Question

Given ABCD is a parallelogram, prove AB CD and BC & AD.

Answer 1

The missing statement and reason in the table is given below:

Statement Reason

1. ∠ABD ≅ ∠CDB; ∠ADB ≅ ∠CBD

Opposite angles in a parallelogram are congruent.

2. ΔABD ≅ ΔCDB

Side-Angle-Side (SAS) postulate of congruence. |

3. AB ≅ CD and BC ≅ AD

Corresponding parts of congruent triangles are congruent (CPCTC).

1. In a parallelogram, opposite sides are parallel by definition. Therefore, angle
`\( \angle ABD \)` is congruent to angle
`\( \angle CDB \)` and angle
`\( \angle ADB \)` is congruent to angle
`\( \angle CBD \)` because they are alternate interior angles formed by a transversal with two parallel lines.

2. We also know that side
`\( BD \)` is shared between triangles
`\( \triangle ABD \)` and
`\( \triangle CDB \)`, so it is congruent to itself by the reflexive property of equality.

3. With two angles and the side between them congruent (SAS), we can say that
`\( \triangle ABD \)` is congruent to
`\( \triangle CDB \)`.

4. Because the triangles are congruent, all corresponding parts are congruent. Therefore, side
`\( AB \)` is congruent to side
`\( CD \)`, and side
`\( BC \)` is congruent to side
`\( AD \)`, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer 2

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