Finding the Largest Angle in a Triangle Given a Ratio

The question asks us to find the value of the largest angle in a triangle where the measures of its angles are in the ratio 2 : 5 : 8. To solve this, we need to use a fundamental property of triangles.

Understanding Triangle Angle Properties

A key property of any triangle is that the sum of the measures of its interior angles is always equal to 180 degrees. This fact is crucial when dealing with problems involving triangle angles, especially when they are given in a ratio.

Setting up the Equation for Triangle Angles

Since the angles are in the ratio 2 : 5 : 8, we can represent the angles as multiples of a common factor, let's call it \(x\). So, the three angles of the triangle can be written as:

• First angle = \(2x\)
• Second angle = \(5x\)
• Third angle = \(8x\)

Using the property that the sum of the angles in a triangle is \(180^\circ\), we can set up the following equation:

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

Combine the terms on the left side of the equation:

\(15x = 180^\circ\)

Solving for the Unknown Value (x)

Now, we need to find the value of \(x\). We can do this by dividing both sides of the equation by 15:

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

Performing the division:

\(x = 12^\circ\)

So, the common factor \(x\) is 12 degrees.

Calculating Each Angle of the Triangle

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 = \(8x = 8 \times 12^\circ = 96^\circ\)

Identifying the Largest Angle

The three angles of the triangle are \(24^\circ\), \(60^\circ\), and \(96^\circ\). We are asked to find the value of the largest angle. Comparing the three values, the largest angle is \(96^\circ\).

To double-check, let's make sure the sum of these angles is \(180^\circ\): \(24^\circ + 60^\circ + 96^\circ = 84^\circ + 96^\circ = 180^\circ\). This confirms our calculations are correct.

Therefore, the value of the largest angle is \(96^\circ\).

Revision Table: Triangle Angle Ratios

Additional Information: Angles and Triangles

Understanding angle properties helps classify triangles. Based on angles, triangles can be:

• Acute Triangle: All three angles are less than \(90^\circ\). (Example: \(24^\circ, 60^\circ, 96^\circ\) is not acute because \(96^\circ > 90^\circ\)).
• Right Triangle: One angle is exactly \(90^\circ\).
• Obtuse Triangle: One angle is greater than \(90^\circ\). (Example: Our triangle with angles \(24^\circ, 60^\circ, 96^\circ\) is an obtuse triangle because \(96^\circ\) is obtuse).

The ratio of angles provides information about the relative size of the angles, allowing us to determine the exact angle measures using the \(180^\circ\) sum property.