https://file.aiquickdraw.com/mathbot/file/1729789867653ahjuiwpy.jpg

Let's break down and solve the problems based on the image you've uploaded. I'll start with the first one, and work through each in sequence. Here is the detailed explanation:

### Problem 11:
- **Given:**
  - \( \triangle ABC \) is isosceles.
  - \( \triangle ADBE \) is equilateral.

- **Steps:**
  - Since \( \triangle ADBE \) is equilateral, all angles in this triangle are \( 60^\circ \).
    - \( m\angle 2 = m\angle 3 = m\angle 5 = 60^\circ \).
  - \( \triangle ABC \) is isosceles, which means the base angles \( \angle 1 \) and \( \angle 6 \) are equal.
    - \( m\angle 1 = m\angle 6 \).
  - The sum of angles in \( \triangle ABC \) is \( 180^\circ \), so:
    \[
    m\angle 1 + m\angle 6 + m\angle 2 = 180^\circ
    \]
    \[
    m\angle 1 + m\angle 1 + 60^\circ = 180^\circ
    \]
    \[
    2m\angle 1 = 120^\circ
    \]
    \[
    m\angle 1 = 60^\circ
    \]
    Hence, \( m\angle 1 = m\angle 6 = 60^\circ \).

Thus, the measures are:
- \( m\angle 1 = 60^\circ \)
- \( m\angle 2 = 60^\circ \)
- \( m\angle 3 = 60^\circ \)
- \( m\angle 4 = 60^\circ \)
- \( m\angle 5 = 60^\circ \)

Now, do you want me to continue with problem 12 or provide further details on this? Also, here's a set of related questions to expand on the concepts:

### 5 Related Questions:
1. What defines an isosceles triangle, and how does it affect its angles?
2. Why are all angles in an equilateral triangle equal?
3. Can an isosceles triangle have angles different from \(60^\circ\)?
4. What is the general formula for finding the interior angles of a triangle?
5. How does the property of congruent angles help in solving triangles?

### Tip:
In any triangle, the sum of interior angles is always \(180^\circ\). This is a key rule in solving for unknown angles.

If △ABC is an isosceles triangle and △DBE is an equilateral triangle, find each angle measure. Other problems involve solving for angles and sides using given expressions for triangles △ABC, △QRS, △KLM, and △XYZ.

Find the angle measures in isosceles and equilateral triangles, including solving algebraic expressions for △ABC, △QRS, △KLM, and △XYZ. Learn key concepts like angle sum property and congruent angles.

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