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.
image of two right triangles, abc with right angle at b, and dce with some markings, and dropdown select choice boxes for rotating triangle dce, repositioning, congruent angles, opposite sides, etc.
need help? questions 11

Explanation:

Step1: Analyze Triangle Congruence

Given $\angle 1\cong\angle 4$ and $\angle 2\cong\angle 3$. Let's consider triangles $ABC$ and $DCE$. When we rotate triangle $DCE$ (assuming the triangle is $DCE$ as per the diagram), we want to align $E$ with $A$ and $D$ with a point (probably $B$ or another vertex). Since angles are congruent, the triangles are similar (by AA similarity) and if sides are congruent (from the tick marks, indicating congruent sides), they are congruent.

Step2: Determine the Composite Shape

After rotating triangle $DCE$ (or the given triangle with angle 4) so that $E$ coincides with $A$ and $D$ coincides with the appropriate vertex, the two right triangles (since $\angle B$ is a right angle) will form a quadrilateral. The congruent angles and sides will make opposite sides parallel and equal (since the triangles are congruent right triangles). A quadrilateral with opposite sides parallel and equal is a parallelogram. Also, since one angle is a right angle (from $\angle B$), it will be a rectangle (or a right parallelogram). But more precisely, with the given congruences, when we combine the two triangles, the resulting shape has opposite sides congruent (from triangle congruence) and angles that make it a parallelogram, and with right angles, a rectangle. However, the key is that the rotation and repositioning, using the congruent angles and sides, forms a parallelogram (or rectangle) where opposite sides are congruent (from the triangle congruence, as corresponding sides of congruent triangles are congruent, so when combined, opposite sides of the quadrilateral are congruent).

Step3: Conclusion on the Shape

The rotation aligns the triangles such that their corresponding sides and angles form a quadrilateral with opposite sides congruent (from triangle congruence: $AB\cong DC$ and $AC\cong DE$ or similar, depending on the triangle labels) and opposite angles congruent, hence a parallelogram (or rectangle). The main idea is that the congruent angles (AA) and congruent sides (from the tick marks) make the triangles congruent, so combining them via rotation gives a quadrilateral with opposite sides congruent, so it's a parallelogram (or rectangle, since there's a right angle).

Answer:

The resulting shape is a rectangle (or parallelogram). When we rotate one triangle (e.g., $\triangle DCE$) to align $E$ with $A$ and $D$ with the appropriate vertex, the congruent triangles (by AA and side congruence from tick marks) combine to form a quadrilateral with opposite sides congruent (from triangle congruence) and right angles (from $\angle B$), so it is a rectangle (a type of parallelogram) with opposite sides congruent and all angles right angles.