Question

find the value of x.

Explanation:

Step1: Use the exterior - angle theorem for triangles

The exterior - angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Here, we have a triangle with an exterior angle \((x - 24)^{\circ}\) and two non - adjacent interior angles \((3x + 9)^{\circ}\) (since the two sides of the triangle are equal, the base angles are equal).
So, \(x-24=(3x + 9)+(3x + 9)\).

Step2: Simplify the equation

First, expand the right - hand side: \(x-24 = 3x+9+3x + 9\).
Combine like terms on the right - hand side: \(x-24=6x + 18\).

Step3: Solve for \(x\)

Subtract \(x\) from both sides: \(-24=6x + 18-x\).
Simplify: \(-24 = 5x+18\).
Subtract 18 from both sides: \(-24-18=5x\).
\(-42 = 5x\).
Divide both sides by 5: \(x=\frac{-42}{5}=- 8.4\).

Wait, there is a mistake. Let's re - check.

We should use the property of the isosceles triangle and the linear pair.

The angle adjacent to \((x - 24)^{\circ}\) is \(180-(x - 24)=(204 - x)^{\circ}\).

Since the triangle is isosceles (two equal sides), the two base angles are equal. So, \(3x + 9=204 - x\).

Step1: Set up the equation

\(3x+x=204 - 9\).

Step2: Combine like terms

\(4x=195\).

Step3: Solve for \(x\)

\(x=\frac{195}{4}=48.75\).

Another way:

Since the two sides of the triangle are equal, the base angles are equal.

The sum of an interior angle \((3x + 9)^{\circ}\) and its adjacent exterior angle \((x - 24)^{\circ}\) forms a linear pair (sum to \(180^{\circ}\)).

Step1: Set up the equation

\((3x + 9)+(x - 24)=180\).

Step2: Combine like terms

\(3x+x+9 - 24=180\).
\(4x-15 = 180\).

Step3: Solve for \(x\)

Add 15 to both sides: \(4x=180 + 15\).
\(4x=195\).
Divide both sides by 4: \(x=\frac{195}{4}=48.75\).

Answer:

\(x = 48.75\)