In the first line of a two-column proof, we need to list the given information.
Statements:& ∠ 1≅ ∠ 4, & ∠ 2≅ ∠ 3
Reason:&Given
Renaming the Angles
In the conclusion, the angles are described using the points on the diagram, so let's rename angles ∠ 1, ∠ 2, ∠ 3, and ∠ 4 using the points.
Statements:& ∠ AFB≅ ∠ EFD, & ∠ BFC≅ ∠ DFC
Reason:&Substitution Property of Equality
Using the Definition of Congruence
The congruence of angles implies the equality of their measures.
Statements:& m∠ AFB= m∠ EFD, & m∠ BFC= m∠ DFC
Reason:&Definition of congruence
Joining Adjacent Angles
We will now look at the part of the diagram to the left and to the right of FC separately. We can use that point B is in the interior of angle ∠ AFC.
Statement:& m ∠ AFB+m ∠ BFC=m∠ AFC
Reason:&Angle Addition Postulate
Similarly, point D is in the interior of angle ∠ EFC.
Statement:& m ∠ EFD+m ∠ DFC=m∠ EFC
Reason:&Angle Addition Postulate
Combining the Equations
We can now use that m ∠ EFD=m ∠ AFB, so we can replace one with the other in the equation above. Similarly, we can also replace m ∠ DFC with m ∠ BFC.
Statement:& m ∠ AFB+m ∠ BFC=m∠ EFC
Reason:&Substitution Property of Equality
Notice, in the previous lines we have that m ∠ AFB+m ∠ BFC is equal to both m∠ AFC and m∠ EFC. Hence, these two angle measures must be equal.
Statement:& m∠ AFC=m∠ EFC
Reason:&Transitive Property of Equality
Moving from Equality to Congruence
We now know that angles ∠ AFC and ∠ EFC have the same measure. This means that these angles are congruent.
Statement:& ∠ AFC≅ ∠ EFC
Reason:&Definition of congruence
Through these steps we proved from the given information that ∠ AFC≅ ∠ EFC
