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The problem given involves two triangles, \( \triangle ABC \) and \( \triangle ADE \), with the following information:

1. \( AB = AC \) (isosceles triangle \( ABC \)),
2. \( AD = AE \) (isosceles triangle \( ADE \)),
3. \( \angle 2 = 30^\circ \), 
4. The goal is to find the measure of \( \angle 1 \) (marked on triangle ADE).

### Approach:

1. In the isosceles triangle \( ABC \), since \( AB = AC \), the base angles are equal. Therefore, \( \angle ABC = \angle ACB = \theta \).
   
2. The angle sum property of a triangle states that the sum of the interior angles is \( 180^\circ \). Thus, for \( \triangle ABC \):
   \[
   2\theta + \angle BAC = 180^\circ
   \]
   From the given information, we know that \( \angle 2 = 30^\circ \) is part of \( \angle BAC \). So, \( \angle BAC = 30^\circ \).

3. Now, let's calculate the base angles \( \theta \) of \( \triangle ABC \):
   \[
   2\theta + 30^\circ = 180^\circ \implies 2\theta = 150^\circ \implies \theta = 75^\circ
   \]

4. In the isosceles triangle \( ADE \), we know that \( AD = AE \), so the base angles are also equal: \( \angle ADE = \angle AED = \alpha \).

5. Using the exterior angle theorem, \( \angle 1 \) (which is an exterior angle for triangle \( ADE \)) is equal to the sum of the two non-adjacent interior angles \( \angle ABC \) and \( \angle 2 \). Thus:
   \[
   \angle 1 = \angle ABC + \angle 2 = 75^\circ + 30^\circ = 105^\circ
   \]

Therefore, \( \angle 1 = 105^\circ \).

Would you like more details on any of the steps? Here are some related questions:

1. How do you apply the angle sum property in an isosceles triangle?
2. What is the exterior angle theorem and how is it used here?
3. Can the solution change if triangle dimensions change, and why?
4. What is the relationship between base angles in isosceles triangles?
5. How do you deduce the measure of an angle from given triangle properties?

**Tip:** Always double-check the properties of isosceles triangles, as equal sides imply equal opposite angles, which simplifies many problems!

In the diagram, triangle ABC is isosceles with AB = AC; triangle ADE is also isosceles with AD = AE. If angle 2 = 30 degrees, find the value of angle 1.

Solve for angle 1 in an isosceles triangle problem involving two triangles, ABC and ADE. Using the exterior angle theorem and angle sum property, we calculate the value of angle 1 when angle 2 is 30 degrees.

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