Congruence Challenge
Rationale
This task challenges students to apply what they know about parallel and intersecting lines, parallelograms, and the angles and segments related to these figures to solve a geometric problem in a purely mathematical context. It requires students to think about how to approach the problem and to justify their work.
Instructional Task
In figure ABCD, AB||CD and AD||BC. Point R is in the same plane as ABCD. (Point R can be placed anywhere in the plane.)
Draw a straight line that passes through point R and divides ABCD into two congruent parts. Justify your reasoning that the two parts are congruent.
Discussion/Further Questions/Extensions
Depending on the teacher’s goals, this task can be adapted to require any level of reasoning from an informal justification to a formal proof.
Some students may have difficulty realizing that they must connect point R to the point at the intersection of the parallelogram’s diagonals. While it may be tempting for the teacher to guide students toward connecting point R to this point of intersection, students should first have adequate opportunity to constructively struggle with the task as presented, perhaps working with partners or in small groups.
If additional scaffolding becomes necessary, it might be helpful for students to list all the attributes of parallelograms and the angles formed by parallel and intersecting lines that they know. If necessary, encourage students to place point R at different locations and sketch where they think the line might bisect the parallelogram.
As an extension, ask students to determine whether the method they used to prove that both halves of the bisected parallelogram are congruent would work if point R were located anywhere on the same plane as ABCD—inside, outside, or on the parallelogram itself.
Sample Solution
Students might approach this problem using properties of parallelograms, parallel lines, intersecting lines, and the angles and segments related to these figures. Other solution methods are possible, including the use of transformations, if students can provide adequate justification.
This sample solution uses the following information:
- The diagonals of a parallelogram bisect each other.
- Alternate interior angles formed by two parallel lines and a transversal are congruent.
- Vertical angles formed by two intersecting lines are congruent.
- Opposite sides of a parallelogram are congruent.
- Two triangles can be proven to be congruent using a variety of triangle congruence theorems or postulates (e.g., side-angle-side (SAS), side-side-side (SSS), angle-angle-side (AAS), angle-side-angle(ASA)).
- Draw the two diagonals of the parallelogram.
- Label the intersection of the two diagonals, O.
- The line through R and O intersects the line AB at N and the line CD at M.
- The triangles AON and COM are congruent (by ASA), and triangles MOD and NOB are congruent (also by ASA). Triangles AOD and COB are congruent (by SSS or SAS). Quadrilateral ANMD is composed of triangles AON, MOD, and AOD, and quadrilateral BNMC is composed of triangles COM, NOB, and COB. Therefore, quadrilateral ANMD is congruent to quadrilateral BNMC.
