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Let’s go through the problems step-by-step to solve them. The worksheet appears to consist of triangle-related problems, likely involving trigonometry and the Pythagorean theorem.

### 1A: Find the measure of the side labeled \(x\).
This is a right triangle, and we have the adjacent side (15 cm) and the angle (29°). To find \(x\), the opposite side, we use the tangent function:

\[
\tan(29^\circ) = \frac{x}{15}
\]

Solving for \(x\):
\[
x = 15 \times \tan(29^\circ) \approx 15 \times 0.5543 \approx 8.31 \, \text{cm}
\]

So, \(x \approx 8.31 \, \text{cm}\).

### 2A: Find the measure of the side labeled \(x\).
This is another right triangle, where we have the hypotenuse (35 ft) and the angle (56°). To find the adjacent side \(x\), we use the cosine function:

\[
\cos(56^\circ) = \frac{x}{35}
\]

Solving for \(x\):
\[
x = 35 \times \cos(56^\circ) \approx 35 \times 0.5592 \approx 19.57 \, \text{ft}
\]

So, \(x \approx 19.57 \, \text{ft}\).

### 3A: Find the measure of the angle labeled \(x\).
We are given two sides: the opposite side (9 m) and the adjacent side (17 m). To find the angle, we use the inverse tangent function:

\[
\tan^{-1}\left(\frac{9}{17}\right) = x
\]

\[
x \approx \tan^{-1}(0.5294) \approx 28^\circ
\]

So, \(x \approx 28^\circ\).

### 4A: Find the measure of the side labeled \(x\).
We are given the hypotenuse (380 mm) and the adjacent side (322 mm). To find the angle, we use the cosine function:

\[
\cos(x) = \frac{322}{380}
\]

Solving for \(x\):
\[
x = \cos^{-1}\left(\frac{322}{380}\right) \approx \cos^{-1}(0.8474) \approx 32^\circ
\]

So, \(x \approx 32^\circ\).

### 5A: Solve the telescope problem.
We are given that the horizontal distance is 20 ft, and the angle of elevation is 48°. We want to find the height of the flagpole. Using the tangent function:

\[
\tan(48^\circ) = \frac{h}{20}
\]

Solving for \(h\):
\[
h = 20 \times \tan(48^\circ) \approx 20 \times 1.1106 \approx 22.21 \, \text{ft}
\]

So, the height of the flagpole is approximately \(22.21 + 5 = 27.21 \, \text{ft}\).

Now, these solutions cover the left side (partner A) of the worksheet. Would you like help solving the right side as well (partner B)? 

Here are 5 related questions:
1. How do you apply trigonometric ratios to find unknown sides of a triangle?
2. What is the difference between using sine, cosine, and tangent in trigonometry?
3. How can you solve for an angle if you only have side lengths?
4. Why do we use inverse trigonometric functions to find angles?
5. How does the Pythagorean theorem relate to trigonometric functions?

**Tip:** When solving trigonometry problems involving right triangles, always start by identifying which sides you know (adjacent, opposite, or hypotenuse) and which trigonometric function applies (sine, cosine, or tangent).

What Do You Call a Row of Large Animals Separating Two Yards? Solve trigonometric problems involving right triangles and word problems related to geometry and real-world applications like flagpoles, ski slopes, and submarines.

Learn how to solve right triangle problems using trigonometric functions such as tangent, sine, and cosine. Explore real-world applications of trigonometry with examples including telescopes, ski slopes, and submarines. Suitable for Grades 8-10.

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