(ostion Watch Vide hoose ALL answers that describe the quadrilateral \( G H I J \) if \( \overline{G H} \| \overline{I J}, G H=I J \), and diagonals are perpendicular: \( \overline{G I} \perp \overline{H J} \). Answer Attempt out of 2 Parallelogram Rectangle Rhombus Square Trapezoid stbmansmer

To determine which quadrilaterals \( G H I J \) can be described based on the given conditions, let's analyze each property step-by-step. ### Given: 1. **Parallel Sides**: \( \overline{G H} \parallel \overline{I J} \) 2. **Equal Lengths**: \( G H = I J \) 3. **Perpendicular Diagonals**: \( \overline{G I} \perp \overline{H J} \) ### Options to Consider: 1. Parallelogram 2. Rectangle 3. Rhombus 4. Square 5. Trapezoid ### Analysis: 1. **Parallelogram**: - **Definition**: Both pairs of opposite sides are parallel. - **Given Conditions**: Only one pair (\( \overline{G H} \parallel \overline{I J} \)) is specified as parallel. - **Conclusion**: Not necessarily a parallelogram unless the other pair is also parallel. 2. **Rectangle**: - **Definition**: Opposite sides are equal and parallel, and all angles are right angles. - **Diagonals**: In a rectangle, diagonals are equal but **not** perpendicular. - **Given Conditions**: Diagonals are perpendicular. - **Conclusion**: **Cannot** be a rectangle. 3. **Rhombus**: - **Definition**: All sides are equal in length, and opposite sides are parallel. - **Diagonals**: Diagonals are **perpendicular**. - **Given Conditions**: Satisfies all conditions (since \( G H = I J \) and diagonals are perpendicular). - **Conclusion**: **Yes**, it can be a rhombus. 4. **Square**: - **Definition**: All sides are equal, opposite sides are parallel, and all angles are right angles. - **Diagonals**: Diagonals are both **equal** and **perpendicular**. - **Given Conditions**: Satisfies all conditions. - **Conclusion**: **Yes**, it can be a square. 5. **Trapezoid**: - **Definition**: At least one pair of opposite sides is parallel. - **Given Conditions**: Only one pair is specified as parallel, and diagonals are perpendicular. - **Diagonal Constraint**: Standard trapezoids do **not** have perpendicular diagonals unless they fit into more specific categories like an isosceles trapezoid with additional properties. - **Conclusion**: Generally, **no**, it isn't a trapezoid under given conditions. ### **Final Answer:** The quadrilateral \( G H I J \) can be described as both a **Rhombus** and a **Square** based on the given properties. **Correct Choices:** - Rhombus - Square