Let's fill in the blanks for the given two-column proof starting with the first row that has a blank.
Third row
Consider the first and second steps that are completed.
Statements
Reasons
1.
AB=DE, BC=CD
1.
Given
2.
AB+BC=BC+DE
2.
Addition Property of Equality
Let's take a look at the given diagram.
We can see that BC and CD are congruent segments. Therefore, by the definition of congruent segments, they have the same length and we can conclude that BC=CD. Using the Substitution Property of Equality, we can substitute CD for BC in the equation in the second row.
Statement3)& AB+BC=CD+DE
Reason3)& Substitution Property of Equality
Fourth row
Looking at the diagram again, by the Segment Addition Postulate we can state that AC is equal to the sum of AB and BC. Similarly, by this postulate, we can also say that the sum of CD and DE is equal to CE.
Statement4)& AB+BC=AC,
& CD+DE=CE
Reason4)& Segment Addition Postulate
Fifth row
In the fifth row, we can substitute the left-hand side and the right-hand side of the equation in the third row with the identities in the fourth row.
Statement5)& AC=CE
Reason5)& Substitution Property of Equality
Sixth row
Finally, if two segments have the same length, they are congruent.
Statement6)& AC ≅ CE
Reason6)& Definition of congruent segments
Completed Two-Column Proof
We will now write the complete two-column proof.
Statements
Reasons
1.
AB=DE, BC=CD
1.
Given
2.
AB+BC=BC+DE
2.
Addition Property of Equality
3.
AB+BC=CD+DE
3.
Substitution Property of Equality
4.
AB+BC=AC, CD+DE=CE
4.
Segment Addition Postulate
5.
AC=CE
5.
Substitution Property of Equality
6.
AC ≅ CE
6.
Definition of congruent segments
Flowchart Proof
Let's create a flowchart using the information in the two-column proof.
