Proving Statements About Segments and Angles
Your first formal proofs using what you know
Have you ever had to convince someone of something by walking them through your reasoning step by step? Maybe you explained to a friend why a certain route is faster, or showed a coworker why a particular solution to a problem makes sense. If so, you have already done the essential work of mathematical proof: taking what you know for certain and building a logical chain of reasoning to reach a conclusion.
Proofs in geometry are exactly this kind of careful, step-by-step argument. The difference is that we write them down in a structured way so that anyone reading can follow our logic and see that our conclusion must be true. There is something deeply satisfying about a well-constructed proof. It is not about memorizing steps; it is about understanding why something is true and being able to show others that your reasoning is airtight.
Core Concepts
What is a Proof?
A proof is a logical argument that demonstrates why a mathematical statement must be true. Think of it as building a bridge from what you already know (your given information and established facts) to what you want to show (your conclusion). Every step along that bridge must be supported by a valid reason.
In everyday life, we make arguments all the time, but we often skip steps or rely on things “everyone knows.” In a mathematical proof, we are more careful. Every claim must be backed up by a definition, a property, a postulate (something we accept as true without proof), or a theorem (something that has been proven before).
Why bother being so careful? Because mathematical proofs give us certainty. When you prove something in geometry, you know it is true in all cases, not just the ones you have checked. That kind of certainty is rare and powerful.
The Two-Column Proof Format
The most common format for writing geometry proofs is the two-column proof. It looks like a table with two columns:
| Statements | Reasons |
|---|---|
| What you write down | Why you can write it |
The left column contains your statements - the mathematical claims you are making. The right column contains your reasons - the justification for each claim.
A two-column proof always starts with what you are given and ends with what you wanted to prove. Here is the basic structure:
| Statements | Reasons |
|---|---|
| (Given information) | Given |
| (Next logical step) | (Why this step is valid) |
| … | … |
| (What you wanted to prove) | (Final justification) |
The beauty of this format is its clarity. Anyone can trace your reasoning from start to finish and see exactly why each step follows from what came before.
Given and Prove: Understanding the Setup
Every proof problem starts with two pieces of information:
Given: This is your starting point - the information the problem tells you to accept as true. It might say something like “Given: $M$ is the midpoint of $\overline{AB}$” or “Given: $\angle 1$ and $\angle 2$ are supplementary.”
Prove: This is your destination - the statement you need to show must be true. It might say “Prove: $\overline{AM} \cong \overline{MB}$” or “Prove: $m\angle 1 + m\angle 2 = 180°$.”
Your job is to build a logical bridge from the Given to the Prove, one step at a time.
Properties of Equality
When working with segment lengths and angle measures (which are numbers), we can use the properties of equality. These might seem obvious, but naming them explicitly lets us use them as valid reasons in proofs.
Reflexive Property of Equality: Any quantity equals itself. $$a = a$$
This might seem pointless, but it comes up more often than you would expect. Sometimes you need to establish that two expressions share a common term.
Symmetric Property of Equality: If one quantity equals another, you can flip the equation. $$\text{If } a = b, \text{ then } b = a$$
This lets you rewrite equations in whichever direction is more useful.
Transitive Property of Equality: If one quantity equals a second, and that second equals a third, then the first equals the third. $$\text{If } a = b \text{ and } b = c, \text{ then } a = c$$
This is the “chain” property - it lets you connect equalities together.
Substitution Property of Equality: If two quantities are equal, you can substitute one for the other in any expression. $$\text{If } a = b, \text{ then } a \text{ can replace } b \text{ in any equation or expression}$$
This is incredibly useful. Once you know two things are equal, you can swap them freely.
Addition Property of Equality: If you add the same quantity to both sides of an equation, the equation remains true. $$\text{If } a = b, \text{ then } a + c = b + c$$
Subtraction Property of Equality: If you subtract the same quantity from both sides of an equation, the equation remains true. $$\text{If } a = b, \text{ then } a - c = b - c$$
Multiplication Property of Equality: If you multiply both sides of an equation by the same quantity, the equation remains true. $$\text{If } a = b, \text{ then } ac = bc$$
Division Property of Equality: If you divide both sides of an equation by the same nonzero quantity, the equation remains true. $$\text{If } a = b \text{ and } c \neq 0, \text{ then } \frac{a}{c} = \frac{b}{c}$$
Properties of Congruence
When working with segments and angles themselves (not just their measures), we use congruence instead of equality. The symbol $\cong$ means “is congruent to” - same size and shape.
The properties of congruence mirror the properties of equality:
Reflexive Property of Congruence: Any segment is congruent to itself. Any angle is congruent to itself. $$\overline{AB} \cong \overline{AB}$$ $$\angle A \cong \angle A$$
Symmetric Property of Congruence: If one segment is congruent to another, you can state it in either order. $$\text{If } \overline{AB} \cong \overline{CD}, \text{ then } \overline{CD} \cong \overline{AB}$$ $$\text{If } \angle A \cong \angle B, \text{ then } \angle B \cong \angle A$$
Transitive Property of Congruence: If the first is congruent to the second, and the second is congruent to the third, then the first is congruent to the third. $$\text{If } \overline{AB} \cong \overline{CD} \text{ and } \overline{CD} \cong \overline{EF}, \text{ then } \overline{AB} \cong \overline{EF}$$ $$\text{If } \angle A \cong \angle B \text{ and } \angle B \cong \angle C, \text{ then } \angle A \cong \angle C$$
Definitions as Reasons in Proofs
One of the most common types of reasons in a proof is a definition. Definitions work in two directions:
- If you know something fits a definition, you can conclude it has the properties of that definition.
- If you know something has the properties of a definition, you can conclude it fits that definition.
Here are some key definitions you will use:
Definition of Midpoint: A point $M$ is the midpoint of $\overline{AB}$ if and only if $\overline{AM} \cong \overline{MB}$ (or equivalently, $AM = MB$).
Definition of Segment Bisector: A line, segment, or ray that passes through the midpoint of a segment bisects that segment.
Definition of Angle Bisector: A ray $\overrightarrow{BD}$ is the bisector of $\angle ABC$ if and only if $\angle ABD \cong \angle DBC$ (or equivalently, $m\angle ABD = m\angle DBC$).
Definition of Congruent Segments: $\overline{AB} \cong \overline{CD}$ if and only if $AB = CD$ (they have the same length).
Definition of Congruent Angles: $\angle A \cong \angle B$ if and only if $m\angle A = m\angle B$ (they have the same measure).
Definition of Complementary Angles: Two angles are complementary if and only if the sum of their measures is $90°$.
Definition of Supplementary Angles: Two angles are supplementary if and only if the sum of their measures is $180°$.
Definition of Right Angle: An angle is a right angle if and only if its measure is $90°$.
Key Theorems About Segments and Angles
Once something has been proven, it becomes a theorem that you can use as a reason in future proofs. Here are some important ones:
Segment Addition Postulate: If $B$ is between $A$ and $C$, then $AB + BC = AC$.
Angle Addition Postulate: If $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.
Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.
Congruent Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then they are congruent to each other.
Congruent Complements Theorem: If two angles are complementary to the same angle (or to congruent angles), then they are congruent to each other.
Vertical Angles Theorem: Vertical angles are congruent.
Writing Proofs Step by Step
Here is a strategy for approaching proofs:
Step 1: Understand the problem. Read the Given and Prove carefully. What do you know? What do you need to show?
Step 2: Draw a diagram if one is not provided. Label everything. Mark congruent segments with tick marks and congruent angles with arcs.
Step 3: Plan your approach. Ask yourself: What definitions, properties, or theorems connect my Given information to what I need to Prove? Sometimes it helps to work backward - what would I need to know to prove my conclusion?
Step 4: Write the proof. Start with your Given information. Each new statement should follow logically from previous statements, and each must have a valid reason.
Step 5: Check your work. Does every statement have a reason? Does the logic flow smoothly from Given to Prove? Have you actually reached your conclusion?
Common Proof Strategies
As you write more proofs, you will develop an intuition for which approaches work. Here are some common strategies:
Use definitions. If your Given says “$M$ is the midpoint of $\overline{AB}$,” immediately apply the definition to get $\overline{AM} \cong \overline{MB}$.
Look for shared parts. When two figures share a common segment or angle, the Reflexive Property tells you that part is congruent to itself.
Work with what you have. If you have angle measures, use properties of equality. If you have congruent segments or angles, use properties of congruence.
Connect equals to congruence. Remember that $AB = CD$ means the same thing as $\overline{AB} \cong \overline{CD}$. Use the definition of congruent segments to switch between them.
Look for supplementary or complementary relationships. Angles that form a linear pair are supplementary. Angles that together form a right angle are complementary.
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Proof | Logical argument showing why a statement must be true | Two-column format |
| Given | Starting information you accept as true | What you know |
| Prove | Statement you must demonstrate is true | Your goal |
| Theorem | Statement that has been proven true | Can be used as a reason |
| Postulate | Statement accepted as true without proof | Segment Addition Postulate |
| Reflexive | $a = a$ or $\overline{AB} \cong \overline{AB}$ | Anything equals/is congruent to itself |
| Symmetric | If $a = b$, then $b = a$ | Order can be reversed |
| Transitive | If $a = b$ and $b = c$, then $a = c$ | Chain equalities together |
| Midpoint | Point that divides a segment into two congruent parts | $M$ is midpoint of $\overline{AB}$ means $AM = MB$ |
| Angle Bisector | Ray that divides an angle into two congruent angles | $\overrightarrow{BD}$ bisects $\angle ABC$ means $\angle ABD \cong \angle DBC$ |
| Linear Pair | Two adjacent angles that form a straight line | Measures sum to $180°$ |
Examples
Complete the following proof by filling in the missing reasons.
Given: $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$
Prove: $\overline{AB} \cong \overline{EF}$
| Statements | Reasons |
|---|---|
| 1. $\overline{AB} \cong \overline{CD}$ | 1. ? |
| 2. $\overline{CD} \cong \overline{EF}$ | 2. ? |
| 3. $\overline{AB} \cong \overline{EF}$ | 3. ? |
Solution:
| Statements | Reasons |
|---|---|
| 1. $\overline{AB} \cong \overline{CD}$ | 1. Given |
| 2. $\overline{CD} \cong \overline{EF}$ | 2. Given |
| 3. $\overline{AB} \cong \overline{EF}$ | 3. Transitive Property of Congruence |
The first two statements come directly from the Given information. The third statement uses the Transitive Property: since $\overline{AB}$ is congruent to $\overline{CD}$, and $\overline{CD}$ is congruent to $\overline{EF}$, we can conclude $\overline{AB}$ is congruent to $\overline{EF}$.
Think of it like a chain: if A links to B, and B links to C, then A links to C.
Given: $M$ is the midpoint of $\overline{PQ}$
Prove: $\overline{PM} \cong \overline{MQ}$
Solution:
| Statements | Reasons |
|---|---|
| 1. $M$ is the midpoint of $\overline{PQ}$ | 1. Given |
| 2. $\overline{PM} \cong \overline{MQ}$ | 2. Definition of Midpoint |
This proof is short because the definition of midpoint directly gives us what we need to prove. The definition says: a point is the midpoint of a segment if and only if it divides the segment into two congruent parts. Since $M$ is the midpoint of $\overline{PQ}$, it must divide $\overline{PQ}$ into two congruent segments, $\overline{PM}$ and $\overline{MQ}$.
Notice how definitions serve as powerful bridges in proofs. Once you know something satisfies a definition, you immediately know it has all the properties that definition guarantees.
Given: $\overrightarrow{BD}$ bisects $\angle ABC$
Prove: $m\angle ABD = m\angle DBC$
Solution:
| Statements | Reasons |
|---|---|
| 1. $\overrightarrow{BD}$ bisects $\angle ABC$ | 1. Given |
| 2. $\angle ABD \cong \angle DBC$ | 2. Definition of Angle Bisector |
| 3. $m\angle ABD = m\angle DBC$ | 3. Definition of Congruent Angles |
Let us trace the logic:
- We start with the Given: the ray $\overrightarrow{BD}$ bisects angle $ABC$.
- By the definition of an angle bisector, a ray that bisects an angle divides it into two congruent angles. So $\angle ABD \cong \angle DBC$.
- Finally, what does it mean for two angles to be congruent? By the definition of congruent angles, they have equal measures. So $m\angle ABD = m\angle DBC$.
This example shows how we often need to “unpack” a geometric fact into a numerical one. We started with bisection (a geometric relationship), moved to congruence, and ended with equal measures (a numerical relationship).
Given: $B$ is between $A$ and $C$; $AB = BC$
Prove: $AB = \frac{1}{2}AC$
Solution:
| Statements | Reasons |
|---|---|
| 1. $B$ is between $A$ and $C$ | 1. Given |
| 2. $AB + BC = AC$ | 2. Segment Addition Postulate |
| 3. $AB = BC$ | 3. Given |
| 4. $AB + AB = AC$ | 4. Substitution Property of Equality |
| 5. $2 \cdot AB = AC$ | 5. Simplification (combining like terms) |
| 6. $AB = \frac{1}{2}AC$ | 6. Division Property of Equality |
Let us walk through the reasoning:
- Steps 1 and 3 state our Given information.
- Step 2 applies the Segment Addition Postulate: when a point is between two others, the whole segment equals the sum of the parts.
- Step 4 uses substitution: since $AB = BC$, we can replace $BC$ with $AB$ in the equation from step 2.
- Step 5 simplifies $AB + AB$ to $2 \cdot AB$.
- Step 6 divides both sides by 2 to isolate $AB$, giving us our desired conclusion.
This proof shows a common pattern: start with a geometric relationship (point between two others), convert to an equation (Segment Addition Postulate), then use algebra (substitution, simplification, division) to reach your conclusion.
Given: $\angle 1$ and $\angle 2$ are supplementary; $\angle 3$ and $\angle 2$ are supplementary
Prove: $\angle 1 \cong \angle 3$
Solution:
| Statements | Reasons |
|---|---|
| 1. $\angle 1$ and $\angle 2$ are supplementary | 1. Given |
| 2. $m\angle 1 + m\angle 2 = 180°$ | 2. Definition of Supplementary Angles |
| 3. $\angle 3$ and $\angle 2$ are supplementary | 3. Given |
| 4. $m\angle 3 + m\angle 2 = 180°$ | 4. Definition of Supplementary Angles |
| 5. $m\angle 1 + m\angle 2 = m\angle 3 + m\angle 2$ | 5. Transitive Property of Equality (both equal $180°$) |
| 6. $m\angle 1 = m\angle 3$ | 6. Subtraction Property of Equality |
| 7. $\angle 1 \cong \angle 3$ | 7. Definition of Congruent Angles |
The key insight in this proof is step 5: since both $m\angle 1 + m\angle 2$ and $m\angle 3 + m\angle 2$ equal $180°$, they must equal each other. From there, we subtract $m\angle 2$ from both sides to get that the remaining angles have equal measures, which means they are congruent.
This proof actually demonstrates the Congruent Supplements Theorem: if two angles are supplementary to the same angle, they are congruent to each other. Now that we have proven this, we could use “Congruent Supplements Theorem” as a reason in future proofs.
Notice the structure: we converted the geometric concept (supplementary) to equations, manipulated the equations algebraically, then converted back to a geometric conclusion (congruent).
Key Properties and Rules
Properties of Equality (for segment lengths and angle measures)
| Property | Statement | Use in Proofs |
|---|---|---|
| Reflexive | $a = a$ | Establish that a quantity equals itself |
| Symmetric | If $a = b$, then $b = a$ | Reverse the order of an equation |
| Transitive | If $a = b$ and $b = c$, then $a = c$ | Connect chains of equalities |
| Substitution | If $a = b$, then $a$ can replace $b$ | Replace one expression with an equal one |
| Addition | If $a = b$, then $a + c = b + c$ | Add the same amount to both sides |
| Subtraction | If $a = b$, then $a - c = b - c$ | Subtract the same amount from both sides |
| Multiplication | If $a = b$, then $ac = bc$ | Multiply both sides by the same amount |
| Division | If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$ | Divide both sides by the same nonzero amount |
Properties of Congruence (for segments and angles)
| Property | For Segments | For Angles |
|---|---|---|
| Reflexive | $\overline{AB} \cong \overline{AB}$ | $\angle A \cong \angle A$ |
| Symmetric | If $\overline{AB} \cong \overline{CD}$, then $\overline{CD} \cong \overline{AB}$ | If $\angle A \cong \angle B$, then $\angle B \cong \angle A$ |
| Transitive | If $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$, then $\overline{AB} \cong \overline{EF}$ | If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$ |
Essential Definitions for Proofs
Midpoint: $M$ is the midpoint of $\overline{AB}$ $\Leftrightarrow$ $AM = MB$ (or $\overline{AM} \cong \overline{MB}$)
Angle Bisector: $\overrightarrow{BD}$ bisects $\angle ABC$ $\Leftrightarrow$ $m\angle ABD = m\angle DBC$ (or $\angle ABD \cong \angle DBC$)
Congruent Segments: $\overline{AB} \cong \overline{CD}$ $\Leftrightarrow$ $AB = CD$
Congruent Angles: $\angle A \cong \angle B$ $\Leftrightarrow$ $m\angle A = m\angle B$
Supplementary Angles: $\angle A$ and $\angle B$ are supplementary $\Leftrightarrow$ $m\angle A + m\angle B = 180°$
Complementary Angles: $\angle A$ and $\angle B$ are complementary $\Leftrightarrow$ $m\angle A + m\angle B = 90°$
Key Theorems
Segment Addition Postulate: If $B$ is between $A$ and $C$, then $AB + BC = AC$.
Angle Addition Postulate: If $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.
Linear Pair Postulate: If two angles form a linear pair, they are supplementary.
Congruent Supplements Theorem: If two angles are each supplementary to the same angle (or congruent angles), then they are congruent.
Congruent Complements Theorem: If two angles are each complementary to the same angle (or congruent angles), then they are congruent.
Vertical Angles Theorem: Vertical angles are congruent.
Real-World Applications
Building Logical Arguments in Debates
The structure of a two-column proof mirrors how strong arguments work in any field. You start with facts everyone accepts (Given), then build a chain of reasoning where each step follows logically from what came before. Lawyers do this in courtrooms. Scientists do this when presenting research. Anyone trying to convince others of something needs to show their reasoning clearly and justify each step.
Troubleshooting and Systematic Elimination
When you troubleshoot a problem - whether it is a car that will not start, a computer program with a bug, or a recipe that did not turn out right - you are essentially doing proof-style reasoning. You start with what you know (the symptoms), apply your knowledge of how things work (definitions and properties), and logically work toward identifying the cause. Each step eliminates possibilities or confirms facts, just like steps in a proof.
Quality Assurance Processes
In manufacturing and software development, quality assurance involves proving that a product meets specifications. Testers document what is given (the specifications), what they need to prove (the product works correctly), and the logical steps showing that the product satisfies each requirement. This systematic verification mirrors geometric proof.
Mathematical Foundations for Computer Science
Computer programming relies heavily on logical thinking and proof-like reasoning. Program verification, algorithm correctness, and debugging all use the same skills you develop writing geometric proofs. If you continue into computer science, you will encounter formal proofs about programs that look remarkably similar to two-column proofs. The transitive, reflexive, and symmetric properties appear constantly in database theory and programming language design.
Architecture and Engineering
Before a building is constructed, engineers must prove their designs are sound. They start with given conditions (load requirements, material properties, environmental factors) and demonstrate through logical steps that the structure will be safe. Every calculation must be justified, every assumption documented - just like a geometric proof.
Self-Test Problems
Problem 1: Fill in the missing reason.
| Statements | Reasons |
|---|---|
| 1. $\angle A \cong \angle B$ | 1. Given |
| 2. $\angle B \cong \angle A$ | 2. ? |
Show Answer
Symmetric Property of Congruence
If $\angle A \cong \angle B$, we can state the same relationship in reverse order: $\angle B \cong \angle A$. This is the symmetric property - it lets us flip the direction of a congruence statement.
Problem 2: What property justifies the statement $\overline{XY} \cong \overline{XY}$?
Show Answer
Reflexive Property of Congruence
Any segment is congruent to itself. This might seem trivial, but the reflexive property is essential when you need to establish that two expressions or figures share a common part.
Problem 3: Complete the proof.
Given: $\angle 1$ and $\angle 2$ are complementary; $m\angle 1 = 35°$
Prove: $m\angle 2 = 55°$
Show Answer
| Statements | Reasons |
|---|---|
| 1. $\angle 1$ and $\angle 2$ are complementary | 1. Given |
| 2. $m\angle 1 + m\angle 2 = 90°$ | 2. Definition of Complementary Angles |
| 3. $m\angle 1 = 35°$ | 3. Given |
| 4. $35° + m\angle 2 = 90°$ | 4. Substitution Property of Equality |
| 5. $m\angle 2 = 55°$ | 5. Subtraction Property of Equality |
Problem 4: Write a complete two-column proof.
Given: $\overrightarrow{BD}$ bisects $\angle ABC$; $m\angle ABC = 84°$
Prove: $m\angle ABD = 42°$
Show Answer
| Statements | Reasons |
|---|---|
| 1. $\overrightarrow{BD}$ bisects $\angle ABC$ | 1. Given |
| 2. $\angle ABD \cong \angle DBC$ | 2. Definition of Angle Bisector |
| 3. $m\angle ABD = m\angle DBC$ | 3. Definition of Congruent Angles |
| 4. $m\angle ABD + m\angle DBC = m\angle ABC$ | 4. Angle Addition Postulate |
| 5. $m\angle ABC = 84°$ | 5. Given |
| 6. $m\angle ABD + m\angle DBC = 84°$ | 6. Substitution Property of Equality |
| 7. $m\angle ABD + m\angle ABD = 84°$ | 7. Substitution Property of Equality (from step 3) |
| 8. $2 \cdot m\angle ABD = 84°$ | 8. Simplification |
| 9. $m\angle ABD = 42°$ | 9. Division Property of Equality |
Problem 5: Complete the proof.
Given: $M$ is the midpoint of $\overline{AB}$; $N$ is the midpoint of $\overline{MB}$
Prove: $MN = \frac{1}{4}AB$
Show Answer
| Statements | Reasons |
|---|---|
| 1. $M$ is the midpoint of $\overline{AB}$ | 1. Given |
| 2. $AM = MB$ | 2. Definition of Midpoint |
| 3. $AM + MB = AB$ | 3. Segment Addition Postulate |
| 4. $MB + MB = AB$ | 4. Substitution Property of Equality |
| 5. $2 \cdot MB = AB$ | 5. Simplification |
| 6. $MB = \frac{1}{2}AB$ | 6. Division Property of Equality |
| 7. $N$ is the midpoint of $\overline{MB}$ | 7. Given |
| 8. $MN = NB$ | 8. Definition of Midpoint |
| 9. $MN + NB = MB$ | 9. Segment Addition Postulate |
| 10. $MN + MN = MB$ | 10. Substitution Property of Equality |
| 11. $2 \cdot MN = MB$ | 11. Simplification |
| 12. $MN = \frac{1}{2}MB$ | 12. Division Property of Equality |
| 13. $MN = \frac{1}{2} \cdot \frac{1}{2}AB$ | 13. Substitution Property of Equality (from step 6) |
| 14. $MN = \frac{1}{4}AB$ | 14. Simplification |
Problem 6: What is the difference between using “Definition of Congruent Angles” and “Transitive Property of Congruence” in a proof?
Show Answer
Definition of Congruent Angles connects congruence to measurement. It says: $\angle A \cong \angle B$ means the same thing as $m\angle A = m\angle B$. Use this when you need to convert between “congruent” statements and “equal measure” statements.
Transitive Property of Congruence connects three congruences together. It says: if $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$. Use this when you have a chain of congruence relationships and want to connect the first and last items.
The definition switches between congruence and equality. The transitive property chains congruences together.
Summary
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A proof is a logical argument that demonstrates why a mathematical statement must be true. It builds a chain of reasoning from known facts to a desired conclusion.
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Two-column proofs organize your reasoning with Statements in one column and Reasons in the other. Every statement must be justified by a valid reason.
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Every proof starts with Given information (what you know) and works toward the Prove statement (what you must show).
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The properties of equality (reflexive, symmetric, transitive, addition, subtraction, multiplication, division, substitution) apply to numerical quantities like segment lengths and angle measures.
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The properties of congruence (reflexive, symmetric, transitive) apply to geometric objects like segments and angles.
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Definitions are powerful tools in proofs. They work in both directions: if something satisfies a definition, it has the properties; if something has the properties, it satisfies the definition.
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Key definitions include: midpoint (divides a segment into two congruent parts), angle bisector (divides an angle into two congruent angles), congruent segments (equal lengths), congruent angles (equal measures), complementary angles (sum to $90°$), and supplementary angles (sum to $180°$).
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Theorems are statements that have been proven true. Once proven, they can be used as reasons in future proofs. Important ones include the Segment Addition Postulate, Angle Addition Postulate, and the theorems about congruent supplements and complements.
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When writing proofs, start with what you know, apply definitions and properties, and work step by step toward your conclusion. Check that every statement has a valid reason.
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Proof-writing skills transfer beyond geometry: building logical arguments, troubleshooting systematically, verifying quality, and programming all use the same careful, step-by-step reasoning.