Question

given abcd is a parallelogram. prove ab ≅ cd and bc ≅ da. assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Draw diagonal AC

In parallelogram ABCD, draw diagonal AC.

Step2: Use properties of parallelograms

Since ABCD is a parallelogram, AB || CD and BC || DA. Then, ∠BAC ≅ ∠DCA (alternate - interior angles) and ∠BCA ≅ ∠DAC (alternate - interior angles).

Step3: Consider triangle congruence

In △ABC and △CDA, we have ∠BAC ≅ ∠DCA, AC ≅ CA (reflexive property), and ∠BCA ≅ ∠DAC. So, △ABC ≅ △CDA by the ASA (Angle - Side - Angle) congruence criterion.

Step4: Use congruent - triangle properties

Since △ABC ≅ △CDA, then by corresponding parts of congruent triangles (CPCTC), AB ≅ CD and BC ≅ DA.

Answer:

Statements	Reasons
2. Draw diagonal AC	Construction
3. ∠BAC ≅ ∠DCA; ∠BCA ≅ ∠DAC	Alternate - interior angles of parallel lines (AB		CD and BC		DA)
4. AC ≅ CA	Reflexive property
5. △ABC ≅ △CDA	ASA congruence criterion
6. AB ≅ CD; BC ≅ DA	CPCTC