Two-column and paragraph
Two-Column Proof:
A two-column proof is a type of mathematical proof that organizes the statements and reasons in two columns
Two-Column Proof:
A two-column proof is a type of mathematical proof that organizes the statements and reasons in two columns. The left column lists the statements, step by step, while the right column explains the reasons or justifications for each statement. This format allows for a clear and logical presentation of the steps taken to arrive at a mathematical conclusion. Two-column proofs are commonly used in geometry to prove theorems or solve geometric problems. It is important to provide accurate statements and valid reasons to ensure a valid proof.
Example of a Two-Column Proof:
Statement Reason
——————————————–
1. ∠ABD ≅ ∠CBD Given
2. AB ≅ CB Given
3. ∆ABD ≅ ∆CBD Angle-Side-Angle (ASA)
4. AD ≅ CD Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
5. DB ≅ DB Reflexive Property
6. ∆ABD ≅ ∆CBD Side-Angle-Side (SAS)
7. ∠ADB ≅ ∠CDB Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
8. ∠ADB ≅ ∠ADB Reflexive Property
9. AD ≅ CD Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
10. ∠ADB ≅ ∠CDB Angle-Side-Angle (ASA)
11. ∠ADB ≅ ∠ADB Reflexive Property
12. BD ≅ BD Reflexive Property
13. ∆ADB ≅ ∆CDB Side-Angle-Side (SAS)
14. ∠ADB ≅ ∠CDB Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
Paragraph Proof:
A paragraph proof is a type of mathematical proof that presents the reasoning and steps in a paragraph form rather than using a two-column format. In a paragraph proof, the statements and reasons are written together coherently in sentences, explaining the logical progression of the proof. This format allows for a more narrative-style presentation of the proof. A well-written paragraph proof should be clear, concise, and logically organized.
Example of a Paragraph Proof:
Given: Triangle ABC is isosceles with AB ≅ AC.
To Prove: ∠B ≅ ∠C.
Proof: Since triangle ABC is isosceles with AB ≅ AC, we can conclude that the base angles, ∠B and ∠C, are congruent. This is because in an isosceles triangle, the base angles are always congruent. By the Isosceles Triangle Theorem, if two sides of a triangle are congruent, then the angles opposite those sides are also congruent. Therefore, we can say that ∠B ≅ ∠C, as required.
In this paragraph proof, we clearly state the given information and the desired conclusion. We then provide the logical reasoning behind the congruence of the angles B and C based on the properties of isosceles triangles and the Isosceles Triangle Theorem. The proof is written in a clear and coherent manner, explaining each step concisely.
More Answers:
Understanding the Congruent Supplements Theorem | Vertical Angles and Angle Relationships in GeometryUnderstanding Biconditional Statements | Exploring the If and Only If Connective in Math
Understanding the Congruent Complements Theorem | Explained with Examples and Applications