Rhombus is a quadrilaterala)in which diagonals are at right angleb)in ...
This is because in a rhombus, the diagonals always bisect each other at right angles (90 degrees). While the diagonals do bisect each other (option C) and also bisect the opposite angles (option D), the defining characteristic in the context of a rhombus's diagonal properties is that they are perpendicular.
So, if the question specifically asks about the nature of the diagonals' intersection, option A is the most accurate. Option D, "in which diagonals bisect opposite angles," is also true but is not specifically about the angle at which they bisect each other.
Rhombus is a quadrilaterala)in which diagonals are at right angleb)in ...
A rhombus is a quadrilateral with certain unique properties. The correct answer is option 'D', which states that a rhombus is a quadrilateral in which the diagonals bisect opposite angles. Let's understand why this is the correct answer.
Diagonals of a Rhombus:
- A rhombus has two pairs of opposite sides that are equal in length.
- The diagonals of a rhombus are the line segments connecting opposite vertices.
Bisecting Opposite Angles:
- When we say that the diagonals of a rhombus bisect opposite angles, it means that the diagonals divide the rhombus into four congruent triangles.
- Each diagonal cuts the rhombus into two equal parts, and these parts are congruent.
- As a result, the opposite angles formed by the diagonals are equal in measure.
Proof of Diagonal Bisecting Opposite Angles:
To prove that the diagonals of a rhombus bisect opposite angles, we can use the concept of congruent triangles.
Let's consider a rhombus ABCD with diagonals AC and BD.
Proof:
1. In rhombus ABCD, opposite sides are equal. Therefore, AB = BC = CD = DA.
2. Diagonals of a rhombus bisect each other. Therefore, AC and BD intersect at point O such that AO = OC and BO = OD.
3. Now, consider triangle ABO and triangle BCO.
- AB = BC (opposite sides of the rhombus)
- AO = OC (diagonals bisect each other)
- BO = BO (common side)
By side-side-side congruence, we can conclude that triangle ABO is congruent to triangle BCO.
4. Congruent triangles have equal corresponding angles. Therefore, angle AOB = angle BOC.
5. Similarly, we can prove that angle COD = angle DOA.
6. Hence, the diagonals of a rhombus bisect opposite angles.
Conclusion:
Based on the properties of a rhombus and the proof provided, it is clear that option 'D' is the correct answer. A rhombus is a quadrilateral in which the diagonals bisect opposite angles.
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