Let's examine the given diagram and the information we can learn from it.
To show that the distance from B to F is three times the distance from D to F, 3DF=BF, we will construct a two-column proof. The first step is to state the given information and the statement to be proven.
Given: Each triangle is isosceles, BG≅ HC,
HD≅ JF, ∠ G ≅ ∠ H and ∠ H ≅ ∠ J
Prove: 3DF=BF
The Transitive Property of Congruence can help us write the second line of our proof. We know that ∠ G ≅ ∠ H and ∠ H ≅ ∠ J. Therefore, ∠ G ≅ ∠ J.
2. Transitive Property of Congruence
∠ G ≅ ∠ J
Using the Transitive Property of Congruence again, this time with BG≅ HC and HC≅ HD, we can see that BG≅ HD.
4. Transitive Property of Congruence
BG≅ HD
We will use the Transitive Property of Congruence one more time and get CG≅ HD from CG≅ BG and BG≅ HD.
5. Transitive Property of Congruence
CG≅ HD
Now we see that two sides and the included side of each triangle are congruent to each other. By the Side-Angle-Side Theorem, we can conclude that the triangles are congruent to each other.
6. SAS Theorem
△ BCG ≅ △ CDH ≅ △ DFJ
Because the corresponding parts of congruent triangles are congruent, the base lengths of the triangle will be congruent.
7. CPCTC
BC≅ CD≅ DF
Remember that the segments are congruent if and only if they have the same length by the Definition of Congruence.
8. Definition of Congruence
BC=CD=DF
Since the points C and D are both between B and F, we can write BF as the sum of three segments.
9. Segment Addition Postulate
BC+CD+DF=BF
Now by the eighth step, we will substitute DF for BC and CD into the equation in the previous step.
10. Substitution Property
DF+DF+DF=BF
Finally, we will add the terms and finish our proof.
11. Addition Property
3DF=BF
Combining these steps, we can construct the two column proof.
