Question

given abcd is a parallelogram. prove: ab ≅ cd and bc ≅ da. statements: 1. abcd is a parallelogram (1. given); 2. ab || cd (2. def. of parallelogram); 3. draw ac (3. unique line postulate); 4. ∠bca and ∠dac are alt interior angles (4. def. of alt. interior angles); 5. ∠dca and ∠bac are alt interior angles (5. def. of alt. interior angles); 6. bc || da (6. def. of parallelogram); 7. ∠bca ≅ ∠dac (7. alternate interior angles theorem)

Explanation:

Step1: Recall properties of parallelogram

Since ABCD is a parallelogram, by definition, AB || CD and BC || DA.

Step2: Draw diagonal

Draw AC. By the unique - line postulate, we can draw a line segment between two points A and C.

Step3: Identify alternate - interior angles

Because AB || CD, ∠DCA and ∠BAC are alternate - interior angles, and because BC || DA, ∠BCA and ∠DAC are alternate - interior angles.

Step4: Apply alternate - interior angles theorem

By the alternate - interior angles theorem, ∠BCA ≅ ∠DAC and ∠DCA ≅ ∠BAC.

Step5: Consider triangles

In △ABC and △CDA, we have ∠BCA ≅ ∠DAC, AC ≅ CA (reflexive property), and ∠DCA ≅ ∠BAC.

Step6: Prove triangles congruent

By the Angle - Side - Angle (ASA) congruence criterion, △ABC ≅ △CDA.

Step7: Use CPCTC

Since △ABC ≅ △CDA, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), AB ≅ CD and BC ≅ DA.

Answer:

AB ≅ CD and BC ≅ DA are proved by showing △ABC ≅ △CDA using properties of parallelograms, alternate - interior angles, and the ASA congruence criterion.