Deductive Reasoning and Geometry Proofs Study Guide

Lesson Overview and Objectives

• Central Theme: Writing proofs is a process of using deductive reasoning to prove statements based on established truths such as postulates, axioms, and previously proven theorems.
• Prior Knowledge: Foundations in using deductive reasoning for proving statements.
• New Objectives:
  • Analyze figures to identify and use postulates about points, lines, and planes.
  • Analyze and construct viable arguments in several proof formats, including flow proofs, algebraic proofs, and paragraph proofs.
• Physics Connection: The lesson draws a parallel to Newton's laws of gravity and inertia. For example, if a rock and a feather are dropped from the same height in a vacuum chamber, they fall at the same rate. This law is accepted as a fundamental truth of physics, much like postulates in geometry must be accepted as true to build further arguments.
• Mathematical Practices:
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.

Key Vocabulary

• Postulate: A statement that is accepted as true without proof.
• Axiom: Another term for a postulate; a statement accepted as true without evidence.
• Proof: A logical argument in which each statement made is supported by a statement that is accepted as true.
• Flow Proof: A proof format that uses statements written in boxes and arrows to show the logical progression of an argument, with reasons written below each box.
• Deductive Argument: An argument built on a chain of logical statements leading from a hypothesis to a conclusion.
• Algebraic Proof: A proof composed of a series of algebraic statements justified by properties of real numbers.
• Theorem: A statement or conjecture that has been proven true.
• Paragraph Proof: An informal proof written in a narrative format explaining why a conjecture is true.
• Informal Proof: Another term for a paragraph proof.
• Formal Proof: Often refers to a two-column proof containing statements and reasons organized in a systematic table.

Postulates: Points, Lines, and Planes

Postulates 2.1 through 2.7 define the fundamental relationships between points, lines, and planes that serve as the building blocks for geometric proofs.

• Postulate 2.1: Through any two points, there is exactly one line.
  • Example: Line $n$ is the only line through points $P$ and $R$.
• Postulate 2.2: Through any three noncollinear points, there is exactly one plane.
  • Example: Plane $K$ is the only plane through noncollinear points $A$, $B$, and $C$.
• Postulate 2.3: A line contains at least two points.
  • Example: Line $n$ contains points $P$, $Q$, and $R$.
• Postulate 2.4: A plane contains at least three noncollinear points.
  • Example: Plane $K$ contains noncollinear points $L$, $B$, $C$, and $E$.
• Postulate 2.5: If two points lie in a plane, then the entire line containing those points lies in that plane.
  • Example: Points $A$ and $B$ lie in plane $K$, and line $m$ contains points $A$ and $B$; therefore, line $m$ is in plane $K$.
• Postulate 2.6: If two lines intersect, then their intersection is exactly one point.
  • Example: Lines $s$ and $t$ intersect at point $P$.
• Postulate 2.7: If two planes intersect, then their intersection is a line.
  • Example: Planes $F$ and $G$ intersect in line $w$.

Properties of Real Numbers

Algebraic proofs rely on several properties of equality that are true for any real numbers $a$, $b$, and $c$. These properties provide the justification for solving equations step-by-step.

• Addition Property of Equality: If $a = b$, then $a + c = b + c$.
• Subtraction Property of Equality: If $a = b$, then $a - c = b - c$.
• Multiplication Property of Equality: If $a = b$, then $a \times c = b \times c$.
• Division Property of Equality: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.
• Reflexive Property of Equality: $a = a$.
• Symmetric Property of Equality: If $a = b$, then $b = a$.
• Transitive Property of Equality: If $a = b$ and $b = c$, then $a = c$.
• Substitution Property of Equality: If $a = b$, then $a$ may be replaced by $b$ in any equation or expression.
• Distributive Property of Equality: $a(b + c) = ab + ac$.

Flow Proofsand the Midpoint Theorem

A flow proof visualizes the deduction process. It can be written vertically or horizontally.

How to Write a Flow Proof

• Step 1: Write the given statement in a box and the word "Given" underneath.
• Step 2: Create a deductive argument by forming a logical chain of statements linking the given information to what you are trying to prove. Write these in linked boxes and justify each with a reason underneath.
• Step 3: State the final proved conclusion.

Theorem 2.1: Midpoint Theorem

• Definition: If $M$ is the midpoint of $AB$, then $AM \cong MB$.
• Practical Application (Example 4):
  • Given: $Q$ is the midpoint of $PR$.
  • Prove: $PQ = QR$.
  • Proof Logic: Start with the box "$Q$ is the midpoint of $PR$" (Given). An arrow leads to the next box: "$PQ \cong QR$" (Definition of midpoint). A final arrow leads to "$PQ = QR$" (Definition of congruence).

Paragraph Proofs

Paragraph proofs are narrative explanations that are logically equivalent to formal proofs but written in sentences. They are also known as informal proofs.

How to Write a Paragraph Proof

• Step 1: Write the "Given" and "Prove" statements.
• Step 2: Draw a diagram and label any given information.
• Step 3: Write the proof as a deductive argument, forming a logical chain of statements. Justify each statement with definitions, postulates, algebraic properties, or theorems.
• Step 4: State the final conclusion of what has been proved.

Example 5: Midpoint Verification

• Given: $C$ is between $A$ and $B$ and $AC = CB$.
• Prove: $C$ is the midpoint of $AB$.
• Structure: The proof uses the definition of midpoint to conclude that because $C$ is between the points and the segments are equal in length, $C$ must be the midpoint.

Ruler and Segment Addition Postulates

• Postulate 2.8: Ruler Postulate: The points on any line or line segment can be put into one-to-one correspondence with real numbers. Given any two points $A$ and $B$ on a line, if $A$ corresponds to zero, then $B$ corresponds to a positive real number.
• Postulate 2.9: Segment Addition Postulate: If $A$, $B$, and $C$ are collinear, then point $B$ is between $A$ and $C$ if and only if $AB + BC = AC$.

Properties of Segment Congruence

Theorem 2.2 identifies that the properties of equality also apply to the congruence of segments.

• Reflexive Property of Congruence: $AB \cong AB$.
• Symmetric Property of Congruence: If $AB \cong CD$, then $CD \cong AB$.
• Transitive Property of Congruence: If $AB \cong CD$ and $CD \cong EF$, then $AB \cong EF$.

Proving the Transitive Property of Congruence (Paragraph Proof)

• Given: $AB \cong CD$ and $CD \cong EF$.
• Prove: $AB \cong EF$.
• Logic: Because $AB \cong CD$ and $CD \cong EF$, it follows that $AB = CD$ and $CD = EF$ by the definition of congruent segments. By the Transitive Property of Equality, $AB = EF$. Thus, $AB \cong EF$ by the definition of congruence.

Real-World Application: Charity Fitness Run

• Scenario: A route has checkpoints $X$ and $Z$ as midpoints. $X$ is the midpoint between the starting line and Checkpoint $Y$. $Z$ is the midpoint between Checkpoint $Y$ and the finish line $F$.
• Task: If Checkpoint $Y$ is the same distance from Checkpoints $X$ and $Z$, prove the route from $Z$ to the finish line is congruent to the route from the start to $X$.
• Variables:
  • Start to $X = SX$
  • $X$ to $Y = XY$
  • $Y$ to $Z = YZ$
  • $Z$ to Finish = $ZF$
• Logic Chain:
  1. $X$ is midpoint of $SY \implies SX = XY$.
  2. $Z$ is midpoint of $YF \implies YZ = ZF$.
  3. Given $XY = YZ$.
  4. By Transitive Property: if $SX = XY$ and $XY = YZ$, then $SX = YZ$.
  5. If $SX = YZ$ and $YZ = ZF$, then $SX = ZF$.
  6. Therefore, $SX \cong ZF$.