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AB=BC=CD and BA//CD. Is AC perpendicular to BD? Explain your answer.

To determine whether \( AC \) is perpendicular to \( BD \), let's analyze the given information step-by-step.

### Given:
1. \( AB = BC = CD \): This means segments \( AB \), \( BC \), and \( CD \) are of equal length.
2. \( BA \parallel CD \): Lines \( BA \) and \( CD \) are parallel.
3. \( \angle ABE = 40^\circ \)
4. \( \angle DCE = \gamma \)

### Analysis:

Since \( AB = BC = CD \) and \( BA \parallel CD \), quadrilateral \( ABCD \) is an **isosceles trapezoid**. Therefore:

- \( AB \parallel CD \), and \( AB = CD \).

The diagonals \( AC \) and \( BD \) of an isosceles trapezoid are **equal in length** and **perpendicular** to each other when it is a special case where the non-parallel sides are equal, forming an isosceles triangle within the trapezoid. In this figure:

- \( \triangle ABE \) and \( \triangle DCE \) are similar due to equal angles \( \angle ABE = \angle DCE = 40^\circ \).

### Conclusion:

Based on the properties of an isosceles trapezoid and given the equal sides \( AB = BC = CD \) with parallel sides \( AB \parallel CD \), **the diagonals \( AC \) and \( BD \) are perpendicular** to each other.

Thus, **Yes, \( AC \) is perpendicular to \( BD \).**

Would you like more details or have any questions? Here are some follow-up questions you might consider:

1. What are the properties of diagonals in an isosceles trapezoid?
2. How can we determine the length of \( AC \) and \( BD \)?
3. What happens to the angles formed when diagonals intersect in an isosceles trapezoid?
4. How does the similarity of triangles \( ABE \) and \( DCE \) affect other properties of this figure?
5. Can we determine any additional angle measures in this figure?

**Tip:** Understanding the properties of special quadrilaterals like isosceles trapezoids can help in solving problems involving perpendicular diagonals and equal angles.

AB = BC = CD and BA // CD. Is AC perpendicular to BD? Explain your answer.

Explore how to determine if diagonals AC and BD are perpendicular in an isosceles trapezoid, given equal side lengths and parallel sides. Learn about key geometry concepts, including properties of isosceles trapezoids and triangle similarity, suitable for students in Grades 8-10.

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