https://file.aiquickdraw.com/mathbot/file/1726065493931fdir34js.jpg

In the image, we have a right triangle \( \triangle PQR \), where:

- \( \angle QPR = 90^\circ \),
- \( \angle PQR = 54^\circ \),
- \( \angle PRQ = 36^\circ \), and
- The expression for \( \angle PQR \) is given as \( 4x + 30^\circ \).

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

We know that the sum of the angles in any triangle is \( 180^\circ \). In this case, since \( \angle QPR = 90^\circ \), the other two angles \( \angle PQR \) and \( \angle PRQ \) must sum to \( 90^\circ \).

Given that \( \angle PRQ = 36^\circ \), we can express the equation for \( \angle PQR \) as follows:

\[
4x + 30^\circ = 54^\circ
\]

Now, solve for \( x \):

\[
4x + 30 = 54
\]
\[
4x = 54 - 30
\]
\[
4x = 24
\]
\[
x = \frac{24}{4} = 6
\]

### Step 2: Find \( \angle P \)

Since \( \angle P = \angle QPR = 90^\circ \), we already have the value of \( \angle P \).

### Final Answer:

- \( x = 6^\circ \)
- \( \angle P = 90^\circ \)

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How do we calculate angles in non-right triangles?
2. How do trigonometric functions apply to right triangles?
3. What are the properties of complementary angles in right triangles?
4. How can we solve for side lengths in right triangles?
5. How can we verify our angle calculations using geometry?

**Tip**: When solving for angles, always check the sum of all angles in a triangle, which should equal 180°.

In the triangle shown above, x = ? and ∠P = ?

Learn how to solve for the variable x and find the value of angle P in a right triangle. This problem involves geometric properties such as the angle sum theorem and algebraic expressions, making it suitable for students in Grades 7-9.

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