22. in the figure, ( riangle ghfcong riangle ehd ). which statement is true by cpctc? ( overline{dh}congoverline{hf} ) ( overline{gh}congoverline{hd} ) ( overline{fh}congoverline{he} ) ( overline{gf}congoverline{hd} )

Question

22. in the figure, ( 	riangle ghfcong	riangle ehd ). which statement is true by cpctc? ( overline{dh}congoverline{hf} ) ( overline{gh}congoverline{hd} ) ( overline{fh}congoverline{he} ) ( overline{gf}congoverline{hd} )

Accepted Answer

# Explanation:
## Step1: Recall CPCTC
CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent". If $\triangle GHF\cong\triangle EHD$, then their corresponding sides are congruent.
## Step2: Identify corresponding sides
In $\triangle GHF$ and $\triangle EHD$:
- $GH$ corresponds to $EH$
- $HF$ corresponds to $HD$
- $GF$ corresponds to $ED$

Now check each option:
- Option 1: $\overline{DH}\cong\overline{HF}$ is incorrect. Since $HF$ corresponds to $HD$ (not $DH$)
- Option 2: $\overline{GH}\cong\overline{HD}$ is incorrect. $GH$ corresponds to $EH$
- Option 3: $\overline{FH}\cong\overline{HE}$ is incorrect. $FH$ corresponds to $HD$
- Option 4: $\overline{GF}\cong\overline{ED}$ is incorrect. Wait, no, actually, if we consider the letters of the triangle names $\triangle GHF$ and $\triangle EHD$, the order of congruence gives:
$\overline{GH}\cong\overline{EH}$, $\overline{HF}\cong\overline{HD}$, $\overline{GF}\cong\overline{ED}$. But if we re - check the options again, maybe there was a mis - label in the problem's options (assuming it's a typo in the problem's options and we go by the side - correspondence in the congruent triangles $\triangle GHF\cong\triangle EHD$). The correct side - correspondence for the segments with $H$ in them: $\overline{FH}$ (from $\triangle GHF$) corresponds to $\overline{HD}$ (from $\triangle EHD$) is wrong. Wait, no, using the order of congruence $\triangle GHF\cong\triangle EHD$, we have:
$\overline{GH}$ (first - second letter of $\triangle GHF$) corresponds to $\overline{EH}$ (first - second letter of $\triangle EHD$)
$\overline{HF}$ (second - third letter of $\triangle GHF$) corresponds to $\overline{HD}$ (second - third letter of $\triangle EHD$)
$\overline{GF}$ (first - third letter of $\triangle GHF$) corresponds to $\overline{ED}$ (first - third letter of $\triangle EHD$). But if we assume the problem has a typo and we consider the segments with $H$ in a different way. Since $\triangle GHF\cong\triangle EHD$, by CPCTC, $\overline{FH}\cong\overline{HD}$ (no, that's not an option). Wait, no, another approach:
The two triangles $\triangle GHF$ and $\triangle EHD$ are congruent. So, $\overline{GH}$ in $\triangle GHF$ and $\overline{EH}$ in $\triangle EHD$ are congruent; $\overline{HF}$ in $\triangle GHF$ and $\overline{HD}$ in $\triangle EHD$ are congruent; $\overline{GF}$ in $\triangle GHF$ and $\overline{ED}$ in $\triangle EHD$ are congruent. But if we look at the options again, and assume that the problem intended $\overline{FH}\cong\overline{HD}$ (not an option) but if we consider the order of the congruence statement $\triangle GHF\cong\triangle EHD$, the sides:
Let $A = G$, $B = H$, $C = F$ for $\triangle ABC=\triangle GHF$ and $D = E$, $E = H$, $F = D$ for $\triangle DEF=\triangle EHD$ (using the congruence notation $\triangle ABC\cong\triangle DEF$ implies $AB\cong DE$, $BC\cong EF$, $AC\cong DF$). So $GH\cong EH$, $HF\cong HD$, $GF\cong ED$. But if we consider the segments with $H$ in the options:
The second option $\overline{GH}\cong\overline{HD}$ is wrong. The first option $\overline{DH}\cong\overline{HF}$ (if $DH = HD$) is wrong. The third option $\overline{FH}\cong\overline{HE}$ is wrong. But if we assume that the problem has a mis - written option and we use the fact that in congruent triangles $\triangle GHF\cong\triangle EHD$, the side $HF$ (from $\triangle GHF$) and $HD$ (from $\triangle EHD$) are congruent. If we consider the option $\overline{FH}\cong\overline{HD}$ (but $FH = HF$ by segment notation). Wait, no, another way:
Since $\triangle GHF\cong\triangle EHD$, then $\overline{GH}\cong\overline{EH}$, $\overline{HF}\cong\overline{HD}$, $\overline{GF}\cong\overline{ED}$. If we look at the options:
- Option 1: $\overline{DH}\cong\overline{HF}$ ($DH = HD$, so $\overline{HD}\cong\overline{HF}$ is correct as $HF$ and $HD$ are corresponding sides of congruent triangles $\triangle GHF$ and $\triangle EHD$)

# Answer:
$\overline{DH}\cong\overline{HF}$ (assuming the first option is the correct one based on CPCTC and side - correspondence in $\triangle GHF\cong\triangle EHD$)

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QUESTION IMAGE

Question

Explanation:

Step1: Recall CPCTC

Step2: Identify corresponding sides

Answer: