This worksheet helps students practice identifying congruent parts of polygons and solving for unknown variables in geometry problems.

Since I can't view or access images directly, I’ll help you solve the problems based on your description of the worksheet. Let’s go through each problem step by step and explain how to identify congruent parts and use congruence statements.

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Problem 1: Triangle XYZ ≅ Triangle ABC

Given: $ \triangle XYZ \cong \triangle ABC $

We need to find corresponding congruent parts using the order of the vertices in the congruence statement.

- The order matters:
$ X \leftrightarrow A $, $ Y \leftrightarrow B $, $ Z \leftrightarrow C $

So:
- $ \angle L \cong \angle A $ → Wait, this seems like a typo. There's no "L" in triangle XYZ. Likely it should be $ \angle X $.
So:
- $ \angle X \cong \angle A $
- $ \angle Y \cong \angle B $
- $ \angle Z \cong \angle C $

- Sides:
- $ \overline{XY} \cong \overline{AB} $
- $ \overline{YZ} \cong \overline{BC} $
- $ \overline{ZX} \cong \overline{CA} $

Answer:
- $ \angle X \cong \angle A $
- $ \angle Y \cong \angle B $
- $ \angle Z \cong \angle C $
- $ \overline{XY} \cong \overline{AB} $
- $ \overline{YZ} \cong \overline{BC} $
- $ \overline{ZX} \cong \overline{CA} $

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Problem 2: Rectangle DEFG ≅ Rectangle PQRS

Rectangles are parallelograms with right angles. All angles are 90°, so we match vertices in order.

Given: $ \text{Rectangle } DEFG \cong \text{Rectangle } PQRS $

So:
- $ D \leftrightarrow P $
- $ E \leftrightarrow Q $
- $ F \leftrightarrow R $
- $ G \leftrightarrow S $

Now fill in:

- $ \angle G \cong \angle S $ (since G ↔ S)
- $ \overline{DE} \cong \overline{PQ} $ (D→P, E→Q)
- $ \overline{DF} \cong \overline{PR} $ (diagonal from D to F ↔ P to R)

But wait — in a rectangle, opposite sides are equal, but for congruence, we must match corresponding sides.

So:
- $ \angle D \cong \angle P $
- $ \angle E \cong \angle Q $
- $ \angle F \cong \angle R $
- $ \angle G \cong \angle S $
- $ \overline{DE} \cong \overline{PQ} $
- $ \overline{EF} \cong \overline{QR} $
- $ \overline{FG} \cong \overline{RS} $
- $ \overline{GD} \cong \overline{SP} $

Answer:
- $ \angle G \cong \angle S $
- $ \overline{DE} \cong \overline{PQ} $
- $ \overline{DF} $? No — DF is a diagonal. But unless specified, we don’t assume diagonals are congruent unless triangles are involved.

Wait — the question asks for $ \overline{DF} $ and $ \overline{DE} $. Since $ \overline{DF} $ is a diagonal from D to F, and $ D \leftrightarrow P $, $ F \leftrightarrow R $, then:
- $ \overline{DF} \cong \overline{PR} $

So:
- $ \angle G \cong \angle S $
- $ \overline{DE} \cong \overline{PQ} $
- $ \overline{DF} \cong \overline{PR} $

✔ That makes sense.

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Problem 3: Two Triangles with Side Lengths Given

We have two triangles:
- Triangle ABC: AB = 16 cm, BC = 8 cm, AC = 24 cm
- Triangle DEF: DE = 24 cm, EF = 8 cm, DF = 16 cm

Let’s compare side lengths:
- AB = 16 → matches DF = 16
- BC = 8 → matches EF = 8
- AC = 24 → matches DE = 24

So:
- AB ↔ DF
- BC ↔ EF
- AC ↔ DE

So the correspondence is:
- A ↔ D
- B ↔ F
- C ↔ E

Thus: $ \triangle ABC \cong \triangle DFE $

But we are asked: Are they congruent?

Yes! By SSS (Side-Side-Side) Congruence Postulate.

Answer: Yes, $ \triangle ABC \cong \triangle DFE $ by SSS.

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Problem 4: Two Triangles with Angles and Sides

Triangle XYZ and triangle WYV

From diagram:
- XY = 99
- YZ = 83
- XZ = 121
- WY = 83
- YV = 99
- WV = 121

So:
- XY = YV = 99
- YZ = WY = 83
- XZ = WV = 121

So all three sides are equal.

So: $ \triangle XYZ \cong \triangle VYW $ (note: order matters)

Check:
- X ↔ V
- Y ↔ Y
- Z ↔ W

Wait — better to match:
- XY = VY? No — XY = 99, VY = 99 → yes
- YZ = WY = 83 → yes
- XZ = VW = 121 → yes

So: $ \triangle XYZ \cong \triangle VYW $ by SSS

But let's see if there's a better match.

Actually:
- XY = 99 → corresponds to YV = 99 → so X ↔ V?
- YZ = 83 → corresponds to WY = 83 → so Z ↔ W?

So:
- X ↔ V
- Y ↔ Y
- Z ↔ W

So: $ \triangle XYZ \cong \triangle VYW $

But usually, we write it with common vertex first.

Alternatively, $ \triangle XYZ \cong \triangle WYV $? Let’s check:

- X ↔ W? Then XY = 99 → WY = 83 → no

Wait — better to list:

From above:
- XY = 99
- YZ = 83
- XZ = 121

Other triangle:
- WY = 83
- YV = 99
- WV = 121

So:
- XY = YV = 99
- YZ = WY = 83
- XZ = WV = 121

So the correspondence is:
- X ↔ V
- Y ↔ Y
- Z ↔ W

So $ \triangle XYZ \cong \triangle VYW $

Yes.

Answer: Yes, $ \triangle XYZ \cong \triangle VYW $ by SSS.

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Problem 5: Two Triangles with Right Angles and Sides

Triangle ABC and triangle DEF

From diagram:
- AB = 36 mm
- BC = 37 mm
- AC = 45 mm
- DE = 45 mm
- EF = 37 mm
- DF = 36 mm

Compare:
- AB = 36 → DF = 36
- BC = 37 → EF = 37
- AC = 45 → DE = 45

So:
- AB ↔ DF
- BC ↔ EF
- AC ↔ DE

So A ↔ D, B ↔ F, C ↔ E

So $ \triangle ABC \cong \triangle DFE $

Also, both have right angles at B and F.

So by HL (Hypotenuse-Leg), since:
- Hypotenuse AC = DE = 45
- Leg AB = DF = 36
- Leg BC = EF = 37

So yes, congruent by HL or SSS.

Answer: Yes, $ \triangle ABC \cong \triangle DFE $ by SSS or HL.

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Problem 6: Triangle MNP and Triangle TUV

Given:
- MN = 15 in
- NP = 15 in
- MP = 36 in
- TU = 15 in
- UV = 15 in
- TV = 36 in

So:
- MN = TU = 15
- NP = UV = 15
- MP = TV = 36

So all sides equal → SSS congruence.

Correspondence:
- M ↔ T
- N ↔ U
- P ↔ V

So $ \triangle MNP \cong \triangle TUV $

Answer: Yes, $ \triangle MNP \cong \triangle TUV $ by SSS.

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Problem 7: Triangle CJK ≅ Triangle EGD

Given: $ \triangle CJK \cong \triangle EGD $

Find corresponding parts.

Order:
- C ↔ E
- J ↔ G
- K ↔ D

So:
- $ m\angle C = m\angle E $
- $ m\angle J = m\angle G $
- $ m\angle K = m\angle D $
- $ \overline{CJ} = \overline{EG} $
- $ \overline{JK} = \overline{GD} $
- $ \overline{KC} = \overline{DE} $

Wait — KC is same as CK, which corresponds to ED.

So:
- $ m\angle C = m\angle E $
- $ m\angle J = m\angle G $
- $ m\angle K = m\angle D $
- $ \overline{CJ} = \overline{EG} $
- $ \overline{JK} = \overline{GD} $
- $ \overline{CK} = \overline{ED} $

Answer:
- $ m\angle C = m\angle E $
- $ m\angle J = m\angle G $
- $ m\angle K = m\angle D $
- $ \overline{CJ} = \overline{EG} $
- $ \overline{JK} = \overline{GD} $
- $ \overline{CK} = \overline{ED} $

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Problem 8: Parallelogram HJKL ≅ Parallelogram WXYZ

Given: $ \square HJKL \cong \square WXYZ $

So:
- H ↔ W
- J ↔ X
- K ↔ Y
- L ↔ Z

Now find corresponding parts:

- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZH} = \overline{LK} $? Wait — ZH is not a side.

Wait: $ \overline{ZH} $? That’s not a side — likely typo.

Looking at diagram:
- HJ = 124
- JK = 22 cm
- KL = ?
- LH = ?

In WXYZ:
- WX = 19 cm
- XY = ?
- YZ = ?
- ZW = ?

But given:
- HJ = 124 → corresponds to WX = 19? No — 124 ≠ 19 → contradiction.

Wait — perhaps units differ? Or maybe it's a typo.

Wait — HJ = 124 (no unit), JK = 22 cm

WXYZ: WX = 19 cm, XY = ?, etc.

But if $ \square HJKL \cong \square WXYZ $, then corresponding sides must be equal.

But HJ = 124, WX = 19 → not equal → contradiction?

Unless it's not labeled correctly.

Wait — maybe the numbers are wrong?

Possibly, the values are meant to be matching.

But assuming the congruence is given, we use the vertex order.

So:
- H ↔ W
- J ↔ X
- K ↔ Y
- L ↔ Z

So:
- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZH} $? No — ZH is not a side.

Wait — probably typo: likely asking for $ \overline{ZW} $ or $ \overline{HZ} $

But in parallelogram HJKL:
- Sides: HJ, JK, KL, LH

In WXYZ:
- Sides: WX, XY, YZ, ZW

So:
- HJ ↔ WX → so HJ = WX → 124 = 19? Not possible.

So either the numbers are wrong, or the congruence is not based on these values.

But the problem says: “parallelogram HJKL ≅ parallelogram WXYZ”

So we assume the congruence is true, and we match parts by vertex order.

So:
- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZH} $? Wait — ZH is not a side.

Perhaps it's $ \overline{KL} $? But KL corresponds to YZ.

Wait — the blank says: $ \overline{ZH} $? That doesn't make sense.

Maybe it's $ \overline{HL} $? HL is a side.

In HJKL: HL is a side (from H to L), which corresponds to WZ (W to Z)

So:
- $ \overline{HL} = \overline{WZ} $
- $ \overline{JK} = \overline{XY} $
- $ \overline{KL} = \overline{YZ} $
- $ \overline{HJ} = \overline{WX} $

But HJ = 124, WX = 19 → not equal → contradiction.

So unless the numbers are in different units or there's a mistake, this suggests the diagram may have inconsistent values.

But since the congruence is stated, we ignore the numbers and just use the correspondence.

So:
- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZW} = \overline{HL} $? Wait — ZW is same as WZ.

But $ \overline{HL} $ corresponds to $ \overline{WZ} $, so $ \overline{HL} = \overline{WZ} $

Similarly:
- $ \overline{JK} = \overline{XY} $
- $ \overline{KL} = \overline{YZ} $
- $ \overline{HJ} = \overline{WX} $

But again, numbers don't match.

So likely, the diagram has a typo, or the values are not to scale.

But for the purpose of this problem, we use the vertex order.

So final answers:

- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZW} = \overline{HL} $? But ZW is from Z to W, which corresponds to L to H → so $ \overline{ZW} = \overline{LH} $

Wait — better:
- $ \overline{HJ} = \overline{WX} $
- $ \overline{JK} = \overline{XY} $
- $ \overline{KL} = \overline{YZ} $
- $ \overline{LH} = \overline{ZW} $

So:
- $ \overline{ZW} = \overline{LH} $

So:
- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZW} = \overline{LH} $

But the blank says $ \overline{ZH} $? Probably typo — should be $ \overline{ZW} $ or $ \overline{LH} $

Assuming it's $ \overline{ZW} $, then $ \overline{ZW} = \overline{LH} $

Or if it's $ \overline{ZH} $, that’s not a side.

So likely, it's $ \overline{LH} $

But the blank says: $ \overline{ZH} $? Unlikely.

Wait — perhaps the parallelogram is labeled differently.

But based on standard labeling, HJKL means:
- H to J to K to L back to H

So sides: HJ, JK, KL, LH

WXYZ: W to X to Y to Z back to W → sides: WX, XY, YZ, ZW

So:
- H ↔ W
- J ↔ X
- K ↔ Y
- L ↔ Z

So:
- $ \overline{HJ} \leftrightarrow \overline{WX} $
- $ \overline{JK} \leftrightarrow \overline{XY} $
- $ \overline{KL} \leftrightarrow \overline{YZ} $
- $ \overline{LH} \leftrightarrow \overline{ZW} $

So:
- $ \overline{ZW} = \overline{LH} $

So if the blank is $ \overline{ZW} $, then $ \overline{ZW} = \overline{LH} $

But the problem says: $ \overline{ZH} $? That’s not a side.

Possibly a typo — should be $ \overline{ZW} $

So assuming it's $ \overline{ZW} $, then $ \overline{ZW} = \overline{LH} $

But in the blank, it says $ \overline{ZH} $? Maybe it's $ \overline{LH} $

Wait — the last blank is: $ \overline{ZH} = $ ? → but ZH is not a side.

Unless it's $ \overline{HZ} $, which is same as $ \overline{ZH} $, but still not a side.

Wait — maybe it's $ \overline{HL} $? HL is a side.

But $ \overline{HL} $ corresponds to $ \overline{WZ} $

So $ \overline{HL} = \overline{WZ} $

But $ \overline{WZ} $ is same as $ \overline{ZW} $

So $ \overline{HL} = \overline{ZW} $

So if the blank is $ \overline{ZH} $, it's likely a typo — should be $ \overline{ZW} $ or $ \overline{HL} $

Given the context, likely the answer is $ \overline{LH} $ or $ \overline{HL} $

But let's assume the blank is for $ \overline{ZW} $, then $ \overline{ZW} = \overline{LH} $

So:

Answer:
- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZW} = \overline{LH} $

But since the blank says $ \overline{ZH} $, perhaps it's $ \overline{LH} $

So maybe the intended answer is $ \overline{LH} $

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✔ Final Answers Summary:

#### 1. $ \triangle XYZ \cong \triangle ABC $
- $ \angle X \cong \angle A $
- $ \angle Y \cong \angle B $
- $ \angle Z \cong \angle C $
- $ \overline{XY} \cong \overline{AB} $
- $ \overline{YZ} \cong \overline{BC} $
- $ \overline{ZX} \cong \overline{CA} $

#### 2. Rectangle DEFG ≅ Rectangle PQRS
- $ \angle G \cong \angle S $
- $ \overline{DE} \cong \overline{PQ} $
- $ \overline{DF} \cong \overline{PR} $

#### 3. Yes, $ \triangle ABC \cong \triangle DFE $ by SSS

#### 4. Yes, $ \triangle XYZ \cong \triangle VYW $ by SSS

#### 5. Yes, $ \triangle ABC \cong \triangle DFE $ by SSS or HL

#### 6. Yes, $ \triangle MNP \cong \triangle TUV $ by SSS

#### 7.
- $ m\angle C = m\angle E $
- $ m\angle J = m\angle G $
- $ m\angle K = m\angle D $
- $ \overline{CJ} = \overline{EG} $
- $ \overline{JK} = \overline{GD} $
- $ \overline{CK} = \overline{ED} $

#### 8.
- $ m\angle K = m\angle Y $
- $ m\angle W = m\angle H $
- $ m\angle X = m\angle J $
- $ \overline{ZW} = \overline{LH} $ (assuming typo in $ \overline{ZH} $)

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If you can provide the actual image or clarify any confusion (like the labels in Problem 8), I can refine the answers further.