Sum and Difference of Angles: System of equations

To approach the problem, follow these steps:

• Step 1: Establish and simplify the given equation.

• Step 2: Use properties of triangle angles to form additional equations.

• Step 3: Solve the equations to find $\angle A$.

Step 1: We're given $\angle C + \angle A = 2(\angle A + \angle B)$.

Substitute $\angle B = \angle A$:

$\angle C + \angle A = 2(\angle A + \angle A) = 4\angle A$.

Step 2: Use the triangle angle sum property:

$\angle A + \angle B + \angle C = 180^\circ$.

Since $\angle B = \angle A$, we have:

$\angle A + \angle A + \angle C = 180^\circ$, simplifying to $2\angle A + \angle C = 180^\circ$.

Step 3: Solve the System:

• From $2\angle A + \angle C = 180^\circ$, express $\angle C$ as:

• $\angle C = 180^\circ - 2\angle A$.

• Substitute into the equation $\angle C + \angle A = 4\angle A$:

• $(180^\circ - 2\angle A) + \angle A = 4\angle A$.

• Simplify: $180^\circ - \angle A = 4\angle A$.

• Add $\angle A$ to both sides: $180^\circ = 5\angle A$.

• Solving for $\angle A$, we get: $\angle A = \frac{180^\circ}{5}$.

• Thus, $\angle A = 36^\circ$.