In trapezoid ABCD,AD ∥BC and AB =BC =AD/2. Find ∠ACD.

We are given that trapezoid \( ABCD \) has \( AD \parallel BC \) (indicating that it is an isosceles trapezoid), and the side lengths satisfy \( AB = BC = \frac{AD}{2} \). We are tasked with finding \( \angle ACD \).

### Step-by-step solution:

1. **Label and define the trapezoid**:
   - \( AB \parallel CD \), with \( AB = BC = \frac{AD}{2} \).
   - Let \( AD = 2x \), meaning \( AB = BC = x \).

2. **Position the trapezoid in a coordinate system**:
   - Place \( D \) at \( (0, 0) \) and \( A \) at \( (2x, 0) \) on the horizontal axis.
   - Since \( AB = BC = x \), place \( B \) at \( (x, h) \), where \( h \) is the height from the base \( AD \) to \( B \) and \( C \).

3. **Use coordinate geometry**:
   Since \( AB = BC \) and the trapezoid is isosceles, point \( C \) must be vertically above point \( D \), so \( C = (0, h) \).

4. **Calculate \( \angle ACD \)**:
   To find \( \angle ACD \), use the slope formula for the lines \( AC \) and \( CD \), and apply the tangent formula to compute the angle.

   - Slope of \( AC \): 
     \[
     \text{slope of } AC = \frac{h - 0}{0 - 2x} = \frac{h}{-2x}
     \]
   - Slope of \( CD \):
     \[
     \text{slope of } CD = \frac{h - 0}{0 - 0} = 0
     \]
   
   Using the formula for the angle between two lines:
   \[
   \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|
   \]
   where \( m_1 \) is the slope of \( AC \) and \( m_2 \) is the slope of \( CD \), we have:
   \[
   \tan(\theta) = \left| \frac{\frac{-h}{2x} - 0}{1 + 0} \right| = \frac{h}{2x}
   \]

   From the given information \( AB = BC = \frac{AD}{2} \), we know the geometry forms an equilateral triangle, so \( \angle ACD = 30^\circ \).

Thus, the measure of \( \angle ACD \) is \( \boxed{30^\circ} \).

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Would you like more details, or do you have any questions?

Here are five related questions for practice:
1. If the height \( h \) of the trapezoid was given, how would it affect the calculations?
2. What is the sum of the interior angles of a trapezoid?
3. How would the angle change if \( AB \neq BC \)?
4. How do you find the area of an isosceles trapezoid?
5. How would the solution change if \( AB = AD \) instead of \( AB = BC \)?

**Tip**: Always consider the properties of parallel lines when working with trapezoids!

In trapezoid ABCD, AD ∥ BC and AB = BC = AD/2. Find ∠ACD.

Explore how to find the angle ∠ACD in trapezoid ABCD, where AB = BC = AD/2 and AD is parallel to BC. Using coordinate geometry and the tangent formula for angles, this solution is suitable for students in Grades 9-12 studying trapezoids and angles.

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