classification: Acquire skills to measure and classify segments and angles, including the application of formulas for midpoint and distance.

• Geometric Relationships: Explore angle relationships, special pairs, and utilize tools such as a compass and straightedge for geometric constructions.

• Logical Reasoning: Analyze logical statements through conditional reasoning and apply both deductive and inductive logic to the construction of proofs.

• Proof Construction: Construct formal algebraic and geometric proofs involving segments, angles, bisectors, and vertical angles.

Lesson 1.1: Basic Geometry Vocabulary

• Undefined Terms: These are the most basic building blocks of geometry. They are not defined by simpler concepts but are established by descriptions.

  • Point: Represents a location in space. It has no dimension (length, width, or thickness). Named using a capital letter.

  • Line: A collection of points that extends infinitely in two opposite directions. It is one-dimensional (length) and has no thickness. Named by any two points on the line (e.g., $\overleftrightarrow{AB}$) or a lowercase script letter.

  • Plane: A flat surface that extends infinitely in all directions. It is two-dimensional (length and width). Named by three non-collinear points or a single capital script letter.

• Geometry Building Blocks:

  • Line Segment: A portion of a line consisting of two endpoints and all the points between them. Represented notationally as $\overline{AB}$.

  • Ray: A part of a line consisting of one endpoint and all points extending infinitely in one direction. Notation: $\overrightarrow{AB}$ (where $A$ is the endpoint).

  • Angle: Formed by two rays sharing a common endpoint called the vertex.

• Types of Lines:

  • Parallel Lines: Lines in the same plane that never intersect. Symbol: $\parallel$.

  • Perpendicular Lines: Lines that intersect at a $90^\circ$ angle. Symbol: $\perp$.

  • Skew Lines: Lines that are not in the same plane and do not intersect.

• Angles Specifics:

  • Parts of an Angle: The vertex (common endpoint) and the sides (the rays).

• Congruence:

  • Definition: Geometric figures are congruent if they have the same shape and size.

  • Notation: $\cong$.

• Special Angle Relationships:

  • Adjacent Angles: Two angles that share a common vertex and a common side but no interior points.

  • Complementary Angles: Two angles whose measures sum to $90^\circ$.

  • Supplementary Angles: Two angles whose measures sum to $180^\circ$.

  • Linear Pair: A pair of adjacent angles whose non-common sides are opposite rays (forming a line). These are always supplementary.

  • Vertical Angles: Two non-adjacent angles formed by two intersecting lines. Vertical angles are always congruent.

Polygons and 2-D Shapes

• Polygon Definition: A closed plane figure formed by three or more line segments that intersect only at their endpoints.

• Types of Polygons:

  • Convex: No line containing a side of the polygon goes through the interior of the polygon.

  • Concave: At least one line containing a side goes through the interior (it "caves in").

  • Regular: A polygon that is both equilateral (all sides equal) and equiangular (all angles equal).

  • Irregular: A polygon that is not regular.

• Names of Polygons by Side Count ($n$):

  • $3$ sides: Triangle.

  • $4$ sides: Quadrilateral.

  • $5$ sides: Pentagon.

  • $6$ sides: Hexagon.

  • $7$ sides: Heptagon.

  • $8$ sides: Octagon.

  • $9$ sides: Nonagon.

  • $10$ sides: Decagon.

  • $12$ sides: Dodecagon.

  • $n$ sides: n-gon.

• General 2D Measurements:

  • Perimeter: The continuous line forming the boundary of a closed geometric figure; the total length of the sides.

  • Area: The size of a surface; the number of unit squares that can fit inside the shape.

• Circles and Curved Figures:

  • Circle: The set of all points in a plane that are a fixed distance from a center point.

  • Arc: A portion of the circumference of a circle.

  • Diameter: A line segment passing through the center of a circle, with endpoints on the circle.

  • Radius: A line segment from the center of the circle to any point on the circle (half the diameter).

  • Circumference: The perimeter of a circle ($C = 2\pi r$ or $C = \pi d$).

  • Arc Length: The distance along an arc.

Solids and 3-D Figures

• Solid: A three-dimensional object.

• Net: A two-dimensional pattern that can be folded to create a three-dimensional solid.

• Types of Solids:

  • Polygon Faces: Includes Prisms (two congruent parallel bases) and Pyramids (one base, other faces meet at a vertex).

  • Curved Surface Faces: Includes Cylinders (two circular bases), Cones (one circular base and a vertex), and Spheres (all points equidistant from the center in 3D).

• Measuring Solids:

  • Surface Area: The total area of all faces and curved surfaces of a solid.

  • Volume: The total amount of space that a solid occupies.

Lesson 1.2: Measuring Segments

• Key Placement Concepts:

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

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

• Segment Addition Postulate: If point $B$ is between $A$ and $C$, then $AB + BC = AC$.

• Examples of Segment Addition:

  1. If $AB = 5$ and $BC = 3$, then $AC = 5 + 3 = 8$.

  2. If $AC = 21$ and $AB = 12$, then $BC = 21 - 12 = 9$.

  3. If $AB = 2x + 4$, $BC = 4x - 3$, and $AC = 3x + 46$. Find $x$, $AB$, $BC$, and $AC$.        Applying postulate: $(2x + 4) + (4x - 3) = 3x + 46$        $6x + 1 = 3x + 46$        $3x = 45 \rightarrow x = 15$        Substituting back: $AB = 2(15) + 4 = 34$, $BC = 4(15) - 3 = 57$, $AC = 125$ (Wait, let's recheck the calculation against the postulate: $34 + 57 = 91$. Note: verify if total matches calculated $3(15) + 46 = 45+46 = 91$).

  4. If $PQ = x$, $QR = 2x$, $RS = x + 5$, and $PS = 33$. Find $x$ and segment lengths.        $x + 2x + (x + 5) = 33$        $4x + 5 = 33$        $4x = 28 \rightarrow x = 7$        Lengths: $PQ = 7$, $QR = 14$, $RS = 12$.

• Bisectors: A point, line, or plane that divides a segment into two congruent segments.

• Midpoint Concept:

  • Definition: The point that divides a segment into two congruent segments.

  • Formula: For endpoints at $(x_1, y_1)$ and $(x_2, y_2)$, Midpoint $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

  • Endpoint Calculation Example: If one endpoint is $(1, 1)$ and the midpoint is $(4, 2)$, the other endpoint $(x, y)$ is found via:       $\frac{1 + x}{2} = 4 \rightarrow 1 + x = 8 \rightarrow x = 7$       $\frac{1 + y}{2} = 2 \rightarrow 1 + y = 4 \rightarrow y = 3$       The endpoint is $(7, 3)$.

• Distance Formula: Used to calculate the length between two points $(x_1, y_1)$ and $(x_2, y_2)$.

  • Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Lesson 1.3: Measuring Angles

• Angle Naming Conventions: Angles can be named by their vertex (if only one angle is present), by three points (vertex in the middle, e.g., $\angle ABC$), or by a number assigned to it.

• Adjacent Angles & Angle Addition Postulate:

  • Definition: If point $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.

  • Example: If $m\angle ABD = 30^\circ$ and $m\angle CBD = 47^\circ$, then $m\angle ABC = 30 + 47 = 77^\circ$.

  • Complex Solving Example: If $m\angle MQJ = 135^\circ$, $m\angle MKL = 6x$, and $m\angle LKJ = 16x + 3$, then $6x + 16x + 3 = 135 \rightarrow 22x + 3 = 135 \rightarrow 22x = 132 \rightarrow x = 6$. Substitute to find $m\angle LKJ = 16(6) + 3 = 99^\circ$.

• Tools and Classification:

  • Protractor: Tools used to measure angles in degrees.

  • Protractor Postulate: For any ray $OA$, there is a real number between $0$ and $180$ corresponding to the ray relative to the straight angle.

  • Classification:

    • Acute: 0^\circ < \theta < 90^\circ.

    • Right: $\theta = 90^\circ$.

    • Obtuse: 90^\circ < \theta < 180^\circ.

    • Straight: $\theta = 180^\circ$.

Lesson 1.4: Angle Pairs and Relationships

• Vertical Angles: Angles opposite each other when two lines intersect. They are congruent ($\angle 1 \cong \angle 3$).

• Linear Pair: Two adjacent angles whose non-common sides form a line. They are supplementary ($m\angle 1 + m\angle 2 = 180^\circ$).

• Angle Bisector: A ray that divides an angle into two congruent angles.

• Measurement Examples:

  1. If $m\angle 1 = 120^\circ$, then its vertical angle $m\angle 3 = 120^\circ$, and its linear pair supplement $m\angle 2 = 180 - 120 = 60^\circ$.

  2. If $\angle ADB$ and $\angle BDC$ are complementary, then their measures add to $90^\circ$. Solve for $x$ given algebraic expressions for the measurements.

Lesson 1.5: Constructions

• Standard Tools:

  • Compass: A tool used to draw circles and arcs, and to transfer distances.

  • Straightedge: A tool with no markings used to draw straight lines (not to be used for measurement).

  • Ruler: A marked tool used to measure lengths (rarely used in classical construction proofs).

• Key Construction Procedures:

  • Copying a segment or angle.

  • Constructing a perpendicular bisector of a segment.

  • Bisecting an existing angle.

  • Constructing a line perpendicular to a line through a point on the line or not on the line.

  • Constructing parallel lines by copying a corresponding angle.

Lesson 1.6: Conditional Statements

• Conditional Form ($p \rightarrow q$):

  • Hypothesis ($p$): The "if" part of the statement.

  • Conclusion ($q$): The "then" part of the statement.

• Forms of Conditionals:

  • If-Then Form: "If it rains ($p$), then the ground is wet ($q$)."

  • Implication Form: "Rain implies the ground will be wet."

  • Only If Form: "You will play the game only if it is Friday."

• Negation ($\sim p$): The opposite of the original statement. A statement that has the opposite truth value.

• Truth Value: Either true (T) or false (F). A Counterexample is a specific instance that proves a statement false.

• Converse, Inverse, Contrapositive:

  • Original Statement: If an animal is a panda ($p$), then it is a bear ($q$). ($p \rightarrow q$)

  • Converse: If an animal is a bear, then it is a panda. ($q \rightarrow p$)

  • Inverse: If an animal is not a panda, then it is not a bear. ($\sim p \rightarrow \sim q$)

  • Contrapositive: If an animal is not a bear, then it is not a panda. ($\sim q \rightarrow \sim p$)

• Logical Equivalence: Two statements are logically equivalent if they always have the same truth value.

  • The Conditional and the Contrapositive are logically equivalent.

  • The Converse and the Inverse are logically equivalent.

• Biconditional Statements ($p \leftrightarrow q$):

  • The combination of a conditional statement and its converse when both are true.

  • Phrased using "if and only if" (iff).

  • Example: "The sum of two angles is $180^\circ$ if and only if they are supplementary."

• The Law of Contrapositive: If the conditional statement is true, the contrapositive must be true.

Lesson 1.7: Inductive and Deductive Reasoning

• Inductive Reasoning: The process of reasoning that a rule or statement is true because specific cases are true. Based on visual patterns and observations. Useful for making conjectures.

• Deductive Reasoning: The process of using logic to draw conclusions from given facts, definitions, and properties. Based on facts, laws, and rules. Accuracy is guaranteed if the premises are true.

• Logic Examples:

  1. If $4 + x = 7$, then $x = 3$: Deductive (Algebraic law).

  2. Sequence 5, 7, 9, 11$,$ then next is 13: **Inductive** (Pattern recognition).\n 3. Conjectures based on observation (Doctors observing weight loss from Dimatrin): **Inductive**.\n* **Law of Detachment:** If p \rightarrow q$is a true statement and$p$is true, then$q is true.\n * Example: Rule: Firefighters must pass training (p \rightarrow q$). Fact: Bryan is a firefighter ($p$). Conclusion: Bryan passed training ($q).\n* **Law of Syllogism:** If p \rightarrow q$and$q \rightarrow r$are true statements, then$p \rightarrow r is a true statement.\n * Example: If a = b$and$b = 3$, then$a = 3.\n* **Logic Puzzle:**\n * Janna, Zeriah, and Bethany vote on Soccer, Volleyball, or Basketball. Each voted for a different sport.\n * Fact 1: If Janna voted, she voted for basketball.\n * Fact 2: If Bethany voted, she did not vote for soccer.\n * **Deduction:** Janna = Basketball. Bethany voted for something other than soccer, so Bethany = Volleyball. This leaves Zeriah = Soccer.\n\n# Lesson 1.8: Algebraic Proofs\n\n* **Fundamental Properties of Equality:**\n * **Reflexive Property:** a = a.\n * **Symmetric Property:** If a = b$, then$b = a.\n * **Transitive Property:** If a = b$and$b = c$, then$a = c.\n* **Algebraic Operations Properties:**\n * **Identity Property:** a + 0 = a$;$a \times 1 = a.\n * **Inverse Property:** a + (-a) = 0$;$a \times \frac{1}{a} = 1.\n * **Associative Property:** (a + b) + c = a + (b + c).\n * **Commutative Property:** a + b = b + a.\n * **Distributive Property:** a(b + c) = ab + ac.\n* **Equality Properties used in Proofs:**\n * **Substitution:** If a = b$, then$a$can be substituted for$b in any expression.\n * **Addition Property:** If a = b$, then$a + c = b + c.\n * **Subtraction Property:** If a = b$, then$a - c = b - c.\n * **Multiplication Property:** If a = b$, then$ac = bc.\n * **Division Property:** If a = b$and$c \neq 0$, then$\frac{a}{c} = \frac{b}{c}.\n * **Square Root Property:** If a^2 = b$, then$a = \sqrt{b}.\n* **Types of Proofs:**\n * **Two-Column Proof:** A formal structure where Statements are in the left column and Reasons (properties, theorems, definitions) are in the right column.\n * **Flow Chart:** A visual representation of logical steps.\n * **Paragraph Proof:** A written explanation of the reasoning.\n\n# Lesson 1.9: Segment and Angle Proofs\n\n* **Proving Segment Congruence Example:**\n * Given: WY = XZ$. Prove:$WX = YZ.\n * Logical process: Use Segment Addition Postulate (WY = WX + XY$and$XZ = XY + YZ$). By substitution,$WX + XY = XY + YZ$. Use subtraction property to remove$XY$from both sides results in$WX = YZ.\n* **Vertical Angles Theorem:**\n * **Theorem:** If two angles are vertical, then they are congruent.\n * **Proof Context:** Given intersecting lines PQ$and$RS$at vertex$B$.$\angle PBR$and$\angle RBS$form a linear pair (sum is$180^\circ$).$\angle SBQ$and$\angle RBS$form a linear pair (sum is$180^\circ$). By subtraction/substitution,$\angle PBR \cong \angle SBQ.\n * **Application Example:** If m\angle RBQ = 82^\circ$, then its vertical angle$m\angle PBS = 82^\circ$. The supplementary angles$m\angle PBR = 180 - 82 = 98^\circ$and$m\angle SBQ = 98^\circ$$.