https://file.aiquickdraw.com/mathbot/file/1727091250182idh7scuo.jpg

The image shows a right-angled triangle, and I can see the following key information:

- Angle \( \angle ABC = 20^\circ \)
- Angle \( \angle BDC = 90^\circ \)
- Angle \( \angle DBC = 65^\circ \)

We are required to find the value of \( x \), which seems to represent the length of one side of the triangle, though I will clarify by working through the triangle and relations.

Let's summarize the angles first:

1. In triangle \( BDC \), the sum of the angles must be \( 180^\circ \) as it's a triangle:
   \[
   \angle DBC + \angle BDC + \angle BCD = 180^\circ
   \]
   Substituting known angles:
   \[
   65^\circ + 90^\circ + \angle BCD = 180^\circ
   \]
   Solving for \( \angle BCD \):
   \[
   \angle BCD = 180^\circ - 155^\circ = 25^\circ
   \]

2. Using these angle relationships and the geometry of the triangle, we can further explore the sides based on trigonometry. However, more information about the side lengths is necessary to calculate \( x \).

Please confirm whether \( x \) is the length of one of the sides (for example, \( BD \) or \( BC \)) or provide any other clarifications. 

Do you want a step-by-step solution once the side \( x \) is confirmed?

Here are five related questions to consider:

1. Is \( x \) the length of side \( BD \) or another side in the diagram?
2. Would you like to calculate any other side of the triangle, such as \( BC \) or \( AB \)?
3. Should we use trigonometric functions (sine, cosine, etc.) to solve for \( x \)?
4. Are there additional lengths given for sides not visible in this portion of the problem?
5. Would you like to explore any other properties of the triangles formed?

**Tip:** Always check for basic geometric relationships such as the sum of angles in a triangle before applying trigonometry!

Find x in the triangle shown, with angle measurements given as \(\angle ABC = 20^\circ\), \(\angle BDC = 90^\circ\), and \(\angle DBC = 65^\circ\).

Learn how to solve for the side length x in a right-angled triangle, using the angles \(\angle ABC = 20^\circ\), \(\angle BDC = 90^\circ\), and \(\angle DBC = 65^\circ\). This detailed step-by-step guide covers trigonometric concepts and key formulas for solving triangle problems. Suitable for Grades 9-11 students.

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