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Let's go through these problems one by one:

### Problem 11: 
#### Given:
- \( \triangle ABC \) is isosceles, and \( \triangle ADBE \) is equilateral. We need to find the angle measures in both triangles. 

For the isosceles triangle \( \triangle ABC \):
- The given angle measures involve expressions like \( (4x + 3)^\circ \) and \( (9x - 47)^\circ \).

For the equilateral triangle \( \triangle ADBE \):
- All angles are \( 60^\circ \), as it's equilateral.

#### Strategy:
1. Use the fact that the sum of angles in a triangle is \( 180^\circ \).
2. Set up equations for the angles given the conditions of the isosceles triangle.

### Problem 12:
#### Given:
- \( AC \cong CB \) and the angle measures are given by:
  - \( m \angle A = (3x + 18)^\circ \)
  - \( m \angle B = (7x - 58)^\circ \)
  - \( m \angle C = (2x - 8)^\circ \)
  
#### Strategy:
1. Since \( AC = CB \), \( \triangle ABC \) is isosceles, meaning two angles must be equal.
2. Use the sum of angles in a triangle \( \triangle ABC = 180^\circ \) to solve for \( x \) and calculate the measures of each angle.

### Problem 13:
#### Given:
- \( \triangle QRS \) with the following angle measures:
  - \( m \angle Q = (8x - 17)^\circ \)
  - \( m \angle R = (19x + 4)^\circ \)
  - \( m \angle S = (5x + 1)^\circ \)

#### Strategy:
1. Use the sum of angles in \( \triangle QRS \) equal to \( 180^\circ \) to solve for \( x \).
2. Once \( x \) is found, substitute to find each angle's measure.

### Problem 14:
#### Given:
- In \( \triangle KLM \), the sides are expressed as:
  - \( KL = 9x - 40 \)
  - \( LM = 7x - 37 \)
  - \( KM = 3x + 23 \)

#### Strategy:
1. Since \( \triangle KLM \) is isosceles, two sides must be equal.
2. Set up the equality between two sides to solve for \( x \), then substitute to find each side's measure.

### Problem 15:
#### Given:
- \( \triangle XYZ \) with the following side lengths:
  - \( XY = 13x - 21 \)
  - \( YZ = 8x - 6 \)
  - \( XZ = x + 4 \)

#### Strategy:
1. Similar to the previous problems, set up equations based on the relationships between the sides of \( \triangle XYZ \).
2. Solve for \( x \) and substitute to find each side's measure.

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Would you like to proceed with solving any specific problem, or should I start solving them one by one? Let me know if you need further details. 

Here are five related questions:
1. What are the key properties of isosceles triangles that help solve such problems?
2. How do we use the sum of angles in a triangle to find unknown angles?
3. Why do we assume equal

Math problems involving isosceles triangles, equilateral triangles, and angle/side measures in various triangles.

Learn how to solve angle and side length problems in isosceles and equilateral triangles. This lesson covers key geometry concepts and includes step-by-step solutions for determining angles and side measures in various triangles.

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