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Writing a Paragraph Proof



Given: $m \angle A + m \angle B = m \angle B + m \angle C$

Prove: $m \angle C = m \angle A$



Complete the paragraph proof.



We are given that the sum of the measures of angles A and B is equal to the sum of the measures of angles B and C. The measure of angle $B$ is equal to itself by the [ ] property, so you can subtract that measure from both sides of the equation. Now the [ ]. By the [ ] property, this means the measure of angle C equals the measure of angle A.

Verified Answer :

  • Use the Reflexive Property of Equality to state that \(m\angle B = m\angle B\).
  • Apply the Subtraction Property of Equality to the given equation \(m\angle A + m\angle B = m\angle B + m\angle C\) to isolate \(m\angle A = m\angle C\).
  • Utilize the Symmetric Property of Equality to rearrange the statement to the required conclusion, \(m\angle C = m\angle A\).
  • The final proof completes the missing logic as: \(\boxed{\text{Reflexive, measure of angle A equals the measure of angle C, Symmetric}}\).

Explanation

  1. Analyze the logical flow and properties of equality.

To prove \(m\angle C = m\angle A\) given the equation \(m\angle A + m\angle B = m\angle B + m\angle C\), we follow a logical sequence of properties of equality.

  1. Reflexive Property: The first step is to recognize that any quantity is equal to itself. The measure of angle \(B\) is equal to the measure of angle \(B\), expressed as \(m\angle B = m\angle B\). This is justified by the Reflexive Property of Equality.
  2. Subtraction Property: We are given \(m\angle A + m\angle B = m\angle B + m\angle C\). By subtracting the same quantity, \(m\angle B\), from both sides of the equation, we maintain the equality. This results in \(m\angle A = m\angle C\).
  3. Symmetric Property: Finally, we have the equality \(m\angle A = m\angle C\). The Symmetric Property of Equality states that if \(x = y\), then \(y = x\). Applying this property allows us to conclude that \(m\angle C = m\angle A\).
  1. Complete the paragraph proof.

We can now fill in the blanks in the provided paragraph proof:

"We are given that the sum of the measures of angles \(A\) and \(B\) is equal to the sum of the measures of angles \(B\) and \(C\). The measure of angle \(B\) is equal to itself by the Reflexive property, so you can subtract that measure from both sides of the equation. Now the measure of angle A equals the measure of angle C. By the Symmetric property, this means the measure of angle \(C\) equals the measure of angle \(A\).

Examples

In a real-life scenario, such as carpentry or engineering, proving that two angles are equal is essential for symmetry. For example, if you are building a symmetrical frame and you know that the total angle formed by two adjacent joints on the left matches the total angle formed by the corresponding joints on the right, you can use the subtraction property of equality to isolate the specific angle you need to measure, ensuring both sides of your structure are perfectly balanced. This logical process is the foundation for verifying precision in design and construction.

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