In ΔBCD,  start overline, B, D, end overline 
BD
  is extended through point D to point E,  m, angle, B, C, D, equals, left parenthesis, 3, x, plus, 11, right parenthesis, degreesm∠BCD=(3x+11) 
∘
 ,  m, angle, D, B, C, equals, left parenthesis, 2, x, plus, 16, right parenthesis, degreesm∠DBC=(2x+16) 
∘
 ,   and m, angle, C, D, E, equals, left parenthesis, 8, x, minus, 15, right parenthesis, degreesm∠CDE=(8x−15) 
∘
 .   What is the value of x, question markx?

We are given a triangle \( \Delta BCD \) where side \( BD \) is extended to point \( E \). The measures of the angles are expressed as algebraic expressions:

- \( m \angle BCD = (3x + 11)^\circ \)
- \( m \angle DBC = (2x + 16)^\circ \)
- \( m \angle CDE = (8x - 15)^\circ \)

### Step 1: Use the exterior angle theorem
The exterior angle theorem states that the exterior angle (here \( \angle CDE \)) is equal to the sum of the two non-adjacent interior angles (here \( \angle BCD \) and \( \angle DBC \)).

Therefore, we have the equation:
\[
m \angle CDE = m \angle BCD + m \angle DBC
\]

Substitute the given expressions:
\[
(8x - 15) = (3x + 11) + (2x + 16)
\]

### Step 2: Solve the equation
First, simplify the right-hand side:
\[
(3x + 11) + (2x + 16) = 5x + 27
\]

Now, set the two sides equal to each other:
\[
8x - 15 = 5x + 27
\]

Subtract \( 5x \) from both sides:
\[
3x - 15 = 27
\]

Add 15 to both sides:
\[
3x = 42
\]

Finally, divide by 3:
\[
x = 14
\]

### Conclusion
The value of \( x \) is \( \boxed{14} \).

Do you want further details or have any questions? Here are 5 related questions to expand your understanding:

1. How would the angle measures change if \( x \) were doubled?
2. Can the exterior angle theorem be applied to all types of triangles?
3. What are other methods to solve for \( x \) in this scenario?
4. How does extending a side of a triangle affect the angle properties?
5. What is the sum of the interior angles of any triangle?

**Tip**: The sum of the interior angles of any triangle is always \( 180^\circ \).

In ΔBCD, BD is extended through point D to point E, m∠BCD=(3x+11)∘, m∠DBC=(2x+16)∘, and m∠CDE=(8x−15)∘. What is the value of x?

In this geometry problem, learn how to solve for x using the Exterior Angle Theorem. The problem involves angles in ΔBCD where BD is extended to point E, and the angle measures are given as algebraic expressions. Detailed step-by-step explanation provided. Suitable for Grades 8-10.

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