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The problem provides the following information:

1. **∠TRS is a straight angle**, which means that its measure is 180°.
2. **∠TRX is a right angle**, which means that its measure is 90°.
3. The measures of the angles are given as:
   - \( m\angle TRS = 2x + 5y \)
   - \( m\angle XRS = 3x + 3y \)

We need to solve for \( x \) and \( y \).

### Solution:

Since \(\angle TRS\) is a straight angle, it can be divided into two parts: \(\angle TRX\) and \(\angle XRS\).

\[
\angle TRS = \angle TRX + \angle XRS
\]

Substitute the given values:

\[
2x + 5y = 90 + (3x + 3y)
\]

Simplifying:

\[
2x + 5y = 90 + 3x + 3y
\]

Rearrange to solve for \( x \) and \( y \):

\[
2x + 5y - 3x - 3y = 90
\]
\[
-x + 2y = 90
\]
\[
x = 2y - 90 \quad \text{(Equation 1)}
\]

Next, we substitute this value of \( x \) into the given \( m\angle TRX = 90° \) and verify.

### Check the value:
We substitute \( x = 2y - 90 \) into original equation:

Result...

Let's calculate directly.The solution to the equations is:

\[
x = 2y - 90
\]

This means \( x \) is expressed in terms of \( y \). There isn't a unique solution for both \( x \) and \( y \) since we need an additional independent equation to solve for both variables.

Would you like to discuss this further or solve additional problems related to this topic?

### Related Questions
1. If \( y = 50 \), what is the value of \( x \)?
2. How do you find the complementary angles to \(\angle TRX\) and \(\angle XRS\)?
3. What are the conditions for two lines to form a straight angle?
4. Can two right angles form a straight line?
5. How do angle measures change with different \( x \) and \( y \) values?

**Tip:** Always check if the number of equations matches the number of variables when solving a system of linear equations.

Given: ∠TRS is a straight angle, ∠TRX is a right angle, m∠TRS = 2x + 5y, and m∠XRS = 3x + 3y. Solve for x and y.

Learn how to solve for x and y using algebraic expressions for angles in a straight line. This problem involves key geometry and algebra concepts, such as angle sum on a straight line and solving systems of linear equations. Suitable for students in Grades 9-10.

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