Question

1. a is the midpoint of \\( \overline{zy} \\) 1) given 2) 2) given 3) 3) definition of midpoint 4) \\( \overline{ax} \cong \overline{ax} \\) 4) 5) \\( \triangle xza \cong \triangle xy a \\) 5) sss vertical angles are congruent sas reflexive definition of midpoint \\( \overline{xz} \cong \overline{xy} \\) \\( \overline{az} \cong \overline{ay} \\) definition of a linear pair

Explanation:

Step1: Fill in statement 2

Since we are proving $\triangle XZA\cong\triangle XYA$, and we know from the goal that we need some side - angle - side or other congruence criteria. Given that we will use the mid - point (which gives equal segments) and a common side. The second given could be $\overline{XZ}\cong\overline{XY}$.

Step2: Fill in statement 3

By the definition of a mid - point, if $A$ is the mid - point of $\overline{ZY}$, then $\overline{AZ}\cong\overline{AY}$.

Step3: Fill in reason 4

For $\overline{AX}\cong\overline{AX}$, the reason is the reflexive property (a segment is congruent to itself).

Step4: Fill in reason 5

We have $\overline{XZ}\cong\overline{XY}$ (statement 2), $\overline{AZ}\cong\overline{AY}$ (statement 3), and $\overline{AX}\cong\overline{AX}$ (statement 4). By the Side - Angle - Side (SAS) congruence criterion, $\triangle XZA\cong\triangle XYA$.

Answer:

1. $\overline{XZ}\cong\overline{XY}$; 3) $\overline{AZ}\cong\overline{AY}$; 4) reflexive; 5) SAS