We are given a triangle where one of the angles is $\frac{2}{5}$ of the second angle and $\frac{1}{4}$ of the third angle. We need to find the measures of all three angles of the triangle.

1. Problem Description

We are given a triangle where one of the angles is $\frac{2}{5}$ of the second angle and $\frac{1}{4}$ of the third angle. We need to find the measures of all three angles of the triangle.

2. Solution Steps

Let the three angles of the triangle be $a, b, c$. We are given that

$a = \frac{2}{5}b$

$a = \frac{1}{4}c$

We also know that the sum of the angles in a triangle is 180 degrees:

$a + b + c = 180$

From the given equations, we can express $b$ and $c$ in terms of $a$:

$b = \frac{5}{2}a$

$c = 4a$

Now we can substitute these expressions into the equation for the sum of the angles:

$a + \frac{5}{2}a + 4a = 180$

Multiply by 2 to eliminate the fraction:

$2a + 5a + 8a = 360$

$15a = 360$

$a = \frac{360}{15}$

$a = 24$

Now we can find $b$ and $c$:

$b = \frac{5}{2}a = \frac{5}{2}(24) = 5 \cdot 12 = 60$

$c = 4a = 4(24) = 96$

So, the three angles are 24, 60, and 96 degrees.

3. Final Answer

The three angles of the triangle are 24°, 60°, and 96°.