Question

writing a two-column proof
given: $\angle abc \cong \angle def$ and
$\angle ghi \cong \angle def$
prove: $m\angle abc = m\angle ghi$
assemble the proof by dragging tiles to
the statements and reasons columns.

Explanation:

Step1: Identify given congruences

We know that $\angle ABC \cong \angle DEF$ and $\angle GHI \cong \angle DEF$. By the definition of congruent angles, congruent angles have equal measures. So, $m\angle ABC = m\angle DEF$ (because $\angle ABC \cong \angle DEF$) and $m\angle GHI = m\angle DEF$ (because $\angle GHI \cong \angle DEF$).

Step2: Use transitive property of equality

The transitive property of equality states that if $a = b$ and $b = c$, then $a = c$. Here, let $a = m\angle ABC$, $b = m\angle DEF$, and $c = m\angle GHI$. Since $m\angle ABC = m\angle DEF$ and $m\angle GHI = m\angle DEF$, we can conclude that $m\angle ABC = m\angle GHI$.

Now, let's structure the two - column proof:

Statements	Reasons
2. $m\angle ABC = m\angle DEF$ <br> $m\angle GHI = m\angle DEF$	2. Definition of congruent angles (If two angles are congruent, their measures are equal)
3. $m\angle ABC = m\angle GHI$	3. Transitive property of equality (If $a = b$ and $b = c$, then $a = c$)

Answer:

The two - column proof is as shown above, with the conclusion $m\angle ABC = m\angle GHI$ proven using the given congruences and the transitive property of equality.