Question: COMPOSITE SHAPE A geometry teacher asks her students to rotate and reposition one of the two triangles shown such that the new shape will be a polygon. Angles $1$ and $4$ are congruent, and angles $2$ and $3$ are congruent. What shape will be obtained? Support your conclusion. A B C D E $1$ $2$ $3$ $4$ The diagram shows two right triangles: triangle $ABC$ on the left and triangle $DCE$ on the right, with $A$ at top-left, $B$ at bottom-left, $C$ at the shared bottom-center point, $D$ at top-center, and $E$ at bottom-right; $BC$ and $CE$ form a straight horizontal line, $AB$ and $DC$ are vertical, $AC$ and $DE$ are slanted, and matching tick marks on $AC$ and $DE$ indicate equal lengths. Select Choice : Rotate triangle $DCE$ Select Choice and reposition it so that point $E$ coincides with point $A$ and point $D$ coincides with point Select Choice. The rotation forms congruent Select Choice angles, which shows that opposite sides of the polygon are Select Choice, so the figure is a Select Choice.

1. **State the problem:** We have two right triangles $ABC$ and $DCE$ sharing a base line $BC$ and $CE$. We want to rotate and reposition triangle $DCE$ so that it forms a polygon with triangle $ABC$. 2. **Given:** Angles $1$ and $4$ are congruent, and angles $2$ and $3$ are congruent. Also, $AC$ and $DE$ are equal in length. 3. **Step 1: Rotate triangle $DCE$ so that point $E$ coincides with point $A$.** 4. **Step 2: Reposition triangle $DCE$ so that point $D$ coincides with point $B$.** 5. **Step 3: Analyze the angles formed:** - Since angles $1$ and $4$ are congruent, and angles $2$ and $3$ are congruent, the adjacent angles at the joined points are equal. 6. **Step 4: Check side lengths:** - $AC$ and $DE$ are equal. - $AB$ and $DC$ are vertical and equal in length. 7. **Step 5: Conclusion:** - The two triangles fit together perfectly to form a quadrilateral with opposite sides equal and parallel. - This quadrilateral has four right angles (since original triangles are right triangles). 8. **Final answer:** The shape obtained is a rectangle. **Summary:** By rotating triangle $DCE$ so that $E$ coincides with $A$ and $D$ coincides with $B$, the two triangles combine to form a rectangle because opposite sides are equal and angles are right angles.