To prove that the diagonals of a parallelogram bisect each other, we can use the concept of congruent triangles. Let's go through the proof step by step.
Step 1: Identify the parallelogram and label its vertices.
Consider a parallelogram [tex]\( ABCD \)[/tex] with vertices labeled in a clockwise or counterclockwise direction.
Step 2: Draw the diagonals.
Draw diagonal [tex]\( AC \)[/tex] and diagonal [tex]\( BD \)[/tex], intersecting at point [tex]\( O \)[/tex].
Step 3: State what is given.
Since [tex]\( ABCD \)[/tex] is a parallelogram, by definition opposite sides are equal and parallel ([tex]\( AB \parallel CD \)[/tex] and [tex]\( AD \parallel BC \)[/tex]).
Step 4: Show triangles formed by diagonals are congruent.
Now, we look at triangles [tex]\( \triangle AOB \)[/tex] and [tex]\( \triangle COD \)[/tex] formed by the diagonals. Because [tex]\( AB \parallel CD \)[/tex], angle [tex]\( AOB \)[/tex] is congruent to angle [tex]\( COD \)[/tex] (alternate interior angles are congruent when lines are parallel). Similarly, angle [tex]\( OAB \)[/tex] is congruent to angle [tex]\( OCD \)[/tex] for the same reason.
Step 5: Use properties of parallelograms.
The sides of the parallelogram [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] are congruent (opposite sides of a parallelogram are equal).
Step 6: Apply Side-Angle-Side (SAS) congruence.
We have two sides and the angle between them congruent in triangles [tex]\( \triangle AOB \)[/tex] and [tex]\( \triangle COD \)[/tex], which means by the SAS Congruence Postulate, [tex]\( \triangle AOB \cong \triangle COD \)[/tex].
Step 7: Draw conclusion about diagonals.
Since the triangles are congruent, their corresponding parts are congruent as well. This means [tex]\( AO \cong CO \)[/tex] and [tex]\( BO \cong DO \)[/tex]. Thus, each diagonal bisects the other; that is, [tex]\( O \)[/tex] is the midpoint of both [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex].
In conclusion, using the concept of congruent triangles, we have proven that the diagonals of a parallelogram bisect each other. The correct answer is "congruent triangles" which corresponds to the choice indicated as 0.
