To solve for $x$ in this triangle, we use the property that the sum of the angles in a triangle equals $180^\circ$.

Step 1: Write the equation for the angle sum.

The angles are: $(4x + 4)^\circ, \; 65^\circ, \; 79^\circ.$ So: $(4x + 4) + 65 + 79 = 180.$

Step 2: Simplify the equation.

$4x + 4 + 144 = 180.$ $4x + 148 = 180.$

Step 3: Solve for $x$.

$4x = 180 - 148,$ $4x = 32,$ $x = \frac{32}{4} = 8.$

Final Answer:

$x = 8.$

Would you like a detailed explanation of any step or assistance with similar problems?

Related Questions:

1. What are the measures of all the angles in this triangle when $x = 8$?
2. How do you verify the angle sum property of a triangle?
3. Can this triangle be classified as acute, right, or obtuse? Why?
4. What happens to the triangle's angles if $x$ is changed to another value?
5. How is the angle sum property derived from geometric principles?

Tip:

Always double-check your solution by substituting the calculated value of $x$ back into the angle expressions.