Given that:

- \( m\angle TRI = 54^\circ \)
- \( m\angle DRI = 3x \)

To find \( x \), we'll need more context, like whether these angles are part of a larger geometric figure (e.g., a triangle, a line, or a polygon). If they are part of a triangle, for instance, the sum of the interior angles of the triangle is always \(180^\circ\). If they're part of a linear pair, their sum would be \(180^\circ\), and if they're part of a quadrilateral, the sum of all angles is \(360^\circ\).

However, based on the information provided:

### Case 1: Both Angles Form a Linear Pair
If \( m\angle TRI \) and \( m\angle DRI \) are supplementary (i.e., they form a linear pair), then:

\[
m\angle TRI + m\angle DRI = 180^\circ
\]

Substituting the values:

\[
54^\circ + 3x = 180^\circ
\]

### Step 1: Solve for \( x \)

\[
3x = 180^\circ - 54^\circ
\]
\[
3x = 126^\circ
\]
\[
x = \frac{126^\circ}{3} = 42^\circ
\]

So, \( x = 42^\circ \).

### Step 2: Verify
Now, \( m\angle DRI = 3x = 3(42^\circ) = 126^\circ \).

Thus, \( 54^\circ + 126^\circ = 180^\circ \), which confirms that the angles form a linear pair.

### Final Answer

- \( x = 42^\circ \)
- \( m\angle DRI = 126^\circ \)

If you have more information about the geometric figure or the relationship between the angles, we can refine or adjust this solution.

Would you like further details or have any other questions? Here are five related questions to consider:

1. **What if these angles were part of a triangle? How would that change the problem?**
2. **Could you find \( x \) if these angles were complementary?**
3. **How does understanding angle pairs help in geometry problems?**
4. **What other angle relationships should be considered in complex figures?**
5. **What are the implications if the angles were part of different polygons?**

**Tip:** When dealing with angle relationships, always identify if they are part of supplementary, complementary, or other defined angle pair relationships to solve efficiently.

Learn how to solve an angle problem involving m<TRI = 54 degrees and m<DRI = 3x. Discover the relationship between these angles and apply the Linear Pair Theorem to find the value of x. Suitable for Grades 7-9.

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