SAS Proof worksheets for triangle congruence practice.

Let’s solve both SAS proofs step by step. We’ll fill in the “Statements” and “Reasons” tables using the given information and geometry rules.

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SAS PROOF #1

We are told:
- X is the midpoint of VZ → so VX = XZ
- X is the midpoint of WY → so WX = XY
We need to prove: ΔVWX ≅ ΔZYX

Looking at the diagram (even though we don’t describe it, we know from context that lines cross at X, forming vertical angles).

Step-by-step reasoning:

1. Start with what’s given.
- Statement: X is the midpoint of VZ → Reason: Given
- Statement: X is the midpoint of WY → Reason: Given

2. Use definition of midpoint to get segment congruences.
- Statement: VX ≅ XZ → Reason: Def. of Midpoint
- Statement: WX ≅ XY → Reason: Def. of Midpoint

3. Look for an angle between those sides — that’s where vertical angles come in.
- At point X, ∠VXW and ∠ZXY are vertical angles → they are congruent.
But wait — in the proof, we’re proving ΔVWX ≅ ΔZYX.
So corresponding vertices: V↔Z, W↔Y, X↔X.
Therefore, the included angle for side VX & WX is ∠VXW.
The included angle for side ZX & YX is ∠ZXY.
And since these are vertical angles, ∠VXW ≅ ∠ZXY.

However, looking at the provided reason boxes, one says “∠WXV ≅ ∠YXZ” — which is the same as ∠VXW ≅ ∠ZXY (just named differently). Also labeled “Vertical Angles”.

4. Now we have:
- Two sides and the included angle congruent → SAS Congruence Postulate.

5. Final statement: triangles are congruent.

So let’s build the table:

| Statements | Reasons |
|-----------------------------|-----------------------|
| X is the midpoint of VZ | Given |
| X is the midpoint of WY | Given |
| VX ≅ XZ | Def. of Midpoint |
| WX ≅ XY | Def. of Midpoint |
| ∠WXV ≅ ∠YXZ | Vertical Angles |
| ΔVWX ≅ ΔZYX | SAS |

✔ This matches all the provided reason/statement boxes.

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SAS PROOF #2

Given:
- XW ≅ XY
- XZ bisects ∠WXY

Prove: ΔWXY ≅ ΔYXZ? Wait — hold on.

Wait — look again: Prove: ΔWXY ≅ ΔYXZ? That doesn’t make sense because ΔWXY includes points W, X, Y — but ΔYXZ includes Y, X, Z. Actually, looking at the diagram description (again, not describing image), it's likely a quadrilateral or triangle split by diagonal XZ.

Actually, re-read: Prove: ΔWXY ≅ ΔYXZ — that seems odd. Let me check the given: XW ≅ XY, and XZ bisects ∠WXY.

But if XZ bisects ∠WXY, then it splits angle at X into two equal parts: ∠WXZ ≅ ∠YXZ.

Also, note: XZ is common to both triangles? Wait — which triangles?

The goal is to prove ΔWXY ≅ ΔYXZ — but that can't be right unless there’s a typo. Looking at the provided answer choices, one box says “ΔWXY ≅ ΔYXZ”, another says “XZ ≅ XZ” (reflexive), and “Def. of Angle Bisector”, etc.

Wait — perhaps it’s supposed to be ΔWXZ ≅ ΔYXZ? Because that would make more sense with the given.

But the problem clearly says: prove: ΔWXY ≅ ΔYXZ

That must be a mistake — because ΔWXY has sides WX, XY, WY — while ΔYXZ has sides YX, XZ, YZ — different triangles.

Alternatively, maybe it’s a typo and should be ΔWXZ ≅ ΔYXZ? That would fit perfectly.

Looking at the provided reason boxes:

Available statements/reasons include:
- ∠WXZ ≅ ∠YXZ
- XZ bisects ∠WXY
- XW ≅ XY
- ΔWXY ≅ ΔYXZ ← this is listed as a possible final statement
- Def. of Angle Bisector
- Reflexive Property (for XZ ≅ XZ)
- SAS

Ah — here’s the key: even though the triangle names seem mismatched, perhaps in the diagram, point Z is connected such that we’re comparing ΔWXZ and ΔYXZ? But the proof says ΔWXY ≅ ΔYXZ — which still doesn’t align.

Wait — let’s think differently.

Maybe the second triangle is ΔXYZ? No.

Another possibility: Perhaps it’s ΔWXZ ≅ ΔYXZ, and the “WXY” is a misprint? But the user wrote exactly what’s on the page.

Looking back at the original text:

> prove: Δ WXY ≅ Δ YXZ

But in standard notation, that would mean triangle with vertices W,X,Y vs Y,X,Z — sharing side XY? Not matching.

However, notice: In the diagram (implied), XZ is drawn, and we have points W, X, Y, Z. If XZ bisects ∠WXY, and XW ≅ XY, then likely we are to prove that ΔWXZ ≅ ΔYXZ — because:

- XW ≅ XY (given)
- ∠WXZ ≅ ∠YXZ (by angle bisector)
- XZ ≅ XZ (reflexive)

→ Then SAS gives ΔWXZ ≅ ΔYXZ.

But the problem says “ΔWXY ≅ ΔYXZ” — which is probably a typo. However, since the provided answer choice includes “ΔWXY ≅ ΔYXZ”, and also “XZ ≅ XZ”, and “SAS”, perhaps in their diagram, triangle WXY is being compared to YXZ via some other path? That doesn’t work.

Wait — another idea: Maybe it’s ΔWXZ ≅ ΔYXZ, and the “WXY” is a mislabel? Or perhaps the second triangle is ΔXYZ? No.

Let me check the available reason boxes again for Proof #2:

Boxes include:
- ∠WXZ ≅ ∠YXZ
- XZ bisects ∠WXY
- XW ≅ XY
- ΔWXY ≅ ΔYXZ ← this is listed as a possible conclusion
- Def. of Angle Bisector
- Given
- SAS
- XZ ≅ XZ
- Reflexive Property

If we force-fit to prove ΔWXY ≅ ΔYXZ, we’d need three pairs of corresponding parts — but WXY and YXZ share only point X and Y? Doesn’t work.

Unless... perhaps it’s a different correspondence? Like W→Y, X→X, Y→Z? Then we’d need WX≅YX (given), XY≅XZ? Not given. Angle at X? ∠WXY vs ∠YXZ — not necessarily related.

This suggests there might be a typo in the problem. BUT — since this is a worksheet with pre-made answer boxes, and one of them is “ΔWXY ≅ ΔYXZ”, and others support SAS with XZ as common side, I suspect the intended proof is for ΔWXZ ≅ ΔYXZ, and “WXY” is a misprint.

However, to match the worksheet exactly, let’s see how the boxes can be arranged logically.

Perhaps the correct interpretation is:

We are to prove ΔWXZ ≅ ΔYXZ, but the worksheet mistakenly wrote ΔWXY ≅ ΔYXZ. Since we must use the provided boxes, and “ΔWXY ≅ ΔYXZ” is among them, maybe in their diagram, point Z is positioned such that triangle WXY corresponds to YXZ? Unlikely.

Alternative approach: Let’s ignore the triangle name confusion and focus on the logic supported by the given and the answer choices.

Given:
- XW ≅ XY
- XZ bisects ∠WXY → so ∠WXZ ≅ ∠YXZ

Common side: XZ ≅ XZ (reflexive)

Then by SAS, ΔWXZ ≅ ΔYXZ.

Now, if the worksheet says “prove ΔWXY ≅ ΔYXZ”, that must be an error. But since we have to complete the table with the given boxes, and “ΔWXY ≅ ΔYXZ” is listed as a possible final statement, perhaps we proceed with the logical steps and put that as the conclusion, even if it’s technically incorrect — assuming it’s a labeling issue.

In many textbooks, sometimes triangle names are written with vertex order indicating correspondence. So ΔWXY ≅ ΔYXZ might mean W↔Y, X↔X, Y↔Z — but then we’d need WX≅YX (true), XY≅XZ (not given), and angle at X: ∠WXY ≅ ∠YXZ? Not necessarily.

This is messy.

Let me try to reconstruct based on standard problems.

Standard problem: Given XW ≅ XY, XZ bisects ∠WXY, prove ΔWXZ ≅ ΔYXZ.

Steps:

1. XW ≅ XY — Given
2. XZ bisects ∠WXY — Given
3. ∠WXZ ≅ ∠YXZ — Def. of Angle Bisector
4. XZ ≅ XZ — Reflexive Property
5. ΔWXZ ≅ ΔYXZ — SAS

Now, looking at the provided boxes for Proof #2, we have:

Statements/Reasons available:
- ∠WXZ ≅ ∠YXZ
- XZ bisects ∠WXY
- XW ≅ XY
- ΔWXY ≅ ΔYXZ ← this is problematic
- Def. of Angle Bisector
- Given
- SAS
- XZ ≅ XZ
- Reflexive Property

Notice: “ΔWXY ≅ ΔYXZ” is listed, but also “XZ ≅ XZ” and “Reflexive Property”, which suggests XZ is a common side — so likely the triangles sharing XZ are ΔWXZ and ΔYXZ.

Therefore, I believe there is a typo in the problem, and it should be “prove: ΔWXZ ≅ ΔYXZ”.

Since the student is to fill the table using the provided boxes, and “ΔWXY ≅ ΔYXZ” is the only congruence statement offered, we may have to use it, but logically it should be ΔWXZ ≅ ΔYXZ.

To resolve this, let’s assume that in the diagram, the triangle is labeled such that “WXY” refers to the triangle including Z? Unlikely.

Another possibility: Perhaps “ΔWXY” is meant to be “ΔWXZ”? Typo in printing.

Given the constraints, I will provide the logically correct proof for ΔWXZ ≅ ΔYXZ, and use the available boxes accordingly, noting that the final statement might be mislabeled.

But to strictly follow the worksheet, let’s arrange the boxes as per the most logical flow, ending with the congruence statement provided.

Here’s the best fit:

| Statements | Reasons |
|-----------------------------|-----------------------|
| XW ≅ XY | Given |
| XZ bisects ∠WXY | Given |
| ∠WXZ ≅ ∠YXZ | Def. of Angle Bisector|
| XZ ≅ XZ | Reflexive Property |
| ΔWXY ≅ ΔYXZ | SAS |

Even though the last line is geometrically questionable, it’s the only congruence statement provided, and the steps before it support SAS for triangles sharing XZ with two sides and included angle — which would be ΔWXZ and ΔYXZ. So likely, “ΔWXY” is a misprint for “ΔWXZ”.

For educational purposes, I’ll present the correct logic, and in the final answer, use the statement as given on the worksheet.

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Final Answer:

SAS PROOF #1 Table:

| Statements | Reasons |
|-----------------------------|-----------------------|
| X is the midpoint of VZ | Given |
| X is the midpoint of WY | Given |
| VX ≅ XZ | Def. of Midpoint |
| WX ≅ XY | Def. of Midpoint |
| ∠WXV ≅ ∠YXZ | Vertical Angles |
| ΔVWX ≅ ΔZYX | SAS |

SAS PROOF #2 Table:

| Statements | Reasons |
|-----------------------------|-----------------------|
| XW ≅ XY | Given |
| XZ bisects ∠WXY | Given |
| ∠WXZ ≅ ∠YXZ | Def. of Angle Bisector|
| XZ ≅ XZ | Reflexive Property |
| ΔWXY ≅ ΔYXZ | SAS |

Note: For Proof #2, the final congruence statement "ΔWXY ≅ ΔYXZ" appears to contain a labeling error; logically, it should be "ΔWXZ ≅ ΔYXZ" based on the given conditions and shared side XZ. However, since the worksheet provides "ΔWXY ≅ ΔYXZ" as the target conclusion, it is used here to match the expected format.