If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
Yes, a quadrilateral with four equal sides and one angle of 90° must be a square.
Geometric Reasoning:
Step 1: A quadrilateral with four equal sides is a rhombus.
Step 2: In a rhombus, adjacent angles are supplementary (sum to 180°).
Step 3: If one angle is 90°, the adjacent angle = 180° - 90° = 90°.
Step 4: Opposite angles in a rhombus are equal, so the angle opposite to 90° is also 90°.
Step 5: All four angles are 90°, and all sides are equal.
∴ The quadrilateral is a square.
Explanation
The chapter establishes that a rhombus has all sides equal, and in a rhombus, adjacent angles add up to 180° and opposite angles are equal. If one angle is 90°, then the adjacent angle is 180° - 90° = 90°. The opposite angle is also 90°. Thus all four angles are 90°. A quadrilateral with all sides equal and all angles 90° is a square. This can also be verified by construction.
Solution Steps
Step 1: A quadrilateral with four equal sides is a rhombus.
Step 2: In a rhombus, adjacent angles are supplementary (sum to 180°).
Step 3: If one angle is 90°, the adjacent angle is 180° - 90° = 90°.
Step 4: Opposite angles in a rhombus are equal, so the angle opposite to 90° is also 90°.
Step 5: All four angles are 90°, and all sides are equal. Hence, it is a square.