4.4 CPCTC and HL - Geometry - Free Printable

It looks like you've uploaded a geometry problem involving proving two triangles congruent using the Side-Angle-Side (SAS) postulate. Let's go through the problem step by step, clarify what’s given, and complete the proof.

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🔷 Given:

- $ \overline{XJ} \cong \overline{HI} $
- $ \overline{YJ} \cong \overline{IH} $ ← This seems to be a typo or mislabeling. Looking at the diagram, it should probably be $ \overline{YJ} \cong \overline{GH} $, or possibly $ \overline{YJ} \cong \overline{GH} $, but let's examine carefully.

Wait — looking at the diagram:

- Points: $ X, Y, J $ form one triangle.
- Points: $ G, H, I $ form another triangle.
- The diagram shows:
- $ XJ \cong HI $ (marked with one tick)
- $ YJ \cong GH $ (marked with two ticks)
- $ \angle XJY \cong \angle GHI $ (marked with arcs)

So likely, the correct given is:
- $ \overline{XJ} \cong \overline{HI} $
- $ \overline{YJ} \cong \overline{GH} $
- $ \angle XJY \cong \angle GHI $

And we are to prove: $ \triangle XJY \cong \triangle GHI $

But in your image, the statement says "Prove: $ XH = JI $" — this seems incorrect or unrelated.

Let’s recheck.

Wait — the diagram shows:

- Triangle $ \triangle XJY $ and triangle $ \triangle GHI $
- Side $ XJ $ ≅ $ HI $
- Side $ YJ $ ≅ $ GH $
- Angle at $ J $ (between $ XJ $ and $ YJ $) ≅ angle at $ H $ (between $ GH $ and $ HI $)

So the included angles are congruent.

Thus, we can use SAS Congruence Postulate.

But the statement to prove says: “Prove: $ XH = JI $” — that doesn’t make sense unless there's a typo.

Wait — perhaps the triangles are $ \triangle XJY $ and $ \triangle HIG $? Or maybe the goal is to prove $ \triangle XJY \cong \triangle HIG $?

But from the diagram, it looks like:

- $ \triangle XJY $ and $ \triangle GHI $
- With:
- $ XJ \cong HI $
- $ YJ \cong GH $
- $ \angle XJY \cong \angle GHI $

That would mean SAS if the sides are adjacent to the angle.

But the order matters in SAS.

Let’s write it properly.

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✔ Correct Interpretation:

We are trying to prove:
> $ \triangle XJY \cong \triangle GHI $
using SAS.

Let’s analyze the statements.

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📋 Proof Table (Completed):

| STATEMENTS | REASONS |
|-----------|--------|
| 1. $ \overline{XJ} \cong \overline{HI} $ | 1. Given |
| 2. $ \overline{YJ} \cong \overline{GH} $ | 2. Given (from diagram) |
| 3. $ \angle XJY \cong \angle GHI $ | 3. Given (from diagram) |
| 4. $ \triangle XJY \cong \triangle GHI $ | 4. SAS Congruence Postulate |

But wait — in your table, you have:

> 2. $ \overline{XJ} \cong \overline{YH} $ → This is wrong.

You wrote:
- Statement 2: $ \overline{XJ} \cong \overline{YH} $ — but $ YH $ isn't even in the diagram.
- Then: $ \overline{GH} \cong \overline{GH} $ — reflexive property? That’s fine, but not helpful.

There seems to be confusion in labeling.

Let’s fix it.

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🔍 Corrected Proof:

We are given:
- $ \overline{XJ} \cong \overline{HI} $ (side)
- $ \overline{YJ} \cong \overline{GH} $ (side)
- $ \angle XJY \cong \angle GHI $ (included angle)

So:

| STATEMENTS | REASONS |
|------------|--------|
| 1. $ \overline{XJ} \cong \overline{HI} $ | 1. Given |
| 2. $ \overline{YJ} \cong \overline{GH} $ | 2. Given |
| 3. $ \angle XJY \cong \angle GHI $ | 3. Given |
| 4. $ \triangle XJY \cong \triangle GHI $ | 4. SAS (Side-Angle-Side) |

✔ This proves the two triangles are congruent.

Now, the question asks: “Are the two triangles congruent?” → Yes, by SAS.

But then it says: “Prove: $ XH = JI $” — this is not necessarily true, and doesn't follow directly.

Unless $ XH $ and $ JI $ are corresponding parts of congruent triangles, but they are not corresponding sides.

In $ \triangle XJY \cong \triangle GHI $, the correspondence is:

- $ X \leftrightarrow G $
- $ J \leftrightarrow H $
- $ Y \leftrightarrow I $

So:
- $ XJ \leftrightarrow GH $
- $ JY \leftrightarrow HI $
- $ XY \leftrightarrow GI $

Therefore, $ XH $ and $ JI $ are not corresponding sides. They're diagonals across different points.

So proving $ XH = JI $ is not valid unless more information is given.

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✘ Conclusion:

There is likely a mistake in the problem statement.

The correct conclusion is:

> $ \triangle XJY \cong \triangle GHI $ by SAS.

But $ XH = JI $ cannot be proven from this.

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✔ Final Answer:

Yes, the two triangles are congruent by SAS.

However, the statement "Prove: $ XH = JI $" appears to be incorrect or based on a mislabeling.

If instead, the goal was to prove $ \triangle XJY \cong \triangle GHI $, then the proof is:

| Statements | Reasons |
|-----------|--------|
| 1. $ \overline{XJ} \cong \overline{HI} $ | 1. Given |
| 2. $ \overline{YJ} \cong \overline{GH} $ | 2. Given |
| 3. $ \angle XJY \cong \angle GHI $ | 3. Given |
| 4. $ \triangle XJY \cong \triangle GHI $ | 4. SAS |

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Let me know if the diagram has different labels or if you meant to prove something else (e.g., $ XY = GI $, which *would* follow from congruence).