https://file.aiquickdraw.com/mathbot/file/1727406656073x7o8xvt9.jpg

The problem presented in the image involves solving for \( x \) using the properties of angles, specifically linear pairs. The diagram shows two angles at a point labeled \( R \), where \( \angle TRV \) is given as \( 60^\circ \), and \( \angle TRS \) is \( (4x)^\circ \). The task is to prove that \( x = 30 \).

### Given:
- \( m\angle TRV = 60^\circ \)
- \( m\angle TRS = (4x)^\circ \)

### To Prove:
- \( x = 30 \)

### Reasoning Steps:
1. **Given**:
   - \( m\angle TRV = 60^\circ \)
   - \( m\angle TRS = (4x)^\circ \)

2. **Statement 2**: \( \angle TRS \) and \( \angle TRV \) are a linear pair.
   - **Reason**: Definition of a linear pair.

3. The missing reason for step 3 likely involves the **Linear Pair Postulate**, which states that the sum of the measures of angles in a linear pair is \( 180^\circ \). Therefore, \( m\angle TRS + m\angle TRV = 180^\circ \).

Let me solve the problem:

\[
m\angle TRV + m\angle TRS = 180^\circ
\]

Substitute the given values:

\[
60^\circ + (4x)^\circ = 180^\circ
\]

Solve for \( x \):

\[
4x = 180^\circ - 60^\circ
\]
\[
4x = 120^\circ
\]
\[
x = \frac{120^\circ}{4} = 30
\]

Thus, \( x = 30 \).

### The Missing Reason in Step 3:
The missing reason in step 3 is the **Linear Pair Postulate**, which states that the sum of the measures of a linear pair of angles is \( 180^\circ \).

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Do you have any questions or need further details? 

Here are 5 related questions you might find interesting:
1. What is the definition of a linear pair of angles?
2. How do we know that angles on a straight line sum up to 180 degrees?
3. Can two angles form a linear pair if they are not adjacent?
4. How would the solution change if \( m\angle TRV \) were \( 70^\circ \) instead of \( 60^\circ \)?
5. What other angle properties (besides linear pairs) can help in solving geometry problems?

**Tip**: Always double-check the definitions and postulates relevant to the problem before proceeding with solving it!

Given that m∠TRV = 60° and m∠TRS = (4x)°, prove that x = 30.

Learn how to solve for x in the given problem involving linear pairs of angles. The problem involves basic geometry principles and algebraic steps to find x when m∠TRV = 60° and m∠TRS = (4x)°.

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