Question

Given ABCD is a parallelogram. Diagonals line segment AC and line segment BD intersect at E. Prove line segment AC is congruent to line segment of CE and line segment BE is congruent line segment DE

Answer 1

Answer: check explanation

Step-by-step explanation:

Since ABCD is a parallelogram, its opposite sides are parallel and congruent. Therefore, we have:

AB || CD and AB ≅ CD

AD || BC and AD ≅ BC

Since AC is a diagonal, it divides the parallelogram into two congruent triangles, namely, ΔABC and ΔACD. Therefore, we have:

∠A ≅ ∠D (corresponding angles of congruent triangles)

∠C ≅ ∠B (corresponding angles of congruent triangles)

AB ≅ CD (opposite sides of parallelogram)

AC ≅ AC (reflexive property of congruence)

Now, consider the triangles ΔAEC and ΔCEB. We have:

∠AEC ≅ ∠CEB (vertical angles)

∠ACE ≅ ∠BCE (corresponding angles of congruent triangles ΔABC and ΔACD)

AC ≅ AC (reflexive property of congruence)

Therefore, by the angle-angle-side (AAS) postulate, we can conclude that ΔAEC ≅ ΔCEB. Hence, we have:

CE ≅ AC (corresponding parts of congruent triangles)

BE ≅ DE (corresponding parts of congruent triangles)

Thus, we have proven that line segment AC is congruent to line segment CE, and line segment BE is congruent to line segment DE.

Answer 2

By using the properties of parallelograms and the congruence postulates, we prove that the diagonals of parallelogram ABCD bisect each other, making AC congruent to CE and BE congruent to DE.

Step-by-step explanation:

To prove that in parallelogram ABCD, the diagonals bisect each other, we need to show that AC is congruent to CE, and BE is congruent to DE. According to the properties of a parallelogram, opposite sides are congruent. Thus, in parallelogram ABCD, AB is congruent to CD, and AD is congruent to BC. Now let us focus on the triangles AEB and CED formed by the diagonals intersecting at E.

Since opposite sides are congruent (AB ≅ CD), triangle AEB is congruent to triangle CED by the Side-Side-Side postulate. Therefore, corresponding parts of congruent triangles are congruent, which means AE ≅ EC and BE ≅ ED. This proves that the diagonals of a parallelogram bisect each other and AC is congruent to CE and BE is congruent to DE.