Question

given abcd is a parallelogram. prove: \\(\overline{ab} \cong \overline{cd}\\) and \\(\overline{bc} \cong \overline{da}\\)
assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Draw Diagonal \( AC \)

In parallelogram \( ABCD \), draw diagonal \( AC \). This creates two triangles, \( \triangle ABC \) and \( \triangle CDA \).

Step2: Identify Parallel Sides and Alternate Angles

Since \( ABCD \) is a parallelogram, \( AB \parallel CD \) and \( BC \parallel DA \) (definition of a parallelogram). By the Alternate Interior Angles Theorem, \( \angle BAC \cong \angle DCA \) (because \( AB \parallel CD \) and \( AC \) is a transversal) and \( \angle BCA \cong \angle DAC \) (because \( BC \parallel DA \) and \( AC \) is a transversal).

Step3: Identify Common Side

The diagonal \( AC \) is common to both \( \triangle ABC \) and \( \triangle CDA \), so \( AC \cong AC \) (Reflexive Property of Congruence).

Step4: Prove Triangles Congruent

By the Angle - Side - Angle (ASA) Congruence Criterion, \( \triangle ABC \cong \triangle CDA \) (since \( \angle BAC \cong \angle DCA \), \( AC \cong AC \), and \( \angle BCA \cong \angle DAC \)).

Step5: Corresponding Parts of Congruent Triangles

Since \( \triangle ABC \cong \triangle CDA \), their corresponding sides are congruent. So, \( AB \cong CD \) and \( BC \cong DA \) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Answer:

To prove \( \overline{AB} \cong \overline{CD} \) and \( \overline{BC} \cong \overline{DA} \) in parallelogram \( ABCD \):

Statements:

1. Draw diagonal \( AC \) in parallelogram \( ABCD \), forming \( \triangle ABC \) and \( \triangle CDA \).
2. \( AB \parallel CD \), \( BC \parallel DA \) (Definition of a parallelogram).
3. \( \angle BAC \cong \angle DCA \), \( \angle BCA \cong \angle DAC \) (Alternate Interior Angles Theorem).
4. \( AC \cong AC \) (Reflexive Property of Congruence).
5. \( \triangle ABC \cong \triangle CDA \) (ASA Congruence Criterion).
6. \( \overline{AB} \cong \overline{CD} \), \( \overline{BC} \cong \overline{DA} \) (CPCTC).

Reasons:

1. Construction.
2. Definition of a parallelogram.
3. Alternate Interior Angles Theorem.
4. Reflexive Property of Congruence.
5. ASA Congruence Criterion.
6. CPCTC.