Question

which is a valid conclusion from these statements? if a quadrilateral is a rhombus, then it is a parallelogram. if a quadrilateral is a parallelogram, then its opposite angles are congruent. opposite angles of a rhombus are congruent. opposite angles of a quadrilateral are congruent. every parallelogram is a rhombus.

Explanation:

Step1: Recall properties of parallelograms and rhombuses

One of the properties of a parallelogram is that its opposite angles are congruent. A rhombus is a special - type of parallelogram. So, if a quadrilateral is a rhombus, since a rhombus is a parallelogram, its opposite angles are congruent.

Step2: Analyze each option

• "Opposite angles of a rhombus are congruent": This is a property of a rhombus, but it is not the most general conclusion based on the given statements.
• "Opposite angles of a quadrilateral are congruent": This is not true for all quadrilaterals.
• "Every parallelogram is a rhombus": This is false. A rhombus is a special parallelogram, not all parallelograms are rhombuses.
• "If a quadrilateral is a rhombus, then it is a parallelogram and its opposite angles are congruent": This is the correct conclusion as a rhombus is a parallelogram and parallelograms have congruent opposite angles.

Answer:

If a quadrilateral is a rhombus, then it is a parallelogram and its opposite angles are congruent.