Worksheet for selecting the correct trigonometric ratio (SOH CAH TOA) based on given right triangles.

Let’s go step by step to figure out which trig ratio (SOH, CAH, or TOA) to use for each triangle.

Remember:
- SOH → sin(θ) = Opposite / Hypotenuse
- CAH → cos(θ) = Adjacent / Hypotenuse
- TOA → tan(θ) = Opposite / Adjacent

We look at what sides are labeled in each triangle and match them to the correct ratio.

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Triangle ①:
Labeled sides: Opposite, Adjacent, Hypotenuse — but we’re being asked to pick a ratio based on what’s given relative to angle θ.
Actually, looking closely: it shows “Opp”, “Adj”, and “Hyp” all labeled. But the blank is next to an equals sign — meaning we need to write the ratio that uses two of these sides for angle θ.

Wait — actually, in triangle ①, the sides shown are:
- Opposite (left side)
- Adjacent (bottom)
- Hypotenuse (slanted side)

But the question is: which ratio would you use? Since all three sides are labeled, but typically we choose based on which two sides we know or are focusing on. However, since this is about choosing the right ratio for the setup, let’s see what’s most direct.

Actually, re-examining: In triangle ①, the angle θ is at the bottom right. The side opposite to θ is the vertical leg. The adjacent is the horizontal leg. The hypotenuse is the slant.

But the problem says: “For each triangle, decide whether you would use... SOH, CAH, or TOA”

So we must pick one based on which two sides are involved in the ratio for angle θ.

In triangle ①, if we were to write a ratio involving θ, and assuming we’re using the standard labeling, perhaps they want us to notice which two sides are marked as known or relevant.

Wait — actually, looking again: in triangle ①, the labels are “Opp”, “Adj”, “Hyp” — so all three are there. But maybe the key is that the blank is after an equals sign — suggesting we’re setting up a ratio.

Perhaps the intention is: for each triangle, identify which trig function relates the two sides that are labeled *in relation to angle θ*.

Let me reinterpret:

In triangle ①: angle θ is at the bottom right. The side opposite to θ is labeled “Opp”. The side adjacent to θ is labeled “Adj”. The hypotenuse is labeled “Hyp”.

But which ratio should we use? It depends on which two sides we’re comparing.

Actually, I think the diagram is showing which sides are present, and we have to choose the ratio that matches the pair of sides that define the trig function for angle θ.

But in triangle ①, all three sides are labeled — so maybe it’s ambiguous? No — let’s look at the others.

Triangle ②: Only “Hyp” and “Adj” are labeled. So we have hypotenuse and adjacent. That means we should use CAH (cosine), because cos(θ) = Adj/Hyp.

Triangle ③: “Hyp” and “Opp” are labeled. So SOH (sine), because sin(θ) = Opp/Hyp.

Triangle ④: “Adj” and “Opp” are labeled. So TOA (tangent), because tan(θ) = Opp/Adj.

Now back to triangle ①: It has all three labeled, but perhaps the intention is that since it’s the reference triangle, and no specific pair is highlighted, but wait — actually, in triangle ①, the way it’s drawn, with θ at the bottom right, and sides labeled Opp, Adj, Hyp — but the blank is there, so likely they expect us to realize that for any right triangle, you can use any, but probably they want the most basic one? Or perhaps it’s a trick.

Wait — looking at the original image description: in triangle ①, it’s identical to the first diagram, which is just defining Opp, Adj, Hyp. Then triangles ②,③,④ are variations.

But in triangle ①, since all three are labeled, and no particular pair is emphasized, but the problem says “decide whether you would use SOH, CAH, or TOA” — implying one per triangle.

Perhaps for triangle ①, since it’s the general case, but that doesn’t make sense.

Another thought: in triangle ①, the angle θ is at the bottom right, same as in the definition. The side opposite is left, adjacent is bottom, hypotenuse is slant. If we were to write a ratio for θ, and if we had to choose one, but actually, I think the key is that in triangle ①, they might be expecting TOA, because often tangent is introduced first? No.

Let’s think differently. Perhaps the blank is where you write the ratio, like "sin θ = ___" etc., but the instruction is to choose SOH, CAH, or TOA.

Looking at triangle ②: only Hyp and Adj are labeled → so CAH.

Triangle ③: Hyp and Opp → SOH.

Triangle ④: Adj and Opp → TOA.

Then triangle ①: it has all three, but perhaps it's meant to be the same as the definition, and since in the definition, they show all, but for calculation, you pick based on known sides. But here, no sides are specified as known; it's just labeled.

Perhaps for triangle ①, since it's the standard position, and θ is at the acute angle, and they label Opp, Adj, Hyp, but the ratio that uses Opp and Adj is TOA, which is common.

I recall that in many textbooks, when they introduce the ratios, they start with sine, but let's see the pattern.

Notice that in triangle ④, the angle θ is at the top, not at the bottom. Let's check that.

In triangle ④: it's a right triangle with right angle at bottom right. Angle θ is at the top vertex. The side labeled "Adj" is the right side (vertical), and "Opp" is the bottom side (horizontal). So for angle θ at the top, the adjacent side is the vertical leg (since it's next to θ), and the opposite side is the horizontal leg (across from θ). And hypotenuse is not labeled. So yes, we have Opp and Adj, so TOA.

Similarly, in triangle ③: angle θ is at the bottom left. Right angle at bottom right. So for θ at bottom left, opposite side is the vertical leg (right side), adjacent is the bottom leg, hypotenuse is slant. Labeled: "Hyp" and "Opp" — so SOH.

Triangle ②: angle θ at bottom right. Right angle at bottom left. So for θ, adjacent is the bottom leg, hypotenuse is slant. Labeled "Hyp" and "Adj" — so CAH.

Now triangle ①: angle θ at bottom right. Right angle at bottom left. Sides: Opp (left vertical), Adj (bottom horizontal), Hyp (slant). All labeled. But which ratio to use? Since all are there, but perhaps the problem expects us to see that for this configuration, if we were to calculate something, but it's ambiguous.

However, looking at the way it's presented, and since the other three are clear, perhaps for triangle ①, they intend for us to recognize that it's the general case, but I think there might be a mistake in my reasoning.

Another idea: in triangle ①, the blank is after an equals sign, and it's probably where you write the ratio, like "tan θ = Opp/Adj" etc. But the instruction is to choose SOH, CAH, or TOA.

Perhaps for each triangle, based on which two sides are indicated as the ones to use, but in triangle ①, all three are shown, so maybe it's not specified.

Let's count the labels:

- Triangle ①: Opp, Adj, Hyp all labeled → but perhaps they want the ratio that involves the two legs, which is TOA.

I recall that in some curricula, they emphasize that tangent is opposite over adjacent, and for the standard position, it's common.

Perhaps the answer for ① is TOA, because in the first diagram, they are defining the terms, and then for practice, but let's see the sequence.

I think I found a better way: in triangle ①, since it's identical to the reference diagram, and the reference diagram is used to define all, but for the purpose of this worksheet, they might expect TOA for ①, as it's the most straightforward for the angle at the corner.

But let's look for clues. In triangle ④, the angle is at the top, and they have Adj and Opp, so TOA.

In triangle ③, angle at bottom left, Hyp and Opp, SOH.

In triangle ②, angle at bottom right, Hyp and Adj, CAH.

For triangle ①, angle at bottom right, same as ②, but in ② they only labeled Hyp and Adj, while in ① they labeled all three. So perhaps for ①, since all are labeled, but the ratio that is most associated with the angle when all sides are considered is not clear.

Perhaps the blank in ① is for the student to write the ratio, and they can choose, but the task is to decide which one to use, implying based on the given information.

Another thought: in triangle ①, the way it's drawn, with θ at the bottom right, and the sides labeled, but perhaps the "Opp" and "Adj" are the legs, and "Hyp" is the hypotenuse, and since no specific pair is highlighted, but in many cases, for such diagrams, they might expect the tangent ratio.

I think I should go with the following logic:

- For a given angle θ in a right triangle, the trig ratio you use depends on which two sides you are relating.

In triangle ①, since all three sides are labeled, but the problem is to "decide whether you would use", perhaps it's implying that for this triangle, if you were to find θ or a side, which ratio is appropriate, but without additional info, it's hard.

Perhaps the key is that in triangle ①, the angle θ is at the vertex where the adjacent and hypotenuse meet, but also opposite is there.

Let's calculate what each ratio would be:

- SOH: sinθ = Opp/Hyp

- CAH: cosθ = Adj/Hyp

- TOA: tanθ = Opp/Adj

All are valid, but the worksheet likely wants us to see which pair is emphasized or which is most direct.

Looking back at the user's image description, in triangle ①, it's the same as the first diagram, which is just for definition. Then triangles 2,3,4 are different configurations.

Perhaps for triangle ①, since it's the standard, and they have all labels, but the blank is there, and in many worksheets, for the first one, they might expect TOA as it's opposite over adjacent, which is intuitive.

I recall that in some sources, for the angle at the bottom right, with opposite on left, adjacent on bottom, they use tangent for slope, etc.

Perhaps the answer is TOA for ①.

Let me assume that, and see if it makes sense.

So:

① TOA (because Opp and Adj are the legs, and often used together)

② CAH (Hyp and Adj)

③ SOH (Hyp and Opp)

④ TOA (Opp and Adj)

But in ④, it's also TOA, which is fine.

Is there a reason to choose differently for ①?

Another idea: in triangle ①, the angle θ is at the bottom right, and the side labeled "Opp" is the one opposite to it, which is correct, "Adj" is adjacent, "Hyp" is hypotenuse. Now, if we were to write a ratio for θ, and if we had to choose one, but perhaps the worksheet intends for us to see that for this triangle, since it's the reference, but I think it's safe to say that for ①, since all sides are given, but the ratio that doesn't involve the hypotenuse is TOA, and it's commonly used.

Perhaps the blank in ① is for the student to fill in the ratio, and they can choose, but the task is to select SOH, CAH, or TOA based on the labels.

I think I found a better approach: in triangle ①, the labels are "Opposite", "Adjacent", "Hypotenuse" written out, while in others, they are abbreviated as "Opp", "Adj", "Hyp". But that shouldn't matter.

Perhaps for triangle ①, because it's the first one and has all, but let's look at the answer choices implied.

I can search for similar worksheets online, but since I can't, I'll go with logic.

Let's consider that in triangle ②, only Hyp and Adj are labeled, so clearly CAH.

In triangle ③, only Hyp and Opp are labeled, so SOH.

In triangle ④, only Adj and Opp are labeled, so TOA.

In triangle ①, all three are labeled, so technically any could be used, but perhaps the problem expects us to use the ratio that involves the two sides that are not the hypotenuse, i.e., TOA, as it's the tangent of the angle.

Maybe for ①, since it's the standard position, and in many contexts, they use sine or cosine, but I think TOA is fine.

Another thought: in the first diagram, they have "Opposite", "Adjacent", "Hypotenuse" labeled, and then for triangle ①, it's the same, so perhaps they want the ratio that is defined by those, but still.

I recall that in some textbooks, when they show the triangle with all sides labeled, and ask for the ratio for θ, they might specify which ratio, but here it's open.

Perhaps the blank is where you write the expression, like "tan θ = \frac{Opp}{Adj}" etc., and for ①, since it's general, but the instruction is to choose SOH, CAH, or TOA.

I think I should conclude that for triangle ①, since the angle θ is at the vertex, and the opposite and adjacent are the legs, and hypotenuse is separate, but to be consistent, let's see the position.

In triangle ①, the right angle is at the bottom left, θ at bottom right, so the side adjacent to θ is the bottom side, opposite is the left side.

If we were to find θ, and if we knew Opp and Adj, we'd use TOA.

Since all are labeled, but perhaps the worksheet assumes that for this triangle, we are to use the ratio that corresponds to the sides that are typically used for that angle.

I think I'm overcomplicating. Let me assign based on the most logical choice:

- ①: TOA (as it's opposite over adjacent, and for the angle at the corner, it's common)

- ②: CAH (hyp and adj)

- ③: SOH (hyp and opp)

- ④: TOA (opp and adj)

And for ④, even though the angle is at the top, it's still TOA because we have opp and adj for that angle.

So final decision:

① TOA

② CAH

③ SOH

④ TOA

But let's double-check triangle ④.

In triangle ④: right angle at bottom right. Angle θ at top vertex. So for angle θ, the side adjacent to it is the vertical side (labeled "Adj"), because it's next to θ. The side opposite to θ is the horizontal side at the bottom (labeled "O" — probably "Opp"). Yes, so Opp and Adj are given, so TOA is correct.

Similarly for others.

For triangle ①, although all are labeled, perhaps the intended answer is TOA, as it's the ratio that uses the two legs, and in many introductory problems, they start with tangent for slope or something.

Perhaps in triangle ①, since it's the same as the definition, and the definition is for all, but I think TOA is acceptable.

Another way: in the first diagram, they have "Opposite", "Adjacent", "Hypotenuse" , and then for triangle ①, it's identical, so maybe they want us to see that for this triangle, if we were to write a ratio, but I think it's fine.

I'll go with that.

So answers:

1. TOA

2. CAH

3. SOH

4. TOA

Now to box the final answer.

The final answer should be the choices for each triangle.

So for ①: TOA

②: CAH

③: SOH

④: TOA

In the response, I need to provide the final answer as per the format.

Since the user didn't specify how to output, but in the context, probably list them.

In the initial request, it's "solve the problem", and the problem is to decide for each triangle which ratio to use.

So final answer is the four choices.

Let me write it clearly.