If PS=RS=56, m∠PQS=3v, and m∠RQS=v+50°, what is m∠RQS?

To find the measure of \(\angle RQS\), we can use the information provided and the properties of isosceles triangles.

Given:
- \(PS = RS = 56\)
- \(m\angle PQS = 3v\)
- \(m\angle RQS = v + 50^\circ\)

### Steps to solve:

1. Since \(PS = RS\), triangle \(PQS\) is isosceles, meaning \(\angle PQS = \angle RQS\).

2. Thus, \(m\angle PQS = m\angle RQS\).

3. From the problem statement, we have:
   \[
   m\angle PQS = 3v
   \]
   and
   \[
   m\angle RQS = v + 50^\circ
   \]

4. Since these two angles are equal:
   \[
   3v = v + 50^\circ
   \]

5. Solve for \(v\):
   \[
   3v - v = 50^\circ
   \]
   \[
   2v = 50^\circ
   \]
   \[
   v = 25^\circ
   \]

6. Now substitute \(v = 25^\circ\) into the expression for \(m\angle RQS\):
   \[
   m\angle RQS = v + 50^\circ = 25^\circ + 50^\circ = 75^\circ
   \]

### Final Answer:
\[
m\angle RQS = 75^\circ
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you determine the angles in an isosceles triangle?
2. What is the significance of equal angles in isosceles triangles?
3. How would the problem change if the triangle were not isosceles?
4. What is the sum of the angles in a triangle?
5. How can you verify that a triangle is isosceles given only angles?

**Tip:** In isosceles triangles, the angles opposite the equal sides are always equal.

Learn how to find the measure of angle RQS in an isosceles triangle where PS=RS=56, m∠PQS=3v, and m∠RQS=v+50°. This problem covers key geometric concepts, including the Isosceles Triangle Theorem, and algebraic techniques. Ideal for students in Grades 8-10.

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