Let's begin with reviewing the idea of a two-column proof. It lists each statement on the left column and its corresponding justification on the right. Each statement must follow logically from its previous steps. In this case, we are given that ∠ GFH is congruent to ∠ GHF. This is how we begin our proof!
Statement1) & ∠ GFH ≅ ∠ GHF
Reason1) & Given
Let's add this information to the diagram.
By the definition of congruent angles, we can conclude that m∠ GFH=m∠ GHF.
Statement2) & m∠ GFH = m∠ GHF
Reason2) & Definition of congruent angles
In the diagram, we can see that ∠ EFG and ∠ GFH form a linear pair.
Statement3) & ∠ EFG and ∠ GFH form
& a linear pair
Reason3) & Given (diagram)
By the definition of a linear pair, ∠ EFG and ∠ GFH are supplementary angles.
Statement4) & ∠ EFG and ∠ GFHare
& supplementary
Reason4) & Definition of linear pair
Since ∠ EFG and ∠ GFH are supplementary angles, their measures add to 180^(∘).
Statement 5) & m∠ EFG + m∠ GFH=180^(∘)
Reason 5) & Definition of supplementary angles
Since m∠ GFH=m∠ GHF, we can substitute m∠ GHF for m∠ GFH in the equation by the Substitution Property of Equality.
Statement 6) & m∠ EFG + m∠ GHF=180^(∘)
Reason 6) & Substitution Property of Equality
Finally, we see that the sum of m∠ EFG and ∠ GHF is 180^(∘). Therefore, these angles are supplementary by the definition of supplementary angles.
Statement 7) & ∠ EFG and ∠ GHF are
& supplementary
Reason 7) & Definition of supplementary angles
Now, we can write our two-column proof.
