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.
rotate triangle dce 90 and reposition it so that point e coincides with point a and point d coincides with point
the rotation forms congruent corresponding angles, which shows that opposite sides of the polygon are parallel, so the figure is a rectangle

Explanation:

Step1: Analyze Triangle Congruence

Given angles 1 ≅ 4 and 2 ≅ 3, and the rotation (which preserves congruence) of one triangle (e.g., DCE) by 90° (since ∠B and the right angle from rotation imply a 90° rotation) to align with triangle ABC.

Step2: Check Quadrilateral Angles/Sides

After rotation, corresponding angles are congruent. The right angles (∠B and the rotated right angle) and congruent angles imply adjacent angles sum to 180° (e.g., ∠1 + ∠2 = ∠4 + ∠3, and with right angles, consecutive angles are supplementary), so opposite sides are parallel. Also, congruent sides from rotation mean opposite sides are equal.

Step3: Identify the Shape

A quadrilateral with opposite sides parallel and equal is a rectangle. The rotation and repositioning (aligning E with A, D with B) forms a rectangle, as the angles (right angles from the triangles and supplementary consecutive angles) and sides (congruent from rotation) satisfy rectangle properties.

Answer:

The shape obtained is a rectangle. The rotation (90°) of one triangle to align with the other creates a quadrilateral with opposite sides parallel (due to congruent angles making consecutive angles supplementary) and equal (from rotational congruence), satisfying the rectangle definition.