Question

in exercises 19–22, find the measure of each acute angle. (see example 4.)

1. right triangle with acute angles labeled 3x° and 2x°
2. right triangle with acute angles labeled (3x + 2)° and x°
3. right triangle with acute angles labeled (11x − 2)° and (6x + 7)°
4. right triangle with acute angles labeled (19x − 1)° and (13x − 5)°

Explanation:

Exercise 19

Step1: Recall right triangle angle sum

In a right triangle, the two acute angles sum to \(90^\circ\). So \(3x + 2x = 90\).

Step2: Solve for \(x\)

Combine like terms: \(5x = 90\). Divide both sides by 5: \(x = 18\).

Step3: Find angle measures

First angle: \(3x = 3\times18 = 54^\circ\). Second angle: \(2x = 2\times18 = 36^\circ\).

Step1: Use right triangle angle sum

The two acute angles sum to \(90^\circ\), so \((3x + 2) + x = 90\).

Step2: Solve for \(x\)

Combine like terms: \(4x + 2 = 90\). Subtract 2: \(4x = 88\). Divide by 4: \(x = 22\).

Step3: Find angle measures

First angle: \(3x + 2 = 3\times22 + 2 = 68^\circ\). Second angle: \(x = 22^\circ\).

Step1: Apply right triangle angle sum

The two acute angles sum to \(90^\circ\), so \((11x - 2) + (6x + 7) = 90\).

Step2: Solve for \(x\)

Combine like terms: \(17x + 5 = 90\). Subtract 5: \(17x = 85\). Divide by 17: \(x = 5\).

Step3: Find angle measures

First angle: \(11x - 2 = 11\times5 - 2 = 53^\circ\). Second angle: \(6x + 7 = 6\times5 + 7 = 37^\circ\).

Answer:

The acute angles are \(54^\circ\) and \(36^\circ\).

Exercise 20