Question

find the measure of each acute angle in a right triangle where the measure of one acute angle is 5 times the measure of the other acute angle. the smaller acute angle measures □°, and the larger acute angle measures □°.

Explanation:

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

Since the measure of one acute angle is 5 times the measure of the other acute angle, the larger acute angle is \(5x\).

Step2: Use the property of right - triangle angles

In a right - triangle, the sum of the two acute angles is \(90^{\circ}\). So, \(x + 5x=90^{\circ}\).

Step3: Solve the equation

Combine like terms: \(6x = 90^{\circ}\). Then, divide both sides by 6: \(x=\frac{90^{\circ}}{6}=15^{\circ}\).

Step4: Find the larger acute angle

The larger acute angle is \(5x\). Substitute \(x = 15^{\circ}\), so \(5\times15^{\circ}=75^{\circ}\).

Answer:

The smaller acute angle measures \(15^{\circ}\) and the larger acute angle measures \(75^{\circ}\).