Geometric Proofs

What is a Geometric Proof?

Proof: Logical argument showing a statement is true using:

• Given information
• Definitions
• Postulates
• Theorems
• Logical reasoning

Goal: Show why something must be true, not just that it appears true

Two-Column Proof Format

Two columns:

1. Statements: What you know or can conclude
2. Reasons: Why each statement is true

Structure:

Statements          | Reasons
--------------------|-------------------
1. Given info       | 1. Given
2. Next step        | 2. Definition/Theorem
3. Conclusion       | 3. Logic rule

Example: Simple Proof

Given: ∠A and ∠B are supplementary, m∠A = 120° Prove: m∠B = 60°

Statements                        | Reasons
----------------------------------|---------------------------
1. ∠A and ∠B are supplementary    | 1. Given
2. m∠A + m∠B = 180°              | 2. Definition of supplementary
3. m∠A = 120°                     | 3. Given
4. 120° + m∠B = 180°             | 4. Substitution
5. m∠B = 60°                      | 5. Subtraction Property

Common Reasons Used in Proofs

Given: Information stated in problem

Definition: Definition of mathematical term

Postulate: Accepted truth (no proof needed)

Theorem: Previously proven statement

Properties:

• Reflexive Property: a = a
• Symmetric Property: If a = b, then b = a
• Transitive Property: If a = b and b = c, then a = c
• Addition/Subtraction/Multiplication/Division Properties

Triangle Congruence

Congruent triangles: Same size and shape (all corresponding parts equal)

Notation: △ABC ≅ △DEF

Means:

• ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F
• AB ≅ DE, BC ≅ EF, AC ≅ DF

SSS (Side-Side-Side) Postulate

If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

Example: SSS Proof

Given: AB ≅ DE, BC ≅ EF, AC ≅ DF Prove: △ABC ≅ △DEF

Statements                        | Reasons
----------------------------------|---------------------------
1. AB ≅ DE                        | 1. Given
2. BC ≅ EF                        | 2. Given
3. AC ≅ DF                        | 3. Given
4. △ABC ≅ △DEF                    | 4. SSS Postulate

SAS (Side-Angle-Side) Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.

Important: Angle must be BETWEEN the two sides

Example: SAS Proof

Given: AB ≅ DE, ∠A ≅ ∠D, AC ≅ DF Prove: △ABC ≅ △DEF

Statements                        | Reasons
----------------------------------|---------------------------
1. AB ≅ DE                        | 1. Given
2. ∠A ≅ ∠D                        | 2. Given
3. AC ≅ DF                        | 3. Given
4. △ABC ≅ △DEF                    | 4. SAS Postulate

ASA (Angle-Side-Angle) Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

Important: Side must be BETWEEN the two angles

Example: ASA Proof

Given: ∠A ≅ ∠D, AB ≅ DE, ∠B ≅ ∠E Prove: △ABC ≅ △DEF

Statements                        | Reasons
----------------------------------|---------------------------
1. ∠A ≅ ∠D                        | 1. Given
2. AB ≅ DE                        | 2. Given
3. ∠B ≅ ∠E                        | 3. Given
4. △ABC ≅ △DEF                    | 4. ASA Postulate

AAS (Angle-Angle-Side) Theorem

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, the triangles are congruent.

Different from ASA: Side is NOT between the angles

Example: AAS Proof

Given: ∠A ≅ ∠D, ∠C ≅ ∠F, BC ≅ EF Prove: △ABC ≅ △DEF

Statements                        | Reasons
----------------------------------|---------------------------
1. ∠A ≅ ∠D                        | 1. Given
2. ∠C ≅ ∠F                        | 2. Given
3. BC ≅ EF                        | 3. Given
4. △ABC ≅ △DEF                    | 4. AAS Theorem

NOT Valid: AAA and SSA

AAA (Angle-Angle-Angle): NOT sufficient

• Triangles with same angles can be different sizes
• Shows similarity, not congruence

SSA (Side-Side-Angle): NOT sufficient

• Two sides and non-included angle
• Can create two different triangles
• Exception: HL for right triangles

HL (Hypotenuse-Leg) for Right Triangles

For right triangles only:

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent.

Example: HL Proof

Given: △ABC and △DEF are right triangles, AB ≅ DE (hypotenuses), AC ≅ DF Prove: △ABC ≅ △DEF

Statements                        | Reasons
----------------------------------|---------------------------
1. △ABC and △DEF are right △s     | 1. Given
2. AB ≅ DE                        | 2. Given
3. AC ≅ DF                        | 3. Given
4. △ABC ≅ △DEF                    | 4. HL Theorem

CPCTC

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Use: After proving triangles congruent, conclude any corresponding parts are congruent

Example: Using CPCTC

Given: △ABC ≅ △DEF Prove: ∠C ≅ ∠F

Statements                        | Reasons
----------------------------------|---------------------------
1. △ABC ≅ △DEF                    | 1. Given
2. ∠C ≅ ∠F                        | 2. CPCTC

Complete Proof Example

Given: Point M is midpoint of AB, AM = MB = 5, MC ⊥ AB, MC = 12 Prove: △AMC ≅ △BMC

Statements                        | Reasons
----------------------------------|---------------------------
1. M is midpoint of AB            | 1. Given
2. AM ≅ MB                        | 2. Definition of midpoint
3. MC ⊥ AB                        | 3. Given
4. ∠AMC and ∠BMC are right ∠s    | 4. Definition of perpendicular
5. ∠AMC ≅ ∠BMC                   | 5. All right angles are congruent
6. MC ≅ MC                        | 6. Reflexive Property
7. △AMC ≅ △BMC                    | 7. SAS Postulate

Proof Strategy

Steps to write a proof:

1. Draw and label diagram
2. Mark given information
3. Identify what to prove
4. Work backwards: What would make conclusion true?
5. Write statements and reasons in logical order
6. Check each step has valid reason

Common Proof Patterns

Proving segments/angles congruent:

• Show triangles congruent first
• Then use CPCTC

Proving triangles congruent:

• Identify three pairs of congruent parts
• Determine which postulate applies (SSS, SAS, ASA, AAS, HL)

Flowchart Proofs

Alternative format: Boxes with arrows showing logical flow

Each box contains:

• Statement (top)
• Reason (bottom)

Arrows: Show which statements lead to next conclusions

Paragraph Proofs

Written in paragraph form

Must still include:

• All statements
• All reasons
• Logical progression

Example: Paragraph Proof

Given: AB ≅ CD, BC ≅ DA Prove: △ABC ≅ △CDA

Proof: Since AB ≅ CD (given) and BC ≅ DA (given), and AC ≅ AC by the Reflexive Property, we have three pairs of congruent sides. Therefore, △ABC ≅ △CDA by the SSS Postulate.

Practice