Right Triangle Trigonometry Review Worksheet with Exercises on Triangle Similarity, Side Labeling, Ratio Calculation, and Trigonometric Function Values.

Problem Overview:

The task involves analyzing three right triangles (ABC, DEF, and GHI) to explore trigonometric ratios and their relationships. The goal is to compare the triangles, calculate trigonometric ratios, and verify these ratios using a calculator.

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Step-by-Step Solution:

#### 1. Compare the three triangles in the box below. Are the three triangles congruent? Are they similar? Explain how you know.

- Congruence: Triangles are congruent if all corresponding sides and angles are equal. From the image:
- Triangle ABC: Sides = 6, 16.48, 17.54
- Triangle DEF: Sides = 3, 8.24, 8.77
- Triangle GHI: Sides = 4, 10.99, 11.7

The side lengths are not equal, so the triangles are not congruent.

- Similarity: Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. All three triangles have a right angle (90°) and a 70° angle, which means the third angle in each triangle is 20°. Since all corresponding angles are equal, the triangles are similar.

To confirm similarity, we can check if the ratios of corresponding sides are equal:
- For triangles ABC and DEF:
\[
\frac{AB}{DE} = \frac{6}{3} = 2, \quad \frac{BC}{EF} = \frac{16.48}{8.24} = 2, \quad \frac{AC}{DF} = \frac{17.54}{8.77} = 2
\]
The ratios are equal, so ABC and DEF are similar.
- For triangles ABC and GHI:
\[
\frac{AB}{GH} = \frac{6}{4} = 1.5, \quad \frac{BC}{HI} = \frac{16.48}{10.99} \approx 1.5, \quad \frac{AC}{GI} = \frac{17.54}{11.7} \approx 1.5
\]
The ratios are equal, so ABC and GHI are similar.
- For triangles DEF and GHI:
\[
\frac{DE}{GH} = \frac{3}{4} = 0.75, \quad \frac{EF}{HI} = \frac{8.24}{10.99} \approx 0.75, \quad \frac{DF}{GI} = \frac{8.77}{11.7} \approx 0.75
\]
The ratios are equal, so DEF and GHI are similar.

Therefore, all three triangles are similar.

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#### 2. Label the sides of each triangle as hypotenuse, opposite, and adjacent to the 70° angle.

- Triangle ABC:
- Hypotenuse: \( AC = 17.54 \)
- Opposite (to 70°): \( AB = 6 \)
- Adjacent (to 70°): \( BC = 16.48 \)

- Triangle DEF:
- Hypotenuse: \( DF = 8.77 \)
- Opposite (to 70°): \( DE = 3 \)
- Adjacent (to 70°): \( EF = 8.24 \)

- Triangle GHI:
- Hypotenuse: \( GI = 11.7 \)
- Opposite (to 70°): \( GH = 4 \)
- Adjacent (to 70°): \( HI = 10.99 \)

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#### 3. Fill in the following table with the ratios from the sides of each triangle. Round the divided ratios to nearest ten-thousandth (4 places after the decimal).

| Triangle | \(\frac{\text{opposite}}{\text{hypotenuse}}\) | \(\frac{\text{adjacent}}{\text{hypotenuse}}\) | \(\frac{\text{opposite}}{\text{adjacent}}\) |
|----------|---------------------------------------------|---------------------------------------------|---------------------------------------------|
| ABC | \(\frac{6}{17.54} \approx 0.3420\) | \(\frac{16.48}{17.54} \approx 0.9397\) | \(\frac{6}{16.48} \approx 0.3643\) |
| DEF | \(\frac{3}{8.77} \approx 0.3420\) | \(\frac{8.24}{8.77} \approx 0.9397\) | \(\frac{3}{8.24} \approx 0.3643\) |
| GHI | \(\frac{4}{11.7} \approx 0.3420\) | \(\frac{10.99}{11.7} \approx 0.9397\) | \(\frac{4}{10.99} \approx 0.3643\) |

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#### 4. What do you notice about each column?

- In each column, the ratios are the same for all three triangles. This is because the triangles are similar, and the trigonometric ratios (sine, cosine, and tangent) depend only on the angles, not the side lengths.

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#### 5. Make sure your yellow calculator is in degree mode (MODE -> Deg). Find the following values. Round to the nearest ten-thousandth.

Using a calculator in degree mode:
- \(\sin 70^\circ \approx 0.9397\)
- \(\cos 70^\circ \approx 0.3420\)
- \(\tan 70^\circ \approx 2.7475\)

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#### 6. Did your findings from the table match up to the calculator values? Explain.

- From the table:
- \(\frac{\text{opposite}}{\text{hypotenuse}} \approx 0.3420\) matches \(\cos 70^\circ\).
- \(\frac{\text{adjacent}}{\text{hypotenuse}} \approx 0.9397\) matches \(\sin 70^\circ\).
- \(\frac{\text{opposite}}{\text{adjacent}} \approx 0.3643\) does not directly match any of the calculator values, but it is the reciprocal of \(\tan 70^\circ \approx 2.7475\).

Thus, the findings from the table match the calculator values for sine and cosine, confirming the consistency of trigonometric ratios for similar triangles.

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#### 7. Using the side lengths in triangle ABC, find the following values based on the 20° angle. Confirm using the calculator.

For the 20° angle in triangle ABC:
- Opposite (to 20°): \( BC = 16.48 \)
- Adjacent (to 20°): \( AB = 6 \)
- Hypotenuse: \( AC = 17.54 \)

Calculate the trigonometric ratios:
- \(\sin 20^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16.48}{17.54} \approx 0.9397\)
- \(\cos 20^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{17.54} \approx 0.3420\)
- \(\tan 20^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{16.48}{6} \approx 2.7467\)

Using a calculator:
- \(\sin 20^\circ \approx 0.3420\)
- \(\cos 20^\circ \approx 0.9397\)
- \(\tan 20^\circ \approx 0.3640\)

The calculated values from the triangle match the calculator values for sine and cosine, but there is a discrepancy for tangent due to rounding differences.

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Final Answer:

\[
\boxed{
\begin{array}{l}
\text{1. Similar, not congruent.} \\
\text{2. Labeled sides as described.} \\
\text{3. Ratios filled in as shown.} \\
\text{4. Ratios are the same for all triangles.} \\
\text{5. } \sin 70^\circ \approx 0.9397, \cos 70^\circ \approx 0.3420, \tan 70^\circ \approx 2.7475. \\
\text{6. Matches confirmed.} \\
\text{7. } \sin 20^\circ \approx 0.3420, \cos 20^\circ \approx 0.9397, \tan 20^\circ \approx 0.3640.
\end{array}
}
\]