Question

is \\( \triangle f h k \\) similar to \\( \triangle g h j \\)?
if so, which postulate or theorem proves these two triangles are similar?
\\( \triangle f h k \\) is similar to \\( \triangle g h j \\) by the sss similarity theorem
\\( \triangle f h k \\) is not similar to \\( \triangle g h j \\)
\\( \triangle f h k \\) is similar to \\( \triangle g h j \\) by the ssa similarity theorem
\\( \triangle f h k \\) is similar to \\( \triangle g h j \\) by the sas similarity theorem

Explanation:

Step1: Identify corresponding sides

First, we need to find the lengths of the sides of both triangles. For $\triangle FHK$ and $\triangle GHJ$, we look at the sides around the common angle $\angle H$.

For $\triangle FHK$:
- $FH = 10 + 12 = 22$? Wait, no, wait. Wait, $FH$ is 10 in? Wait, no, the diagram: $FH$ is from $F$ to $G$ is 12 in, and $G$ to $H$ is 10 in? Wait, no, let's re-examine. Wait, the sides: $HG = 10$ in, $HF$? Wait, no, $F$ to $G$ is 12 in, $G$ to $H$ is 10 in, so $FH = FG + GH$? Wait, no, maybe the triangles share angle $H$, and the sides are $HG = 10$, $HJ = 15$, and $FH = 12 + 10$? Wait, no, maybe I misread. Wait, the sides: $FG = 12$ in, $GH = 10$ in, $HJ = 15$ in, $JK = 18$ in? Wait, no, the triangle $FHK$ has sides: $FH$ (from $F$ to $H$) is $12 + 10$? No, wait, the points: $F$, $G$, $H$ are colinear? Wait, no, the diagram: $G$ is on $FH$, and $J$ is on $HK$. So $FH = FG + GH = 12 + 10 = 22$? No, wait, no, $FG$ is 12 in, $GH$ is 10 in, so $FH = 12 + 10 = 22$? And $HK = HJ + JK = 15 + 18 = 33$? And $HJ = 15$ in, $GH = 10$ in. Wait, no, maybe the sides are $GH = 10$, $FH = 12 + 10 = 22$? No, that can't be. Wait, maybe I made a mistake. Wait, the correct approach: for similar triangles by SAS similarity, we need two sides in proportion and the included angle equal. The included angle is $\angle H$, which is common to both triangles. So we check the ratios of the sides around $\angle H$.

For $\triangle GHJ$ and $\triangle FHK$:
- Side $GH$ (in $\triangle GHJ$) corresponds to side $FH$ (in $\triangle FHK$)
- Side $HJ$ (in $\triangle GHJ$) corresponds to side $HK$ (in $\triangle FHK$)

Wait, $GH = 10$ in, $FH = 10 + 12 = 22$? No, that's not right. Wait, no, $G$ is on $FH$, so $FH = FG + GH = 12 + 10 = 22$? And $HK = HJ + JK = 15 + 18 = 33$? Then $GH = 10$, $FH = 22$; $HJ = 15$, $HK = 33$. Then the ratio of $GH/FH = 10/22 = 5/11$, and $HJ/HK = 15/33 = 5/11$. So the ratios are equal, and the included angle $\angle H$ is common. So by SAS similarity (since two sides in proportion and included angle equal), the triangles are similar. Wait, but let's check the options. The options include SAS Similarity Theorem. Wait, but let's recheck the side lengths. Wait, maybe $FH = 12 + 10$? No, maybe $FG = 12$, $GH = 10$, so $FH = FG + GH = 22$? And $HJ = 15$, $HK = HJ + JK = 15 + 18 = 33$? Then $GH/FH = 10/22 = 5/11$, $HJ/HK = 15/33 = 5/11$. So the ratios are equal, and angle $H$ is common. So by SAS similarity, $\triangle GHJ \sim \triangle FHK$ (or $\triangle FHK \sim \triangle GHJ$? Wait, the order: $\triangle FHK$ and $\triangle GHJ$. So $FH$ corresponds to $GH$, $HK$ corresponds to $HJ$, and angle $H$ is common. Wait, $FH = 12 + 10 = 22$, $GH = 10$; $HK = 15 + 18 = 33$, $HJ = 15$. Then $FH/GH = 22/10 = 11/5$, and $HK/HJ = 33/15 = 11/5$. So the ratios are equal, and the included angle $\angle H$ is equal. So by SAS similarity theorem, the triangles are similar. Wait, but let's check the options. The options are:

- $\triangle FHK$ is similar to $\triangle GHJ$ by the SSS Similarity Theorem
- $\triangle FHK$ is not similar to $\triangle GHJ$
- $\triangle FHK$ is similar to $\triangle GHJ$ by the SSA Similarity Theorem (SSA is not a similarity theorem)
- $\triangle FHK$ is similar to $\triangle GHJ$ by the SAS Similarity Theorem

Wait, SSA is not a valid similarity theorem. SSS requires all three sides in proportion. Let's check the ratios for SSS. If we consider the sides:

For $\triangle GHJ$: sides are $GH = 10$, $HJ = 15$, and $GJ$ (unknown). For $\triangle FHK$: sides are $FH = 12 + 10 = 22$, $HK = 15 + 18 = 33$, and $FK$ (unknown). So the ratios of the sides: $GH/FH = 10/22 = 5/11$, $HJ/HK = 15/33 = 5/11$, and if $GJ/FK$ is also $5/11$, then SSS would apply. But we don't know $GJ$ and $FK$. However, since we have two sides in proportion and the included angle (angle $H$) equal, SAS similarity applies. Wait, but let's check the ratios again. Wait, maybe $FH$ is $10 + 12$? No, maybe $FG$ is 12, $GH$ is 10, so $FH = FG + GH = 22$, and $HK = HJ + JK = 15 + 18 = 33$. Then $GH = 10$, $FH = 22$; $HJ = 15$, $HK = 33$. So $GH/FH = 10/22 = 5/11$, $HJ/HK = 15/33 = 5/11$. So the ratios are equal, and angle $H$ is common. Therefore, by SAS similarity theorem, the triangles are similar. Wait, but the option says "by the SAS Similarity Theorem". Let's confirm: SAS similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. Here, angle $H$ is included between $GH$ and $HJ$ in $\triangle GHJ$, and between $FH$ and $HK$ in $\triangle FHK$. The ratios of $GH/FH = 10/22 = 5/11$ and $HJ/HK = 15/33 = 5/11$, so the sides are proportional, and angle $H$ is congruent. Therefore, SAS similarity applies.

Wait, but let's check the other options. SSS would require all three sides, but we don't know the third side. SSA is not a similarity theorem. So the correct answer is that $\triangle FHK$ is similar to $\triangle GHJ$ by the SAS Similarity Theorem.

Step2: Verify the ratios

Calculate the ratios of the corresponding sides:

- For side $GH$ (10 in) and $FH$ (10 + 12 = 22 in? Wait, no, maybe I misread the diagram. Wait, maybe $FG$ is 12, $GH$ is 10, so $FH = FG + GH = 22$? And $HJ = 15$, $HK = HJ + JK = 15 + 18 = 33$? Then $GH/FH = 10/22 = 5/11$, $HJ/HK = 15/33 = 5/11$. So the ratios are equal. The included angle is $\angle H$, which is common. Therefore, by SAS similarity, the triangles are similar.

Answer:

$\triangle FHK$ is similar to $\triangle GHJ$ by the SAS Similarity Theorem