Question

given: ( mangle a + mangle b = mangle b + mangle c ) prove: ( mangle c = mangle a ) write a paragraph proof to prove the statement.

Explanation:

To prove \( m\angle C = m\angle A \) given \( m\angle A + m\angle B = m\angle B + m\angle C \), we can use the Subtraction Property of Equality. This property states that if we subtract the same quantity from both sides of an equation, the equation remains true. Here, we observe that \( m\angle B \) appears on both sides of the given equation. By subtracting \( m\angle B \) from both sides, we get \( (m\angle A + m\angle B) - m\angle B = (m\angle B + m\angle C) - m\angle B \). Simplifying both sides, the \( m\angle B \) terms cancel out, leaving us with \( m\angle A = m\angle C \), which is equivalent to \( m\angle C = m\angle A \) (by the Symmetric Property of Equality, which states that if \( a = b \), then \( b = a \)).

Answer:

To prove \( m\angle C = m\angle A \) given \( m\angle A + m\angle B = m\angle B + m\angle C \), we apply the Subtraction Property of Equality. Subtract \( m\angle B \) from both sides of the equation: \( (m\angle A + m\angle B) - m\angle B = (m\angle B + m\angle C) - m\angle B \). Simplifying, the \( m\angle B \) terms cancel, resulting in \( m\angle A = m\angle C \). By the Symmetric Property of Equality, this implies \( m\angle C = m\angle A \).