Question

use the given information to prove that △def = △ghf. given: $overline{ef}=overline{hf}$, $overline{df}=overline{gf}$. prove: △def = △ghf. statement reason 1. $overline{ef}=overline{hf}$ given 2. $overline{df}=overline{gf}$ reason? 3. $angle dfe=angle$ vertical angles property 4. △def = △ghf reason?

Explanation:

Step1: Fill in the first - given side

Since $\overline{EF}=\overline{HF}$ is given, the first blank is $\overline{HF}$.

Step2: Reason for $\overline{DF}=\overline{GF}$

It is given in the problem statement, so the reason for $\overline{DF}=\overline{GF}$ is "Given".

Step3: Fill in the vertical - angle

By the vertical - angles property, $\angle DFE=\angle GFH$.

Step4: Prove triangle congruence

We have two pairs of equal sides ($\overline{EF}=\overline{HF}$ and $\overline{DF}=\overline{GF}$) and the included angles are equal ($\angle DFE=\angle GFH$). So, by the Side - Angle - Side (SAS) congruence criterion, $\triangle DEF\cong\triangle GHF$. The reason for $\triangle DEF\cong\triangle GHF$ is "Side - Angle - Side (SAS) Congruence Postulate".

Answer:

1. $\overline{HF}$
2. Given
3. $\angle GFH$
4. Side - Angle - Side (SAS) Congruence Postulate