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 $270$.

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.