Question

1 ( ab cong cd ); ( ab parallel cd ) given
2 ( angle bac cong angle dca ) alternate interior angles are congruent
3 ( overline{ac} cong overline{ac} ) reflexive property
4 ( \triangle bac cong \triangle dca ) side - angle - side (sas) triangle congruence criterion
5 ( angle bca cong angle dac ) alternate interior angles are congruent
6 ( overline{bc} parallel overline{da} ) if two lines are cut by a transversal and a pair of alternate interior angles is congruent, the lines are parallel.
7 ( abcd ) is a parallelogram definition of a parallelogram
joanne made a mistake in her proof. what was her mistake?

• reason 3 should be \symmetric property.\
• reason 2 should be \corresponding parts of congruent triangles are congruent.\
• reason 5 should be \corresponding parts of congruent triangles are congruent.\

Explanation:

• Analyze each option:
• Option 1: Reason 3 uses the reflexive property (a segment is congruent to itself), which is correct. The symmetric property is different (if \(a\cong b\), then \(b\cong a\)), so this is not the mistake.
• Option 2: Reason 2 uses alternate interior angles (since \(AB\parallel CD\) and \(AC\) is a transversal), which is correct. Corresponding parts of congruent triangles (CPCTC) would be used after triangles are proven congruent, not here, so this is not the mistake.
• Option 3: After proving \(\triangle BAC\cong\triangle DCA\) (step 4), \(\angle BCA\) and \(\angle DAC\) are corresponding parts of congruent triangles. So Reason 5 should be CPCTC (Corresponding parts of congruent triangles are congruent), not alternate interior angles (which would be used for parallel lines before triangle congruence). This is the mistake.

Answer:

C. Reason 5 should be "Corresponding parts of congruent triangles are congruent." (Assuming the third option is labeled C; if the original options had labels like A, B, C as per typical multiple - choice, with the third option being the correct one as analyzed above)