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.