To prove that segment AM is congruent to segment CM in parallelogram ABCD, we can use the properties of parallelograms. 1. Given: Parallelogram ABCD. 2. Since ABCD is a parallelogram, opposite sides are parallel and congruent. 3. Therefore, segment AB is congruent to segment CD and segment BC is congruent to segment AD. Now, let's focus on segment AM and segment CM. 4. Draw diagonal BD. This divides the parallelogram into two congruent triangles, triangle ABD and triangleCBD. 5. Since triangle ABD and triangle CBD share the same base (segment BD) and have congruent corresponding sides (segments AB and BC), they are congruent by the Side-Side-Side (SSS) congruence criterion. 6. As a result, angle BAD is congruent to angle BCD. Now, let's examine triangle BAM and triangle CDM. 7.
These two triangles share the same base (segment BM) and have congruent corresponding sides (segments AB and CD) due to the properties of parallelograms.8. By the Side-Angle-Side (SAS) congruence criterion, triangle BAM is congruent to triangle CDM. 9. Therefore, segment AM is congruent to segment CM because corresponding parts of congruent triangles are congruent. In conclusion, by using the properties of parallelograms and congruent triangles, we have proven that segment AM is congruent to segment CM in parallelogram ABCD.parallelograms.