Consider the given information and the desired outcome of the proof.
Given: & ∠ AEB ≅ ∠ DEC
Prove: & ∠ AEC ≅ ∠ DEB
Let's do a Flowchart Proof! We begin by copying the diagram and labeling the congruent angles ∠ AEB and ∠ DEC.
We can begin our proof with this given information.
Statement1
∠ AEB ≅ ∠ DEC
Given
By definition of congruent angles, we know that m∠ AEB and m∠ DEC are equal.
Statement2
m∠ AEB = m∠ DEC
Definition of congruent angles
Using the Angle Addition Postulate, we see that the measure of ∠ DEB is the sum of the measures of ∠ BEC and ∠ DEC. This will be our third statement.
Statement3
m∠ DEB=m∠ BEC+m∠ DEC
Angle Addition Postulate
Since m∠ DEC= m∠ AEB, we can use the Substitution Property of Equality to rewrite the previous equation.
m∠ DEB=m∠ BEC + m∠ DEC
⇕
m∠ DEB=m∠ BEC + m∠ AEB
This will be our fourth statement.
Statement4
m∠ DEB = m∠ BEC + m∠ AEB
Substitution Property of Equality
We can use the Angle Addition Postulate once again to write m∠ AEC as the sum of m∠ BEC and m∠ AEB.
Statement5
m∠ AEC=m∠ BEC+m∠ AEB
Angle Addition Postulate
Since m∠ AEC and m∠ DEB are both equal to the sum of m∠ AEB and m∠ CEB we can use the Transitive Property of Equality to set them equal to each other.
Statement6
m∠ AEC = m∠ DEB
Transitive Property of Equality
Finally, since m∠ DEB and ∠ AEC have equal measures, the definition of congruent angles tells us that they are congruent.
Statement7
∠ AEC ≅ ∠ DEB
Definition of congruent angles
Final Proof
Combining everything from above, we can write our final proof.
Given: & ∠ AEB ≅ ∠ DEC
Prove: & ∠ AEC ≅ ∠ DEB
Proof:
