Question

writing a two - column proof
given: \\( \angle abc \cong \angle def \\) and
\\( \angle ghi \cong \angle def \\)
prove: \\( m\angle abc=m\angle ghi \\)

Explanation:

Step1: Use the definition of congruent angles

If \(\angle ABC\cong\angle DEF\), then \(m\angle ABC = m\angle DEF\) (by the definition of congruent angles: congruent angles have equal measures).

Step2: Use the definition of congruent angles again

If \(\angle GHI\cong\angle DEF\), then \(m\angle GHI=m\angle DEF\) (by the definition of congruent angles).

Step3: Apply the transitive property of equality

Since \(m\angle ABC = m\angle DEF\) and \(m\angle GHI=m\angle DEF\), we can conclude that \(m\angle ABC=m\angle GHI\) (by the transitive property of equality: if \(a = b\) and \(b = c\), then \(a = c\)).

Answer:

Statements	Reasons
\(m\angle GHI = m\angle DEF\)	Definition of congruent angles (\(\angle GHI\cong\angle DEF\))
\(m\angle ABC=m\angle GHI\)	Transitive property of equality