Question

statements

1. abcd is a parallelogram
2. \\(\angle amb\\), \\(\angle bmc\\), \\(\angle cmd\\), and \\(\angle dma\\) are right angles
3. \\(\angle amb \cong \angle bmc \cong \angle cmd \cong \angle dma\\)
4. \\(\overline{ac}\\) bisects \\(\overline{bd}\\);

\\(\overline{bd}\\) bisects \\(\overline{ac}\\);

1. \\(\overline{am} \cong \overline{mc}\\), \\(\overline{mb} \cong \overline{md}\\)
2. ?
3. \\(\overline{ab} \cong \overline{bc} \cong \overline{cd} \cong \overline{da}\\)
4. figure abcd is a rhombus

reasons

1. given
2. given
3. right angles are congruent
4. diagonals of a parallelogram bisect each other
5. definition of a bisector
6. sas congruency theorem
7. cpctc
8. definition of a rhombus

\\(\triangle abc \cong \triangle adc \cong \triangle bad \cong \triangle bcd\\)
\\(\triangle amb \cong \triangle cmb \cong \triangle cmd \cong \triangle amd\\)
\\(\triangle abc \cong \triangle adc \cong \triangle amd \cong \triangle bmc\\)
\\(\triangle amb \cong \triangle cmd \cong \triangle bad \cong \triangle bcd\\)

Explanation:

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<reasoning>

Identify the missing statement

Using the Triangle Congruence Proofs knowledge point
We need to find the statement for step 6 that justifies the use of the SAS (Side-Angle-Side) congruency theorem to eventually prove that \(\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}\) in step 7.

• From step 5, we have congruent side segments: \(\overline{AM} \cong \overline{MC}\) and \(\overline{MB} \cong \overline{MD}\).
• From step 3, we have congruent right angles: \(\angle AMB \cong \angle BMC \cong \angle CMD \cong \angle DMA\).
• Combining these pairs of congruent sides and their included right angles, we can apply SAS to prove the four small triangles sharing vertex \(M\) are congruent.
• Therefore, the statement must be: \(\triangle AMB \cong \triangle CMB \cong \triangle CMD \cong \triangle AMD\).

Verify the CPCTC connection

Using the CPCTC knowledge point

• Once the four triangles are proven congruent in step 6:

\[
\triangle AMB \cong \triangle CMB \cong \triangle CMD \cong \triangle AMD
\]

• Their corresponding parts (hypotenuses) must be congruent by CPCTC:

\[
\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}
\]

• This perfectly matches step 7, confirming that the correct statement for step 6 is indeed the congruence of these four triangles.

</reasoning>

<answer>
<mcq-option>(A) \(\triangle ABC \cong \triangle ADC \cong \triangle BAD \cong \triangle BCD\)</mcq-option>
<mcq-correct>(B) \(\triangle AMB \cong \triangle CMB \cong \triangle CMD \cong \triangle AMD\)</mcq-correct>
<mcq-option>(C) \(\triangle ABC \cong \triangle ADC \cong \triangle AMD \cong \triangle BMC\)</mcq-option>
<mcq-option>(D) \(\triangle AMB \cong \triangle CMD \cong \triangle BAD \cong \triangle BCD\)</mcq-option>
</answer>

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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Triangle Congruence Proofs"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Identify the missing statement

Using the Triangle Congruence Proofs knowledge point
We need to find the statement for step 6 that justifies the use of the SAS (Side-Angle-Side) congruency theorem to eventually prove that \(\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}\) in step 7.

• From step 5, we have congruent side segments: \(\overline{AM} \cong \overline{MC}\) and \(\overline{MB} \cong \overline{MD}\).
• From step 3, we have congruent right angles: \(\angle AMB \cong \angle BMC \cong \angle CMD \cong \angle DMA\).
• Combining these pairs of congruent sides and their included right angles, we can apply SAS to prove the four small triangles sharing vertex \(M\) are congruent.
• Therefore, the statement must be: \(\triangle AMB \cong \triangle CMB \cong \triangle CMD \cong \triangle AMD\).

Verify the CPCTC connection

Using the CPCTC knowledge point

• Once the four triangles are proven congruent in step 6:

\[
\triangle AMB \cong \triangle CMB \cong \triangle CMD \cong \triangle AMD
\]

• Their corresponding parts (hypotenuses) must be congruent by CPCTC:

\[
\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}
\]

• This perfectly matches step 7, confirming that the correct statement for step 6 is indeed the congruence of these four triangles.

</reasoning>

<answer>
<mcq-option>(A) \(\triangle ABC \cong \triangle ADC \cong \triangle BAD \cong \triangle BCD\)</mcq-option>
<mcq-correct>(B) \(\triangle AMB \cong \triangle CMB \cong \triangle CMD \cong \triangle AMD\)</mcq-correct>
<mcq-option>(C) \(\triangle ABC \cong \triangle ADC \cong \triangle AMD \cong \triangle BMC\)</mcq-option>
<mcq-option>(D) \(\triangle AMB \cong \triangle CMD \cong \triangle BAD \cong \triangle BCD\)</mcq-option>
</answer>

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"Mathematics",
"Geometry",
"Triangle Congruence Proofs"
]
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