Finding the Smallest Angle in a Triangle Given its Ratio

Understanding the properties of triangles is fundamental in geometry. One key property is the sum of interior angles of any triangle.

For any triangle, the sum of its three interior angles is always 180 degrees ($180^\circ$).

Setting up the Problem with the Given Ratio

The problem states that the ratio of the angles of the triangle is 1 : 2 : 6. This means that if we represent the angles using a common multiplier, say $x$, the three angles can be written as:

• First angle: $1 \times x = x$
• Second angle: $2 \times x = 2x$
• Third angle: $6 \times x = 6x$

These expressions represent the measures of the three angles of the triangle in degrees.

Using the Sum of Angles Property

According to the property that the sum of the angles in a triangle is $180^\circ$, we can set up an equation:

$x + 2x + 6x = 180^\circ$

Solving for the Unknown Variable

Combine the terms on the left side of the equation:

$9x = 180^\circ$

Now, solve for $x$ by dividing both sides by 9:

$x = \frac{180^\circ}{9}$

$x = 20^\circ$

Calculating Each Angle

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

• First angle: $x = 20^\circ$
• Second angle: $2x = 2 \times 20^\circ = 40^\circ$
• Third angle: $6x = 6 \times 20^\circ = 120^\circ$

Let's verify the sum of these angles:

$20^\circ + 40^\circ + 120^\circ = 60^\circ + 120^\circ = 180^\circ$

The sum is indeed $180^\circ$, so our calculations are correct.

Identifying the Smallest Angle

Comparing the three angle measures ($20^\circ$, $40^\circ$, and $120^\circ$), the smallest angle is the one with the measure of $20^\circ$. This corresponds to the angle represented by $x$, which was based on the smallest part of the ratio (1).

Summary of Angle Measures

Therefore, the smallest angle of the triangle is $20^\circ$.

Revision Table: Triangle Angle Ratio

Additional Information: Types of Triangles by Angle

Triangles can be classified based on their angle measures. Knowing the angles helps identify the triangle type:

• Acute Triangle: All three interior angles are acute (less than $90^\circ$). Our example with angles $20^\circ, 40^\circ, 120^\circ$ is NOT an acute triangle because one angle is greater than $90^\circ$.
• Right Triangle: One interior angle is a right angle (exactly $90^\circ$).
• Obtuse Triangle: One interior angle is obtuse (greater than $90^\circ$ and less than $180^\circ$). Our example with angles $20^\circ, 40^\circ, 120^\circ$ is an obtuse triangle because $120^\circ$ is an obtuse angle.
• Equiangular Triangle: All three interior angles are equal. Each angle measures $60^\circ$. This is also a type of acute triangle.

In this specific problem, since one angle ($120^\circ$) is obtuse, the triangle is classified as an obtuse triangle.