Finding Angles in a Triangle Given a Ratio

The problem provides a relationship between the three angles of a triangle ▵ABC. The relationship is given as $2\angle A = 3\angle B = 6\angle C$. We need to find the value of the largest angle among these three angles.

In any triangle, the sum of the interior angles is always $180^\circ$. This is known as the Angle Sum Property of a triangle. So, we have:

• $\angle A + \angle B + \angle C = 180^\circ$

We are given the relationship: $2\angle A = 3\angle B = 6\angle C$. Let's assume that this common value is equal to some constant, say $k$.

So, we have:

• $2\angle A = k \implies \angle A = \frac{k}{2}$
• $3\angle B = k \implies \angle B = \frac{k}{3}$
• $6\angle C = k \implies \angle C = \frac{k}{6}$

Now, we can substitute these expressions for $\angle A$, $\angle B$, and $\angle C$ into the angle sum property equation:

$\frac{k}{2} + \frac{k}{3} + \frac{k}{6} = 180^\circ$

To solve for $k$, we find a common denominator for the fractions, which is 6:

$\frac{3k}{6} + \frac{2k}{6} + \frac{k}{6} = 180^\circ$

Combine the terms on the left side:

$\frac{3k + 2k + k}{6} = 180^\circ$

$\frac{6k}{6} = 180^\circ$

$k = 180^\circ$

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

• $\angle A = \frac{k}{2} = \frac{180^\circ}{2} = 90^\circ$
• $\angle B = \frac{k}{3} = \frac{180^\circ}{3} = 60^\circ$
• $\angle C = \frac{k}{6} = \frac{180^\circ}{6} = 30^\circ$

Let's check if the sum of these angles is $180^\circ$: $90^\circ + 60^\circ + 30^\circ = 180^\circ$. This confirms our calculations are correct.

The three angles of the triangle are $90^\circ$, $60^\circ$, and $30^\circ$. We need to find the largest angle among these three angles.

Comparing the values: $90^\circ > 60^\circ > 30^\circ$.

The largest angle is $90^\circ$.

Let's summarize the angles in a table:

The largest angle is indeed $90^\circ$.

Revision Table: Triangle Angles and Ratios

Here is a quick look back at the key steps involved in solving problems like this one, dealing with triangle angles given in a ratio or a relationship:

• Understand the Angle Sum Property: The sum of interior angles of any triangle is $180^\circ$.
• Set up a Common Ratio: If angles are given by a relationship like $a\angle A = b\angle B = c\angle C$, express each angle in terms of a single variable or constant using this relationship.
• Formulate an Equation: Use the Angle Sum Property ($\angle A + \angle B + \angle C = 180^\circ$) to set up an equation with the single variable.
• Solve the Equation: Solve for the variable.
• Calculate Each Angle: Substitute the value of the variable back into the expressions for each angle.
• Identify the Required Angle: Find the smallest, largest, or specific angle asked for in the question.

Additional Information: Types of Triangles by Angles

Triangles can be classified based on their angles:

• Acute-angled triangle: All three angles are less than $90^\circ$. (Example: $60^\circ, 60^\circ, 60^\circ$)
• Right-angled triangle: One angle is exactly $90^\circ$. The other two angles are acute and their sum is $90^\circ$. (Example: $90^\circ, 45^\circ, 45^\circ$; or in our case, $90^\circ, 60^\circ, 30^\circ$)
• Obtuse-angled triangle: One angle is greater than $90^\circ$. The other two angles are acute. (Example: $100^\circ, 40^\circ, 40^\circ$)

In this specific problem, since one of the angles is $90^\circ$, the triangle ▵ABC is a right-angled triangle.