To solve the given problem, let’s analyze the situation step by step:

Problem Breakdown:

1. Given:

   • A right angle at point $G$, meaning $\angle BGD = 90^\circ$.
   • $\angle BGC = 53^\circ$.
   • $\angle CGD = (3x + 4)^\circ$.

2. Angles in a triangle: The sum of the angles around $G$ (point where the angles meet) is $90^\circ$. This is because $\angle BGD$ is a right angle.

   Therefore: $\angle BGC + \angle CGD = 90^\circ$

3. Substitute the given values: $53^\circ + (3x + 4)^\circ = 90^\circ$

4. Simplify the equation: Combine like terms: $53 + 3x + 4 = 90$ $3x + 57 = 90$

5. Solve for $x$: Subtract $57$ from both sides: $3x = 33$ Divide by $3$: $x = 11$

Final Answer:

$x = 11$

Let me know if you’d like a detailed explanation for each step or additional clarifications!

Expanding Understanding:

1. How do you identify the relationship between angles at a point?
2. What other properties of triangles can help in solving such problems?
3. Can you calculate $\angle CGD$ using the found $x$?
4. What are other common problems involving angles around a point?
5. How do supplementary angles differ from complementary angles?

Tip: Always verify your solution by substituting the value of $x$ back into the equation to ensure the angles sum correctly!