The sum of the interior angles of a triangle is always $180^\circ$.

Given that the angles of the triangle are in terms of $B$, specifically:

• Angle $A = 2B$
• Angle $B = 3B$
• Angle $C = 4B$

We can use the fact that the sum of the angles in a triangle is 180°:

$2B + 3B + 4B = 180^\circ$

Simplify the left side:

$9B = 180^\circ$

Now solve for $B$:

$B = \frac{180^\circ}{9} = 20^\circ$

Now that we know $B = 20^\circ$, we can find the individual angles:

• $\angle A = 2B = 2 \times 20^\circ = 40^\circ$
• $\angle B = 3B = 3 \times 20^\circ = 60^\circ$
• $\angle C = 4B = 4 \times 20^\circ = 80^\circ$

So, the angles are:

• $\angle A = 40^\circ$
• $\angle B = 60^\circ$
• $\angle C = 80^\circ$

Would you like more details or have any questions about this? Here are some related questions for further exploration:

1. How do you solve for angles in a triangle when the angles are in terms of a variable?
2. What happens if the angles of a triangle are expressed as a linear relationship (e.g., $x$, $2x$, $3x$)?
3. How do you use the Law of Sines or Cosines with non-standard angles?
4. How can you determine the type of triangle based on its angles (e.g., acute, obtuse, right)?
5. What is the significance of the angle sum property in different geometric shapes?

Tip: Always remember that the sum of the interior angles of any triangle is $180^\circ$. This property is fundamental in solving many geometric problems.