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The problem in the image appears to be a geometry question involving a circle with points \( A \), \( B \), \( C \), and \( D \), where \( AB \) is the diameter of the circle. The angle measures are related by the ratio \( m(\triangle CDA) : m(\triangle DAB) = 2:1 \), and we need to determine the measure of angle \( \triangle CDA \).

### Information from the diagram:
1. \( AB \) is the diameter, so \( \triangle ACB \) is a right triangle because any triangle inscribed in a semicircle with the diameter as the hypotenuse forms a right angle.
2. Angles are expressed as:
   - \( \angle CDA = 2x + 40^\circ \)
   - \( \angle DAB = x + 20^\circ \)
3. The ratio \( m(\triangle CDA) : m(\triangle DAB) = 2:1 \).

### Solution steps:
1. Using the ratio of the angles \( \triangle CDA : \triangle DAB = 2:1 \), we know:
   \[
   \frac{2x + 40}{x + 20} = 2
   \]
2. Solve for \( x \):
   \[
   2x + 40 = 2(x + 20)
   \]
   Expanding the right-hand side:
   \[
   2x + 40 = 2x + 40
   \]
   Which is an identity, meaning the equation holds true for any \( x \). However, since we need a specific answer, we'll likely have more constraints or values that will emerge through geometric relationships or additional analysis.

Let's check these values based on the specific question choices. Would you like to explore further details? Here's a set of related questions:

1. What is the significance of the diameter in this geometry problem?
2. How do you calculate the angles formed in a triangle inscribed in a circle?
3. What properties of inscribed angles are used here?
4. Why does the ratio \( 2:1 \) between angles matter in this case?
5. Could there be any additional geometric theorems that help solve this?

Tip: Always verify if a triangle is inscribed in a semicircle, as the angle opposite the diameter will always be 90°.

En la figura, AB es diámetro de la circunferencia, m(∠CDA): m(∠DAB) = 2:1, entonces la medida de ∠CDA es?

Solve a geometry problem involving a circle where AB is the diameter. The angle ratio m(∠CDA): m(∠DAB) = 2:1 is given, and the task is to determine the measure of ∠CDA. The problem involves properties of inscribed angles and geometric ratios.

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