Please Help, will brainliest."How do you connect the ideas of congruency and rigid motion and how do you prove congruency?"
Answer:

"Provide an example of a new theorem related to triangles and describe the steps as to how this theorem can be proven."
Answer:

"Explain how to prove one of the following properties of parallelograms: opposite sides are congruent, opposite angles are congruent, diagonals bisect each other"
Answer:

Answer: One possible way to answer your question is:

To connect the ideas of congruency and rigid motion, we can use the following definition: Two figures are congruent if and only if there exists one or more rigid motions that map one figure onto the other. Rigid motions are transformations that preserve the size and shape of a figure, such as reflections, rotations, and translations. Therefore, congruency means that two figures have the same size and shape, and can be superimposed by applying one or more rigid motions.

To prove congruency, we can use the following criteria: Two triangles are congruent if they satisfy one of the following conditions:

SSS (Side-Side-Side): All three pairs of corresponding sides are equal in length.

SAS (Side-Angle-Side): Two pairs of corresponding sides are equal in length, and the included angles are equal in measure.

ASA (Angle-Side-Angle): Two pairs of corresponding angles are equal in measure, and the included sides are equal in length.

AAS (Angle-Angle-Side): Two pairs of corresponding angles are equal in measure, and a pair of corresponding sides not included between the angles are equal in length.

HL (Hypotenuse-Leg): The hypotenuses and a pair of corresponding legs of two right triangles are equal in length.

To prove one of these conditions, we can use the properties of parallel lines, isosceles triangles, midpoints, bisectors, perpendiculars, etc. For example, to prove that opposite sides of a parallelogram are congruent, we can use the following steps:

Given a parallelogram ABCD, draw a diagonal AC.

By the alternate interior angles theorem, we have ∠BAC = ∠DCA and ∠BCA = ∠DAC.

By the reflexive property, we have AC = AC.

By the ASA criterion, we have ΔABC ≅ ΔCDA.

By the CPCTC4 (Corresponding Parts of Congruent Triangles are Congruent), we have AB = CD and BC = AD.

Therefore, opposite sides of a parallelogram are congruent.