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In this problem, two circles with centers \( O_1 \) and \( O_2 \) are tangent to each other at point \( T \), and the line segments \( [O_1A] \) and \( [O_2B] \) are parallel. It is given that:

- \( m(\angle O_1 A T) = 30^\circ \),
- \( O_1A \parallel O_2B \),
- \( m(\angle T O_2 B) = \alpha \).

We are asked to find the value of \( \alpha \), the measure of the angle \( T O_2 B \).

### Steps to solve:

1. Since the circles are tangent to each other at \( T \), \( O_1 T \) and \( O_2 T \) form a straight line, implying that \( O_1 T O_2 \) is a straight line. Therefore, the angle sum around \( T \) on the line is \( 180^\circ \).

2. The parallel lines \( O_1 A \parallel O_2 B \) imply that the corresponding angles formed by the tangents are equal. Therefore, the angle at \( O_2 \), i.e., \( \angle T O_2 B = \alpha \), will be equal to \( 180^\circ - 2 \times 30^\circ \) (since \( O_1 A \) forms an angle of \( 30^\circ \) with the tangent, the sum of the angles at point \( T \) must reflect this).

3. Calculate \( \alpha \):
   \[
   \alpha = 180^\circ - 2 \times 30^\circ = 120^\circ.
   \]

Thus, \( \alpha = 120^\circ \).

The correct answer is **E) 120**.

Would you like more details or further clarifications?

Here are 5 related questions to practice:

1. What is the sum of the interior angles of a triangle?
2. How do parallel lines affect the angles formed by a transversal?
3. What is the relationship between a tangent line and a radius of a circle?
4. How do you calculate the exterior angle of a polygon?
5. How do you find the central angle in a circle given two points on the circumference?

**Tip**: Always check for symmetry or parallel lines in geometry problems to find congruent angles easily!

O₁ ve O₂ merkezli çemberler birbirine T noktasında teğet, [O₁A] // [O₂B], m(O₁ÂT) = 30°. Buna göre, m(TÔ₂B) = α kaç derecedir?

Solve the geometry problem involving two tangent circles and parallel lines. Given that m(O₁ÂT) = 30°, find the angle m(TÔ₂B). Understand key concepts such as tangent and radius, properties of parallel lines, and angle sums. Suitable for students in Grades 8-10.

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