Given m∠A + m∠B = m∠B + m∠C, Prove m∠C = m∠A: Paragraph Proof

The paragraph proof

Given: m∠A + m∠B = m∠B + m∠C. Prove: m∠C = m∠A.

We are given that m∠A + m∠B = m∠B + m∠C. Because m∠B appears added to both sides of the equation, we can subtract m∠B from each side; the Subtraction Property of Equality guarantees the equation stays balanced when we subtract the same quantity from both sides. Doing so removes m∠B and leaves m∠A = m∠C. Finally, by the Symmetric Property of Equality, which states that if a = b then b = a, we may reverse the equality to write m∠C = m∠A. This is exactly what we wanted to prove.

Why each step is justified

A valid proof cites a reason for every move. This proof uses just two properties of equality:

• Subtraction Property of Equality: if a = b, then a - c = b - c. Here a = m∠A + m∠B, b = m∠B + m∠C, and c = m∠B. Subtracting m∠B cancels it on both sides, giving m∠A = m∠C. This is the key algebraic step - it isolates the two angles we care about.
• Symmetric Property of Equality: if a = b, then b = a. The subtraction step ends with m∠A = m∠C, but the goal asks for the logically identical statement m∠C = m∠A. The Symmetric Property lets us flip the two sides so the conclusion matches the required form.

Some students stop at m∠A = m∠C and think they are done. Mathematically that is the same fact, but because the problem specifically asks to prove "m∠C = m∠A," the Symmetric Property is the clean, formal way to reach that exact wording.

Common wrong reasons

Watch out for mislabeling the justification:

• Addition Property of Equality is wrong because we are removing m∠B, not adding it.
• Reflexive Property (a = a) does not apply; nothing here is being set equal to itself.
• Transitive Property (if a = b and b = c, then a = c) is tempting but needs two separate equations chained through a common middle term. This problem gives a single equation, so transitivity is not what removes m∠B.
• Substitution is not needed either; we are performing an operation on the equation, not swapping equal quantities.

Paragraph vs. two-column form

A paragraph proof states the same logic as a two-column proof but in connected sentences, weaving each statement together with its reason. A two-column proof lists Statements on the left and Reasons on the right. Both are equally valid; the paragraph form simply reads like an explanation. For this problem the two-column version is: (1) m∠A + m∠B = m∠B + m∠C - Given; (2) m∠A = m∠C - Subtraction Property of Equality; (3) m∠C = m∠A - Symmetric Property of Equality.