given: $ \overline{yx} $ bisects $ \angle zxy $, $ \overline{zy} \cong \overline{yy} $
prove: $ \triangle zxy \cong \triangle yxy $

| statements | reasons |
| --- | --- |
| 1. $ \overline{yx} $ bisects $ \angle zxy $ | 1. given |
| 2. | 2. definition of angle bisector |
| 3. $ \overline{zy} \cong \overline{yx} $ | 3. angle - side - angle |
| 4. | 4. reflexive property |
| 5. $ \triangle zxy \cong \triangle yxy $ | 5. alternate... |

Question

given: $ \overline{yx} $ bisects $ \angle zxy $, $ \overline{zy} \cong \overline{yy} $
prove: $ \triangle zxy \cong \triangle yxy $

| statements | reasons |
| --- | --- |
| 1. $ \overline{yx} $ bisects $ \angle zxy $ | 1. given |
| 2.  | 2. definition of angle bisector |
| 3. $ \overline{zy} \cong \overline{yx} $ | 3. angle - side - angle |
| 4.  | 4. reflexive property |
| 5. $ \triangle zxy \cong \triangle yxy $ | 5. alternate... |

Accepted Answer

# Explanation:
## Step1: Analyze Step 2
The reason for step 2 is the definition of an angle bisector. By the definition of an angle bisector, if $ \overline{YX} $ bisects $ \angle ZXY $, then $ \angle ZXY = \angle YXY $ (wait, actually, it should be $ \angle ZYX=\angle VYX $? Wait, the angle is $ \angle ZXY $? Wait, the given is $ \overline{YX} $ bisects $ \angle ZXY $, so the two angles formed are $ \angle ZYX $ and $ \angle VYX $? Wait, no, the angle bisector of $ \angle ZXY $ would split it into $ \angle ZYX $ and $ \angle VYX $? Wait, maybe the angle is $ \angle ZXY $, so bisecting it gives $ \angle ZYX = \angle VYX $? Wait, no, let's correct. If $ \overline{YX} $ bisects $ \angle ZXY $, then $ \angle ZYX=\angle VYX $? Wait, no, the vertex is $ X $? Wait, the angle is $ \angle ZXY $, so the bisector $ \overline{YX} $ would make $ \angle ZYX = \angle VYX $? No, the angle at $ X $: $ \angle ZXY $ is bisected by $ \overline{YX} $, so $ \angle ZXY = \angle YXY $? No, that's not right. Wait, the angle is $ \angle ZXY $, so the bisector is $ \overline{YX} $, so $ \angle ZYX = \angle VYX $? Wait, maybe the angle is $ \angle ZXY $, so the two angles are $ \angle ZXY $ and $ \angle YXY $? No, I think I messed up the notation. Let's assume the angle is $ \angle ZXY $, and $ \overline{YX} $ bisects it, so $ \angle ZYX = \angle VYX $? Wait, no, the correct statement is: If $ \overline{YX} $ bisects $ \angle ZXY $, then $ \angle ZYX = \angle VYX $? No, the angle at $ X $: $ \angle ZXY $ is split into $ \angle ZYX $ and $ \angle VYX $? Wait, maybe the angle is $ \angle ZXY $, so the bisector $ \overline{YX} $ creates $ \angle ZYX = \angle VYX $. So the statement for step 2 is $ \angle ZYX = \angle VYX $ (or whatever the correct angle notation is, but the key is that the angle bisector creates two equal angles).

## Step2: Analyze Step 3
The statement is $ \overline{ZY} \cong \overline{VY} $ (wait, the original is $ \overline{ZY} \cong \overline{VY} $? Wait, the given is $ \overline{ZY} \cong \overline{VY} $, so the reason for step 3 is "Given", because it's given in the problem. Wait, the problem says "Given: $ \overline{ZY} \cong \overline{VY} $", so step 3's reason should be "Given".

## Step3: Analyze Step 4
The reason for step 4 is the Reflexive Property. The Reflexive Property states that a segment is congruent to itself. So the statement for step 4 is $ \overline{YX} \cong \overline{YX} $ (since it's the same segment).

## Step4: Analyze Step 5
To prove $ \triangle ZXY \cong \triangle VXY $, we can use the SAS (Side-Angle-Side) congruence criterion. We have $ \angle ZYX = \angle VYX $ (from step 2), $ \overline{ZY} \cong \overline{VY} $ (step 3), and $ \overline{YX} \cong \overline{YX} $ (step 4). So the congruence criterion is SAS.

Now, let's fill in the table:

1. Statement: $ \overline{YX} $ bisects $ \angle ZXY $; Reason: Given (correct as is).

2. Statement: $ \angle ZYX = \angle VYX $ (or the correct angle notation based on the diagram); Reason: Definition of Angle Bisector (correct, because angle bisector splits the angle into two equal angles).

3. Statement: $ \overline{ZY} \cong \overline{VY} $; Reason: Given (since it's given in the problem).

4. Statement: $ \overline{YX} \cong \overline{YX} $; Reason: Reflexive Property (correct, because a segment is congruent to itself).

5. Statement: $ \triangle ZXY \cong \triangle VXY $; Reason: SAS (Side-Angle-Side) Congruence Criterion (because we have two sides and the included angle: $ \overline{ZY} \cong \overline{VY} $, $ \angle ZYX = \angle VYX $, $ \overline{YX} \cong \overline{YX} $).

# Answer:
To complete the proof table:

2. Statement: $ \angle ZYX = \angle VYX $ (or correct angle notation from diagram); Reason: Definition of Angle Bisector.
3. Statement: $ \overline{ZY} \cong \overline{VY} $; Reason: Given.
4. Statement: $ \overline{YX} \cong \overline{YX} $; Reason: Reflexive Property.
5. Statement: $ \triangle ZXY \cong \triangle VXY $; Reason: SAS (Side - Angle - Side) Congruence Criterion.

(Note: Adjust angle and segment notations based on the actual diagram's labeling for precision.)

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze Step 2

Step2: Analyze Step 3

Step3: Analyze Step 4

Step4: Analyze Step 5

Answer:

statements	reasons
2.	2. definition of angle bisector
3. \\( \overline{zy} \cong \overline{yx} \\)	3. angle - side - angle
4.	4. reflexive property
5. \\( \triangle zxy \cong \triangle yxy \\)	5. alternate...