We are given the following angles in a triangle:

• $m\angle RST = \frac{3x}{4}^\circ$
• $m\angle TSV = \frac{x}{3}^\circ$
• $m\angle RSV = 78^\circ$

Since these three angles are part of the same triangle, the sum of their measures must equal $180^\circ$. We can express this as:

$m\angle RST + m\angle TSV + m\angle RSV = 180^\circ$

Substituting the values:

$\frac{3x}{4} + \frac{x}{3} + 78 = 180$

To solve for $x$, first eliminate the fractions by finding the least common denominator (LCD) between 4 and 3, which is 12. We rewrite each term accordingly:

$\frac{3x}{4} = \frac{9x}{12}, \quad \frac{x}{3} = \frac{4x}{12}$

Now, rewrite the equation:

$\frac{9x}{12} + \frac{4x}{12} + 78 = 180$

Combine the $x$-terms:

$\frac{13x}{12} + 78 = 180$

Now, subtract 78 from both sides:

$\frac{13x}{12} = 102$

Multiply both sides by 12 to eliminate the denominator:

$13x = 1224$

Now, divide by 13:

$x = 94.15$

Now that we know $x$, we can find the angles:

• $m\angle RST = \frac{3(94.15)}{4} \approx 70.61^\circ$
• $m\angle TSV = \frac{94.15}{3} \approx 31.38^\circ$
• $m\angle RSV = 78^\circ$ (as given)

So, the three angles are approximately $70.61^\circ$, $31.38^\circ$, and $78^\circ$.

Do you want further details or have any questions?

Here are 5 related questions:

1. How do you determine if a triangle is acute, obtuse, or right based on its angles?
2. Can we form a triangle if one of its angles exceeds $90^\circ$?
3. What would happen to the triangle if $m\angle RSV$ was smaller than 78°?
4. How does the Law of Sines relate to finding missing angles in a triangle?
5. Can a triangle have two angles larger than $90^\circ$?

Tip: Always ensure the sum of interior angles of a triangle equals $180^\circ$, regardless of its shape or size.