The question asks us to find the value of the smallest angle in a triangle where the angles are given in the ratio 2 : 5 : 8.

We know a fundamental property of triangles:

• The sum of the interior angles of any triangle is always 180 degrees.

Given the ratio of the angles is 2 : 5 : 8, we can represent the angles using a common multiplier, let's call it $x$. So, the angles can be written as:

• First angle = $2x$
• Second angle = $5x$
• Third angle = $8x$

Now, we use the property that the sum of these angles is 180 degrees. We can set up an equation:

\( 2x + 5x + 8x = 180^\circ \)

Combine the terms on the left side of the equation:

\( (2 + 5 + 8)x = 180^\circ \)

\( 15x = 180^\circ \)

To find the value of $x$, divide both sides of the equation by 15:

\( x = \frac{180^\circ}{15} \)

\( x = 12^\circ \)

Now that we have the value of $x$, we can find the measure of each angle by substituting $x = 12^\circ$ into our expressions for the angles:

• First angle = \( 2x = 2 \times 12^\circ = 24^\circ \)
• Second angle = \( 5x = 5 \times 12^\circ = 60^\circ \)
• Third angle = \( 8x = 8 \times 12^\circ = 96^\circ \)

The three angles of the triangle are \( 24^\circ \), \( 60^\circ \), and \( 96^\circ \). Let's quickly check if their sum is 180 degrees:

\( 24^\circ + 60^\circ + 96^\circ = 84^\circ + 96^\circ = 180^\circ \)

The sum is indeed 180 degrees, which confirms our calculations for the angles are correct.

The question asks for the value of the smallest angle. Looking at the calculated angles \( 24^\circ \), \( 60^\circ \), and \( 96^\circ \), the smallest angle is \( 24^\circ \).

Revision Table: Triangle Angles and Ratio

Additional Information on Triangle Angle Properties

Understanding the properties of triangle angles is crucial in geometry. Here are a few more points:

• Types of Triangles by Angles:
  • Acute Triangle: All three angles are less than \( 90^\circ \). Our example triangle with angles \( 24^\circ, 60^\circ, 96^\circ \) is not acute because one angle is \( 96^\circ \).
  • Right Triangle: One angle is exactly \( 90^\circ \).
  • Obtuse Triangle: One angle is greater than \( 90^\circ \). Our example triangle with angles \( 24^\circ, 60^\circ, 96^\circ \) is an obtuse triangle because \( 96^\circ > 90^\circ \).
• Exterior Angle Property: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• Angle-Side Relationship: In any triangle, the side opposite the largest angle is the longest side, and the side opposite the smallest angle is the shortest side. In our case, the side opposite the \( 24^\circ \) angle would be the shortest side.

The ratio method is a common way to solve problems involving proportional angles in geometric figures like triangles.