m∠RST = (3x/4)° m∠TSV = (x/3)° m∠RSV = 78°

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.

Explore how to solve a geometry problem involving the angles of a triangle where m∠RST = (3x/4)°, m∠TSV = (x/3)°, and m∠RSV = 78°. Follow a step-by-step solution using the Triangle Angle Sum Theorem, finding the value of x and the measures of the triangle's angles. Suitable for Grades 9-11 students.

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