0. Gabriel has a piece of paper shaped like a quadrilateral. When he folds it in half, it forms an equilateral triangle. What shape must the paper be?

To determine the shape of the quadrilateral that Gabriel has, let's analyze the folding process step by step. ### Understanding the Problem - **Initial Shape:** A quadrilateral (a four-sided polygon). - **Folding Action:** Folded in half along a line (the fold line). - **Resulting Shape:** An equilateral triangle (all sides and angles are equal). ### Analyzing the Folding When you fold a quadrilateral in half to form a triangle, the fold line acts as the **axis of symmetry**. For the resulting triangle to be equilateral: 1. **Equal Sides:** The two sides of the quadrilateral adjacent to the fold must be equal in length. 2. **Equal Angles:** The angles adjacent to the fold must allow the two halves to perfectly overlap and form 60° angles, which are characteristic of an equilateral triangle. ### Identifying the Quadrilateral The shape that satisfies these conditions is a **rhombus** with specific properties: - **All Sides Equal:** A rhombus has four sides of equal length. - **Angles of 60° and 120°:** To form an equilateral triangle upon folding, the rhombus must have angles of 60 degrees and 120 degrees. This ensures that when folded along one diagonal (acting as the fold line), the two 60° angles come together to form the three 60° angles of an equilateral triangle. ### Visualization Imagine a rhombus where each side is of length "s" and the angles are 60° and 120°. Folding this rhombus along the diagonal that connects the two 120° angles will bring the two 60° angles together, forming an equilateral triangle with sides of length "s." ### Conclusion **Therefore, the quadrilateral must be a rhombus with all sides equal and each pair of opposite angles measuring 60 degrees and 120 degrees.** ### Diagram for Clarity ``` Rhombus Before Folding: A ________ B \ / \ / \ / \ / \/ C After Folding Along Diagonal AC: A \ \ C / \ / \ /_____\ B D Forming an Equilateral Triangle: ΔACD ``` ### Final Answer **A rhombus with four equal sides and angles of 60° and 120°.**