Question

4
select the correct answer from each drop - down menu.
given: rhombus abcd with diagonal \\(\overline{ac}\\)
prove: \\(\overline{ac}\\) bisects \\(\angle dab\\) and \\(\angle bcd\\)
image of rhombus abcd with diagonal ac
identify the missing statement and reason in the proof.
given rhombus abcd with diagonal \\(\overline{ac}\\), it follows from the definition of a rhombus that \\(\overline{ab} \cong \overline{bc} \cong \overline{cd} \cong \overline{ad}\\). by the reflexive property of congruence, \\(\overline{ac} \cong \overline{ac}\\). so, \\(\triangle dac \cong \triangle bac\\) by the drop - down menu with options: asa criterion, sss criterion, aas theorem, sas criterion since corresponding parts of congruent triangles are congruent, \\(\angle dac \cong \angle\\) drop - down menu and \\(\angle dca \cong \angle\\) drop - down menu. so by the definition of segment bisector \\(\overline{ac}\\) bisects \\(\angle dab\\) and \\(\angle bcd\\).
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Explanation:

Step1: Determine the congruence criterion

We have \( \overline{AB}\cong\overline{BC}\cong\overline{CD}\cong\overline{AD}\) (sides of a rhombus) and \( \overline{AC}\cong\overline{AC}\) (reflexive property). So, \(\triangle DAC\cong\triangle BAC\) by the SSS (Side - Side - Side) criterion.

Step2: Use corresponding parts of congruent triangles

Since \(\triangle DAC\cong\triangle BAC\), by the property of corresponding parts of congruent triangles (CPCTC), \(\angle DAC\cong\angle BAC\) and \(\angle DCA\cong\angle BCA\)

Answer:

The first blank (congruence criterion) is filled with "SSS criterion". The second blank (angle congruence for \(\angle DAC\)) is filled with "BAC". The third blank (angle congruence for \(\angle DCA\)) is filled with "BCA"