Question

select the correct answer from each drop - down menu. given: kite abdc with diagonals $overline{ad}$ and $overline{bc}$ intersecting at e. prove: $overline{ad}$ bisects $overline{bc}$. determine the missing reasons in the proof. statement reason abdc is a kite given $overline{cd}congoverline{bd}$ and $overline{ac}congoverline{ab}$ definition of a kite

Explanation:

Step1: Recall properties of a kite

A kite has two pairs of adjacent - congruent sides. Given \(ABDC\) is a kite, so \(\overline{CD}\cong\overline{BD}\) and \(\overline{AC}\cong\overline{AB}\) by the definition of a kite. Also, \(\overline{AD}\cong\overline{AD}\) by the reflexive property of congruence (any segment is congruent to itself).

Step2: Prove triangles congruent

Consider \(\triangle ACD\) and \(\triangle ABD\). We have \(\overline{CD}\cong\overline{BD}\), \(\overline{AC}\cong\overline{AB}\), and \(\overline{AD}\cong\overline{AD}\). So, \(\triangle ACD\cong\triangle ABD\) by the SSS (Side - Side - Side) congruence criterion.

Step3: Use congruent - triangle properties

Since \(\triangle ACD\cong\triangle ABD\), \(\angle CAD\cong\angle BAD\) (corresponding parts of congruent triangles are congruent, CPCTC).

Step4: Prove smaller triangles congruent

Now consider \(\triangle ACE\) and \(\triangle ABE\). We have \(\overline{AC}\cong\overline{AB}\), \(\angle CAD\cong\angle BAD\), and \(\overline{AE}\cong\overline{AE}\) (reflexive property). So, \(\triangle ACE\cong\triangle ABE\) by the SAS (Side - Angle - Side) congruence criterion.

Step5: Get the conclusion

Since \(\triangle ACE\cong\triangle ABE\), \(\overline{CE}\cong\overline{BE}\) (CPCTC). This means that \(\overline{AD}\) bisects \(\overline{BC}\) (by the definition of a bisector, a line that divides a segment into two congruent segments).

Answer:

The missing reasons in the proof (continuing from the given part):

1. \(\overline{AD}\cong\overline{AD}\): Reflexive property of congruence
2. \(\triangle ACD\cong\triangle ABD\): SSS congruence criterion
3. \(\angle CAD\cong\angle BAD\): CPCTC
4. \(\triangle ACE\cong\triangle ABE\): SAS congruence criterion
5. \(\overline{CE}\cong\overline{BE}\): CPCTC; \(\overline{AD}\) bisects \(\overline{BC}\): Definition of a bisector