Review the idea of a two-column proof. Remember to list each step and explain your reasoning.
Statements
Reasons
1.
AB ≅ FG, BF bisects AC and DG.
1.
Given
2.
AB ≅ BC, DF ≅ FG
2.
Definition of segment bisector
3.
BC ≅ AB, FG ≅ DF
3.
Symmetric Property of Segment Congruence
4.
BC ≅ FG
4.
Transitive Property of Segment Congruence
5.
BC ≅ DF
5.
Transitive Property of Segment Congruence
Practice makes perfect
Let's begin with reviewing the idea of a two-column proof. It lists each statement on the left and its corresponding justification on the right. Each statement must follow logically from its previous steps. We are given that AB and FG are congruent and that BF bisects AC and DG. This is how we begin our proof!
Statement1)& AB≅FG,
& BF bisects ACandDG.
Reason1)& Given
This tells us that BF is a bisector of AC and DG. By the definition of a segment bisector AB is congruent to BC and DF is congruent to FG.
Knowing this, we can write our second statement.
Statement2)& AB≅BC, DF≅FG
Reason2)& Definition of segment bisector
By the Symmetric Property of Segment Congruence, we can rewrite the above statement.
Statement3)& BC≅AB, FG≅DF
Reason3)& Symmetric Property of
& Segment Congruence
Remember we are given that AB≅FG. By using the Transitive Property of Segment Congruence, we obtain that BC is congruent to FG.
BC≅ AB AB≅ FG ⇒ BC≅ FG
This is our fourth statement.
Statement4)& BC≅FG
Reason4)& Transitive Property of
& Segment Congruence
Finally, we will use the Transitive Property of Segment Congruence once more.
BC≅ FG FG≅ DF ⇒ BC≅ DF
This is our last statement.
Statement5)& BC≅DF
Reason5)& Transitive Property of
& Segment Congruence
