This problem involves solving for $x$ using properties of angles and triangles. Here's the solution:

Step 1: Identify the relationships

From the diagram:

1. The two triangles form vertically opposite angles, meaning the angles at the intersection are equal.
2. The angles along a straight line sum to $180^\circ$.

The smaller triangle has angles:

• $x - 9$ (top-left),
• $5$ (top-right),
• The straight-line continuation is part of the larger triangle.

The larger triangle's angles are:

• $x$ (bottom-right),
• $x + 3$ (bottom-left).

Step 2: Use the triangle angle sum property

In any triangle, the sum of the angles is $180^\circ$.

Small triangle:

$(x - 9) + 5 + 90 = 180 \quad \text{(right angle at the base of small triangle)}$ Simplify: $x - 9 + 5 + 90 = 180$ $x + 86 = 180$ $x = 94$

Step 3: Verify for the large triangle

For the larger triangle, the angles are $x$, $x + 3$, and $90$. Substitute $x = 94$ to check: $94 + (94 + 3) + 90 = 180$ Simplify: $94 + 97 + 90 = 180$ $180 = 180$

This verifies the solution.

Final Answer:

$x = 94$

Would you like more details or a breakdown of the process? Here are five related questions to expand on this:

1. How do vertically opposite angles help in geometry problems like this?
2. Can the angle sum property be applied to polygons other than triangles?
3. How do you solve problems when two variables are involved instead of one?
4. What are some examples of real-world problems where this concept is applied?
5. How would the problem change if one angle was given as a function of $y$ instead?

Tip: Always confirm whether the given diagram assumes certain angle properties like perpendicularity or symmetry!