Finding Angles and Their Ratio in a Triangle

Understanding the properties of triangles is fundamental in geometry. A key property is that the sum of the interior angles of any triangle is always 180 degrees. When the angles are given in a ratio, we can represent the angles using a common multiple and then use this property to find their actual measures.

Analyzing the Triangle Angle Ratio Problem

The problem states that the angles of a triangle are in the ratio of 2 : 3 : 5. This means that if we divide each angle by a common factor, we get these numbers. Let the common factor be $x$.

So, the angles of the triangle can be represented as:

• First angle = $2x$
• Second angle = $3x$
• Third angle = $5x$

Calculating the Actual Angle Measures

We know that the sum of the interior angles of a triangle is $180^\circ$. Therefore, we can set up an equation:

$\qquad 2x + 3x + 5x = 180^\circ$

Combine the terms on the left side:

$\qquad (2 + 3 + 5)x = 180^\circ$

$\qquad 10x = 180^\circ$

Now, solve for $x$ by dividing both sides by 10:

$\qquad x = \frac{180^\circ}{10}$

$\qquad x = 18^\circ$

Now that we have the value of $x$, we can find the measure of each angle:

• First angle = $2x = 2 \times 18^\circ = 36^\circ$
• Second angle = $3x = 3 \times 18^\circ = 54^\circ$
• Third angle = $5x = 5 \times 18^\circ = 90^\circ$

We can check if the sum is correct: $36^\circ + 54^\circ + 90^\circ = 180^\circ$. The angles are correct.

Identifying the Greatest and Smallest Angles

Looking at the calculated angles ($36^\circ$, $54^\circ$, $90^\circ$):

• The smallest angle is $36^\circ$.
• The greatest angle is $90^\circ$.

Finding the Ratio of the Greatest Angle to the Smallest Angle

The problem asks for the ratio of the greatest angle to the smallest angle. This is the greatest angle divided by the smallest angle.

Ratio = $\frac{\text{Greatest Angle}}{\text{Smallest Angle}} = \frac{90^\circ}{36^\circ}$

To simplify the ratio, we can divide both numbers by their greatest common divisor. Both 90 and 36 are divisible by 18.

Ratio = $\frac{90 \div 18}{36 \div 18} = \frac{5}{2}$

So, the ratio of the greatest angle to the smallest angle is 5 : 2.

Conclusion

The ratio of the greatest angle to the smallest angle in the triangle is 5 : 2.

Revision Table: Triangle Angle Ratio

Additional Information: Types of Triangles Based on Angles

Triangles can be classified based on their angle measures:

• Acute Triangle: All three angles are less than $90^\circ$. In our problem, if the angles were $36^\circ, 54^\circ, 90^\circ$, it's not an acute triangle because one angle is $90^\circ$.
• Right Triangle: One of the angles is exactly $90^\circ$. In our problem, the angles are $36^\circ, 54^\circ, 90^\circ$, which means it is a right triangle. The ratio 2:3:5 represents the angles of a right triangle.
• Obtuse Triangle: One of the angles is greater than $90^\circ$.

Also, recall the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. While not directly used here, it's another fundamental triangle property.