Let's break down each of the triangles and solve for the missing sides and angles step by step.

Problem 7:

You have a right triangle with:

• Hypotenuse $\overline{RS} = 18$
• Side $\overline{RQ} = 6$
• You need to find $\overline{QS}$, $m\angle Q$, and $m\angle S$.

Since this is a right triangle, we can apply the Pythagorean theorem and trigonometric ratios to find the missing values.

Step 1: Find $\overline{QS}$

Use the Pythagorean theorem: $QS^2 = RS^2 - RQ^2$ $QS^2 = 18^2 - 6^2 = 324 - 36 = 288$ $QS = \sqrt{288} \approx 16.97$

Step 2: Find $m\angle Q$

Using trigonometry (since we know the opposite side and adjacent side relative to $\angle Q$): $\tan(Q) = \frac{RQ}{QS} = \frac{6}{16.97} \approx 0.354$ $m\angle Q = \tan^{-1}(0.354) \approx 19.5^\circ$

Step 3: Find $m\angle S$

Since the sum of the angles in a triangle is 180°, and $\angle Q$ and $\angle S$ are complementary (because it's a right triangle): $m\angle S = 90^\circ - m\angle Q = 90^\circ - 19.5^\circ = 70.5^\circ$

Problem 8:

This is another right triangle, where we are given:

• Hypotenuse $\overline{XY} = 26.3$
• Angle $m\angle X = 46^\circ$
• We need to find the side $\overline{WX}$, side $\overline{XY}$, and $m\angle X$.

Step 1: Find $WX$

We can use the sine function: $\sin(46^\circ) = \frac{WX}{26.3}$ $WX = 26.3 \times \sin(46^\circ) \approx 26.3 \times 0.7193 \approx 18.9$

Step 2: Find $m\angle Y$

Since it's a right triangle: $m\angle Y = 90^\circ - 46^\circ = 44^\circ$

Problem 9:

For triangle $\triangle ABC$, we are tasked with finding the correct equations to solve for the missing sides or angles.

From the options provided:

• Option A: $BC = 16 \times \tan(29^\circ)$ is correct.
  • This would help solve for $BC$, the side opposite to $\angle B$.

Other options are incorrect based on the standard trigonometric relationships for solving right triangles.

Would you like further details or explanations on any part of the solution?

Here are some related questions you might find interesting:

1. How does the Pythagorean theorem relate to trigonometric functions?
2. When do we use sine, cosine, or tangent in solving right triangles?
3. How do you solve for an unknown angle if you are given two sides of a right triangle?
4. What happens if you’re given all sides of a right triangle but no angles?
5. Can you calculate missing side lengths without using trigonometric ratios?

Tip: Always check whether you are working with a right triangle when solving for angles or sides. The relationships between the sides and angles are simpler and more direct in right triangles!