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We are given an isosceles triangle \( ABC \) with the following angle relationships:

- \(\angle A = x^\circ\)
- \(\angle B = y^\circ\)
- \(\angle C = x - 30^\circ\)

Additionally, we know that in an isosceles triangle, two angles must be equal. Here, we can assume \(\angle B = \angle C\), as indicated by the isosceles triangle property.

### Step 1: Set up the equation for \(\angle B = \angle C\)
Since \( \angle B = y^\circ \) and \( \angle C = x - 30^\circ \), we have:
\[
y = x - 30
\]
This is our first equation.

### Step 2: Use the angle sum property of a triangle
The sum of the interior angles of a triangle is always \(180^\circ\). Thus:
\[
x + y + (x - 30) = 180
\]

### Step 3: Substitute \( y = x - 30 \) from equation 1 into the second equation
\[
x + (x - 30) + (x - 30) = 180
\]

Simplifying the equation:
\[
3x - 60 = 180
\]
\[
3x = 240
\]
\[
x = 80
\]

### Step 4: Find \( y \)
Substitute \( x = 80 \) into the equation \( y = x - 30 \):
\[
y = 80 - 30 = 50
\]

So, \( x = 80^\circ \) and \( y = 50^\circ \).

### Step 5: Calculate the perimeter
To find the perimeter, we need the lengths of the sides of the triangle. The side lengths are given as \( 2x - y \) for \( AB \) and \( AC \), and \( BC = x - 30 \).

Substitute \( x = 80 \) and \( y = 50 \):

- \( AB = AC = 2x - y = 2(80) - 50 = 160 - 50 = 110 \)
- \( BC = x - 30 = 80 - 30 = 50 \)

The perimeter is the sum of all the side lengths:
\[
\text{Perimeter} = AB + AC + BC = 110 + 110 + 50 = 270
\]

### Final Answers:
a. The value of \( x \) is \( 80^\circ \) and \( y \) is \( 50^\circ \).  
b. The perimeter of the triangle is \( 270 \).

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Would you like further details or explanations on any part of the solution?

Here are some related questions for practice:
1. What are the properties of an isosceles triangle?
2. How does the angle sum property of a triangle help in solving for unknown angles?
3. What other methods can be used to find the perimeter of a triangle?
4. How can the Pythagorean theorem be applied in isosceles triangles if they are right triangles?
5. Can a triangle be both isosceles and equilateral? Explain.

**Tip:** Always remember to verify that the angles add up to \(180^\circ\) when solving triangle problems.

The picture above shows an isosceles triangle. Determine: a. The value of x and y. b. The perimeter.

Learn how to solve for the unknown angles and perimeter of an isosceles triangle using algebra and the triangle angle sum property. This step-by-step solution covers key concepts like the Isosceles Triangle Theorem and is suitable for Grades 8-10.

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