Question

find the measures of the angles of a right triangle where one of the two acute angles measures 2 times the other.

Explanation:

Step1: Let the smaller acute angle be \(x\)

Since one acute angle is \(2\) times the other, the larger acute angle is \(2x\).

Step2: Use the angle - sum property of a triangle

In a right - triangle, the sum of all angles is \(180^{\circ}\), and one angle is \(90^{\circ}\). So \(x + 2x+90^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(3x+90^{\circ}=180^{\circ}\).
Subtract \(90^{\circ}\) from both sides: \(3x=180^{\circ}-90^{\circ}=90^{\circ}\).
Divide both sides by \(3\): \(x = 30^{\circ}\).

Step3: Find the other acute angle

The larger acute angle is \(2x\). Substitute \(x = 30^{\circ}\), then \(2x=2\times30^{\circ}=60^{\circ}\).

Answer:

\(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\)