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

We are to find the measure of all marked angles \( m\angle 1 \) through \( m\angle 9 \).

#### Steps for solving:
1. **Equilateral Triangle \( \triangle DBE \)**:
   Since it's equilateral, each angle is \( 60^\circ \). Therefore:
   - \( m\angle 2 = 60^\circ \)
   - \( m\angle 3 = 60^\circ \)
   - \( m\angle 5 = 60^\circ \)

2. **Isosceles Triangle \( \triangle ABC \)**:
   The angles at the base of an isosceles triangle are equal.
   - \( m\angle 1 = m\angle 6 \)

   The angles for \( \angle 4 \), \( \angle 7 \), \( \angle 8 \), and \( \angle 9 \) are based on the provided expressions:
   - \( m\angle 4 = (4x + 3)^\circ \)
   - \( m\angle 7 = (9x - 47)^\circ \)
   - \( m\angle 8 = 68^\circ \)
   - \( m\angle 9 = (7x + 1)^\circ \)

At this point, without knowing \( x \), the full solution for \( \triangle ABC \) remains open. You may need further steps to calculate \( x \).

### Problem 12:
In this problem, we have a triangle with the following angle measures:
- \( m\angle A = (3x + 18)^\circ \)
- \( m\angle B = (7x - 58)^\circ \)
- \( m\angle C = (2x - 8)^\circ \)

Since the sum of the angles in any triangle is \( 180^\circ \), we set up the equation:
\[
(3x + 18) + (7x - 58) + (2x - 8) = 180
\]

Simplifying:
\[
3x + 7x + 2x + 18 - 58 - 8 = 180
\]
\[
12x - 48 = 180
\]
\[
12x = 228
\]
\[
x = 19
\]

Now, substitute \( x = 19 \) into the expressions for the angles:
- \( m\angle A = 3(19) + 18 = 57 + 18 = 75^\circ \)
- \( m\angle B = 7(19) - 58 = 133 - 58 = 75^\circ \)
- \( m\angle C = 2(19) - 8 = 38 - 8 = 30^\circ \)

Thus, \( m\angle A = 75^\circ \), \( m\angle B = 75^\circ \), and \( m\angle C = 30^\circ \).

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How do we solve for the variable \( x \) in isosceles triangles?
2. What is the significance of base angles in an isosceles triangle?
3. How are angle measures related in an equilateral triangle?
4. Can the measures of angles provide information about the type of triangle?
5. How would the problem change if one of the triangle's sides was given?

**Tip:** Always remember that the sum of the angles in a triangle is \( 180^\circ \). This helps in solving for unknown angles.

Find the values of x and angle measures in triangles based on the given expressions.

Learn how to solve triangle problems involving isosceles and equilateral triangles with algebraic expressions for angle measures. This problem covers key geometric and algebraic concepts, including the sum of angles in a triangle and properties of isosceles and equilateral triangles.

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