Find the value of x.

Steps to Solve

1. Identify angle relationships in the triangle

The angle given is $(9x - 3)^\circ$. The other two angles in the triangle must also be defined. Since the triangle has two equal sides, it is an isosceles triangle, meaning the other two angles are equal.

2. Set up the equation based on the triangle's angle sum property

We know that the sum of the angles in a triangle must equal $180^\circ$. Therefore, if we let each of the two equal angles be represented as $y$, we can write the equation: $$ (9x - 3) + y + y = 180 $$

This can be simplified to: $$ (9x - 3) + 2y = 180 $$

3. Solve for $y$ in terms of $x$

Rearranging the equation gives us: $$ 2y = 180 - (9x - 3) $$

This simplifies to: $$ 2y = 183 - 9x $$ $$ y = \frac{183 - 9x}{2} $$

4. Use the property of isosceles triangles

Since the triangle is isosceles, we can equate the angle $(9x - 3)^\circ$ to the angle $y$: $$ 9x - 3 = \frac{183 - 9x}{2} $$

5. Eliminate the fraction

To eliminate the fraction, multiply both sides by 2: $$ 2(9x - 3) = 183 - 9x $$

Distributing gives us: $$ 18x - 6 = 183 - 9x $$

6. Combine like terms

Adding $9x$ to both sides and $6$ to both sides gives: $$ 27x = 189 $$

7. Solve for $x$

Dividing both sides by 27: $$ x = \frac{189}{27} = 7 $$