https://file.aiquickdraw.com/mathbot/file/1729194206157zd9baape.jpg

Let's solve each of these problems in turn:

### Problem 13:
**Given:**
- \(\triangle QRS\) is equilateral, so all sides are equal.
- \(QR = 2x - 7\)
- \(RS = 5x - 61\)
- \(QS = x + 11\)

We need to find the value of \(x\) and the length of each side.

**Solution:**

Since the triangle is equilateral:
\[
QR = RS = QS
\]

1. Set \(QR = RS\):
\[
2x - 7 = 5x - 61
\]
Solve for \(x\):
\[
2x - 5x = -61 + 7
\]
\[
-3x = -54
\]
\[
x = 18
\]

2. Now, check the lengths of the sides using \(x = 18\):
- \(QR = 2(18) - 7 = 36 - 7 = 29\)
- \(RS = 5(18) - 61 = 90 - 61 = 29\)
- \(QS = 18 + 11 = 29\)

Thus, all sides are 29, which confirms that the triangle is equilateral.

### Problem 14:
**Given:**
- \(\triangle CDE\) is isosceles with \(CD = DE\)
- \(CD = 9x - 25\), \(DE = 6x + 8\), \(CE = 10x - 73\)

We need to find \(x\) and the measure of each side.

**Solution:**

Since \(CD = DE\):
\[
9x - 25 = 6x + 8
\]
Solve for \(x\):
\[
9x - 6x = 8 + 25
\]
\[
3x = 33
\]
\[
x = 11
\]

Now, find the lengths of the sides:
- \(CD = 9(11) - 25 = 99 - 25 = 74\)
- \(DE = 6(11) + 8 = 66 + 8 = 74\)
- \(CE = 10(11) - 73 = 110 - 73 = 37\)

So, the sides are:
- \(CD = DE = 74\)
- \(CE = 37\)

### Problem 15:
**Given:**
- \(\triangle WXY\) is isosceles with \(WX = WY\)
- \(WX = 4x + 3\), \(XY = 5x - 7\), \(WY = 7x - 66\)

We need to find \(x\) and the length of each side.

**Solution:**

Since \(WX = WY\):
\[
4x + 3 = 7x - 66
\]
Solve for \(x\):
\[
4x - 7x = -66 - 3
\]
\[
-3x = -69
\]
\[
x = 23
\]

Now, find the lengths of the sides:
- \(WX = 4(23) + 3 = 92 + 3 = 95\)
- \(WY = 7(23) - 66 = 161 - 66 = 95\)
- \(XY = 5(23) - 7 = 115 - 7 = 108\)

So, the sides are:
- \(WX = WY = 95\)
- \(XY = 108\)

---

Do you need any further details or clarification? Here are some related questions you could explore:

1. What is the definition of an equilateral triangle?
2. How do you solve for \(x\) in a system of linear equations?
3. What are the properties of an isosceles triangle?
4. How do you check if a triangle is isosceles using side lengths?
5. Can an equilateral triangle have different side lengths?

**Tip:** When solving for \(x\) in geometric problems, always substitute the value back into the expressions to verify the consistency of your solution.

Solve for x and find the measure of each side for the following triangles: 1) An equilateral triangle QRS where QR = 2x - 7, RS = 5x - 61, and QS = x + 11. 2) An isosceles triangle CDE with CD = DE and CD = 9x - 25, DE = 6x + 8, and CE = 10x - 73. 3) An isosceles triangle WXY with WX = WY and WX = 4x + 3, XY = 5x - 7, and WY = 7x - 66.

Learn how to solve for x in different triangle problems, including an equilateral triangle and two isosceles triangles. Detailed step-by-step solutions for each side measurement are provided, with algebra and geometry concepts. Suitable for high school students in grades 9-11.

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