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To find the measures of the two acute angles in the right triangle, we’ll use the fact that the sum of all angles in a triangle is \(180^\circ\). Since we have a right triangle, one of the angles is \(90^\circ\). Thus, the two acute angles will sum up to \(90^\circ\).

Let’s set up an equation using this information.

### Given:
1. Acute angle \(A = (19x - 1)^\circ\)
2. Acute angle \(B = (13x - 5)^\circ\)
3. Right angle = \(90^\circ\)

### Steps:
Since the two acute angles must sum up to \(90^\circ\), we can write:
\[
(19x - 1) + (13x - 5) = 90
\]

1. Simplify the equation:
   \[
   32x - 6 = 90
   \]

2. Add 6 to both sides:
   \[
   32x = 96
   \]

3. Divide by 32:
   \[
   x = 3
   \]

### Substitute \(x = 3\) back into each angle expression:

1. \( (19x - 1)^\circ = 19(3) - 1 = 57 - 1 = 56^\circ \)
2. \( (13x - 5)^\circ = 13(3) - 5 = 39 - 5 = 34^\circ \)

### Answer:
- The measure of the first acute angle is \(56^\circ\).
- The measure of the second acute angle is \(34^\circ\).

Would you like further clarification on this solution? 

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Here are some additional questions to explore:

1. What would the values of the angles be if \(x\) had a different value?
2. How can we verify that the calculated angles make this a right triangle?
3. Can we set up a similar problem with different expressions for the angles?
4. How does knowing the sum of angles in a triangle help in solving similar problems?
5. Why must the sum of the acute angles in a right triangle always be \(90^\circ\)?

**Tip:** When solving for angles in triangles, always verify that their sum aligns with \(180^\circ\) for consistency.

Find the measure of each acute angle in the right triangle with angles labeled as (19x - 1)° and (13x - 5)°.

Learn how to find the measures of acute angles in a right triangle where the angles are represented by algebraic expressions (19x - 1)° and (13x - 5)°. This solution covers the triangle angle sum theorem, setting up equations, and solving for x. Suitable for students in Grades 8-10.

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