Question

1. fill in the missing statements and reasons to complete the following proof.

given: $overline{wx} \cong \overline{wz}$, $overline{wy}$ bisects $\angle xwz$
prove: $\angle x \cong \angle z$
(there is a diagram of a quadrilateral or two triangles with vertices w, x, y, z, where w is the top vertex, x and z are the left and right lower vertices, and y is the bottom vertex, with segments wx, wz, wy, xy, and zy.)

statements	reasons
2.	2. given
3.	3. definition of angle bisector
4. $overline{wy} \cong \overline{wy}$	4.
5. $\triangle yxw \cong \triangle yzw$	5.
6.	6.

Explanation:

Step 1: Analyze Statement 1

The first statement is $\overline{WX} \cong \overline{WZ}$, and the reason for this is that it is given in the problem. So the reason for statement 1 is "Given".

Step 2: Analyze Statement 2

The second reason is "Given", and from the problem, the given information related to this is that $\overline{WY}$ bisects $\angle XWZ$. So the statement for reason 2 is $\overline{WY}$ bisects $\angle XWZ$.

Step 3: Analyze Statement 3

The reason for statement 3 is "Definition of Angle Bisector". By the definition of an angle bisector, if a ray bisects an angle, it divides the angle into two congruent angles. Since $\overline{WY}$ bisects $\angle XWZ$, we can say that $\angle XWY \cong \angle ZWY$. So the statement for reason 3 is $\angle XWY \cong \angle ZWY$.

Step 4: Analyze Statement 4

The statement is $\overline{WY} \cong \overline{WY}$. A segment is congruent to itself by the Reflexive Property of Congruence. So the reason for statement 4 is "Reflexive Property of Congruence".

Step 5: Analyze Statement 5

We have $\overline{WX} \cong \overline{WZ}$ (statement 1), $\angle XWY \cong \angle ZWY$ (statement 3), and $\overline{WY} \cong \overline{WY}$ (statement 4). This satisfies the Side - Angle - Side (SAS) Congruence Postulate, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. So the two triangles $\triangle YXW$ and $\triangle YZW$ are congruent by SAS. So the reason for statement 5 is "SAS Congruence Postulate".

Step 6: Analyze Statement 6

Since $\triangle YXW \cong \triangle YZW$ (statement 5), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, the corresponding angles $\angle X$ and $\angle Z$ are congruent. So the statement is $\angle X \cong \angle Z$ and the reason is "CPCTC".

Answer:

1. Reason for Statement 1: Given
2. Statement for Reason 2: $\overline{WY}$ bisects $\angle XWZ$
3. Statement for Reason 3: $\angle XWY \cong \angle ZWY$
4. Reason for Statement 4: Reflexive Property of Congruence
5. Reason for Statement 5: SAS (Side - Angle - Side) Congruence Postulate
6. Statement for Reason 6: $\angle X \cong \angle Z$; Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)