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Let's solve both triangles step by step.

---

### **First Triangle (Left Side)**  
The triangle is **isosceles** since two sides are equal (11 and 11).  

#### Step 1: Solve for angle \( x \)  
- In any triangle, the sum of all angles is \( 180^\circ \).  
- Given \( \angle 60^\circ \) and the isosceles property, the other two angles are equal.  
Let each equal angle be \( 3x \).  

\[
3x + 3x + 60^\circ = 180^\circ
\]
\[
6x = 120^\circ
\]
\[
x = 20^\circ
\]

#### Step 2: Solve for \( y \)  
The side opposite \( \angle 60^\circ \) is 11, and we know the opposite sides of equal angles are proportional. Given \( 5y \) corresponds to one of the equal sides:

\[
5y = 11 \quad \Rightarrow \quad y = \frac{11}{5} = 2.2
\]

---

### **Second Triangle (Right Side)**  
This triangle is also isosceles, with two equal sides (\( 3x + 8 \) and \( 23 \)).

#### Step 1: Solve for \( x \)  
Since the two sides are equal:

\[
3x + 8 = 23
\]
\[
3x = 15
\]
\[
x = 5
\]

#### Step 2: Solve for \( y \)  
We know one angle is \( 60^\circ \), and the sum of all angles is \( 180^\circ \). Since this triangle is isosceles, the two base angles are equal. Let each base angle be \( 6y \):

\[
6y + 6y + 60^\circ = 180^\circ
\]
\[
12y = 120^\circ
\]
\[
y = 10
\]

---

### **Final Answers**  

1. **First Triangle:**  
   - \( x = 20 \)  
   - \( y = 2.2 \)  

2. **Second Triangle:**  
   - \( x = 5 \)  
   - \( y = 10 \)  

---

### Questions for You:
1. What is the property of an isosceles triangle regarding angles and sides?  
2. How is the sum of internal angles in a triangle determined?  
3. Why are equal angles opposite to equal sides in a triangle?  
4. Can we use trigonometry (sine, cosine) for triangles like this?  
5. How do you solve for unknown angles algebraically?  

**Tip**: For isosceles triangles, remember that equal sides always correspond to equal opposite angles.

Find the missing sides or angles in the given triangles. Solve for x and y.

Learn how to solve for missing angles and sides in isosceles triangles using algebra and the triangle angle sum theorem. This problem includes detailed step-by-step calculations for variables x and y, suitable for students in Grades 8-10.

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