What changes would you make to turn this formal proof into a paragraph proof?
Given: Angles A and 15 are complementary.
Prove: [tex]A=75[/tex] degrees.
Proof:
Statements & Reason
Angles [tex]A[/tex] and 15 are complementary & Given
[tex]m \angle A+15=90[/tex] & Definition of complementary angles
[tex]m \angle A=75[/tex]. & By subtraction
A. Leave out all symbols and write out the words.
B. Leave out the reasons and just write the statements.
C. Write the reasons first and then the statements.
D. Write the statements and reasons in sentence format.
Answer (2)
We begin with the given information: angles A and 15 are complementary.
Apply the definition of complementary angles: m ∠ A + 15 = 90 .
Solve for m ∠ A by subtracting 15 from both sides: m ∠ A = 75 .
Conclude that A = 75 degrees.
Explanation
Understanding the Task The problem provides a two-column proof and asks for the changes needed to convert it into a paragraph proof. A paragraph proof presents the argument in sentence form, integrating the statements and their justifications into a coherent narrative.
Strategy for Conversion To transform the given two-column proof into a paragraph proof, we need to combine each statement with its corresponding reason into a sentence. The paragraph should start with the given information and proceed logically to the conclusion.
Constructing the Paragraph Proof Here's how the paragraph proof would look:
Given that angles A and 15 are complementary, this means that the sum of their measures is 90 degrees, according to the definition of complementary angles. m ∠ A + 15 = 90
Subtracting 15 from both sides of the equation, we find that the measure of angle A is 75 degrees. m ∠ A = 75
Concluding the Proof Therefore, we have successfully shown that A = 75 degrees, as required.
Examples
Paragraph proofs are commonly used in geometry and other mathematical fields to present arguments in a more readable and narrative format. For example, when explaining geometric relationships in architecture or engineering, a paragraph proof can help to clearly communicate the logic behind design choices or structural calculations to stakeholders who may not be familiar with formal proof structures. This makes the reasoning more accessible and easier to follow.
To convert the formal proof into a paragraph proof, we begin with the information that angles A and 15 degrees are complementary. We then explain that the sum of their measures equals 90 degrees, isolate angle A, and conclude that A equals 75 degrees. This format presents the proof in a more coherent narrative without needing a two-column layout.
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