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The problem provided involves two triangles, ΔRSV and ΔTVS, where the triangles are stated to be congruent (ΔRSV ≅ ΔTVS). You are tasked with solving for \( x \), finding the measure of angle \( \angle R \), and finding the measure of angle \( \angle RVS \).

### Given Information:
- The side opposite \( \angle R \) in ΔRSV is labeled \( (x + 3) \) units.
- The side opposite \( \angle T \) in ΔTVS is labeled as \( x \) units.
- The hypotenuse of the right triangle ΔTVS is given as \( 12 \) units.
- \( \angle S \) is marked as \( 71^\circ \), and the right angles are present in both triangles.

### Solution Outline:
1. **Congruence of Triangles**: Since ΔRSV ≅ ΔTVS, corresponding sides and angles are equal. This will help us set up relationships between the sides and angles.

2. **Using Pythagoras' Theorem**: The triangles appear to be right triangles, so we can use the Pythagorean theorem in ΔTVS to solve for \( x \).

3. **Angle Calculations**: Use the information given and angle sum properties in triangles to find \( \angle R \) and \( \angle RVS \).

Let's begin by applying the Pythagorean theorem to ΔTVS to solve for \( x \).

\[
(x + 3)^2 + x^2 = 12^2
\]
I will now solve this equation for \( x \).The solution to the equation for \( x \) yields two possible values:

\[
x = \frac{-3}{2} + \frac{3\sqrt{31}}{2} \quad \text{or} \quad x = \frac{-3\sqrt{31}}{2} - \frac{3}{2}
\]

Since side lengths must be positive, we take the positive value:

\[
x = \frac{-3}{2} + \frac{3\sqrt{31}}{2}
\]

This simplifies to an approximate value, but let's keep it exact for now.

Next, we will use this value of \( x \) to find the angle measures. Since \( \angle S = 71^\circ \), we can use the properties of triangles and the congruence relationship to calculate \( \angle R \) and \( \angle RVS \). I'll proceed with that calculation now.Given that both triangles \( \Delta RSV \) and \( \Delta TVS \) are right triangles, we can use the fact that the sum of angles in a triangle is \( 180^\circ \).

### Step 1: Find \( \angle R \)
- In triangle \( \Delta RSV \), \( \angle S = 71^\circ \) and there is a right angle at \( \angle V \).
- The sum of the angles in a triangle is \( 180^\circ \), so:

\[
\angle R = 180^\circ - 90^\circ - 71^\circ = 19^\circ
\]

### Step 2: Find \( \angle RVS \)
- Since \( \angle RVS \) is congruent to \( \angle TVS \), and both triangles are right triangles, \( \angle RVS \) is the same as the given \( 71^\circ \) from \( \Delta TVS \).

Thus:
- \( \angle R = 19^\circ \)
- \( \angle RVS = 71^\circ \)

Let me summarize the results:

- \( x \approx \frac{-3}{2} + \frac{3\sqrt{31}}{2} \)
- \( \angle R = 19^\circ \)
- \( \angle RVS = 71^\circ \)

Would you like me to further simplify the value of \( x \) or provide any additional details?

---

Here are five related questions to expand your understanding:
1. How can the Pythagorean theorem be applied to different types of triangles?
2. What are the criteria for congruence of triangles?
3. How does one determine the angles in a right triangle using trigonometric functions?
4. Why is it important for side lengths to be positive in geometric problems?
5. What other methods could be used to verify the congruence of triangles?

**Tip:** In geometry problems involving right triangles, it's often helpful to confirm results using trigonometric ratios, such as sine, cosine, and tangent!

ΔRSV ≅ ΔTVS. Solve for x, find ∠R, and find ∠RVS.

Learn how to solve for x and find the angles ∠R and ∠RVS in congruent triangles ΔRSV ≅ ΔTVS using the Pythagorean theorem and angle sum property. This problem is perfect for students in Grades 9-10.

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