In an isosceles trapezium ______.

Options

A. Diagonals are congruent
B. Opposite sides are parallel
C. Diagonals are bisectors of each other
D. Diagonals bisect opposite angles

Answer

To solve the question regarding the properties of an isosceles trapezium, we will analyze the statements provided and prove which one is correct.

**Step-by-Step Solution:**

1. **Identify the Properties of an Isosceles Trapezium:**
- In an isosceles trapezium, one pair of opposite sides is parallel (let's say AD || BC) and the other pair of sides (AB and CD) are equal in length.

2. **State the Given Information:**
- We are given that the trapezium is isosceles, which means that the non-parallel sides (AB and CD) are equal in length.

3. **Use the Properties of Angles:**
- The base angles of an isosceles trapezium are equal. Therefore, ∠A = ∠B and ∠C = ∠D.

4. **Analyze the Diagonals:**
- The diagonals of the trapezium are AC and BD. To show that they are congruent, we will use the congruence of triangles.

5. **Form Triangles:**
- Consider triangles ABC and DBC. We will show that these two triangles are congruent.
- We have:
- AB = CD (since it is an isosceles trapezium)
- ∠A = ∠B (base angles are equal)
- BC = BC (common side)

6. **Apply the Side-Angle-Side (SAS) Congruence Criterion:**
- Since we have two sides and the included angle of triangle ABC equal to the two sides and included angle of triangle DBC, we can say that:
- Triangle ABC ≅ Triangle DBC (by SAS).

7. **Conclude About the Diagonals:**
- By the property of congruent triangles, the corresponding parts of congruent triangles are equal. Therefore, we have:
- AC = BD.

8. **Final Conclusion:**
- Thus, in an isosceles trapezium, the diagonals are congruent.

**Final Answer:**
In an isosceles trapezium, the diagonals are congruent.

---