Question

1. find the measure of each acute angle.

(2x + 6)°
(5x + 14)°

Explanation:

Step1: Use the property of right - angled triangle angles

In a right - angled triangle, the sum of the two acute angles is \(90^{\circ}\). So, \((2x + 6)+(5x + 14)=90\).

Step2: Solve the equation for \(x\)

Combine like terms: \(2x+5x+6 + 14=90\), which simplifies to \(7x+20 = 90\).
Subtract \(20\) from both sides: \(7x=90 - 20\), so \(7x=70\).
Divide both sides by \(7\): \(x=\frac{70}{7}=10\).

Step3: Find the measure of each acute angle

For the first acute angle \((2x + 6)^{\circ}\):
Substitute \(x = 10\): \(2\times10+6=20 + 6=26^{\circ}\).
For the second acute angle \((5x + 14)^{\circ}\):
Substitute \(x = 10\): \(5\times10+14=50+14 = 64^{\circ}\).

Answer:

The measures of the acute angles are \(26^{\circ}\) and \(64^{\circ}\).