Congruent Triangles worksheet for geometry practice, focusing on CPCTC and triangle congruence.

Let’s solve this worksheet step by step.

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## Part 1: Definitions

Congruent: same size, shape, and measure. (≅)
✔ Already filled.

Corresponding: matching parts or in the same position.
*(In congruent triangles, corresponding angles and sides are those that match up when one triangle is placed on top of the other.)*

Congruence Statement: an equation showing the parts that are congruent.
*(Example: △ABC ≅ △DEF means angle A = angle D, side AB = side DE, etc.)*

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## Part 2: CPCTC

CPCTC stands for: Corresponding Parts of Congruent Triangles are Congruent

So fill in:

- C: Corresponding
- P: Parts in
- C: Congruent
- T: Triangles are
- C: Congruent

> ✔ So: Corresponding Parts Congruent Triangles Congruent

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## Part 3: Are the following triangles congruent? Justify with congruence equations.

We’ll go one pair at a time.

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A. Triangle LJK and Triangle MPQ

Given:
- Triangle LJK:
Angles: ∠L = 50°, ∠J = 52°, ∠K = 78°
Sides: LJ = 50, JK = 39, KL = 40

- Triangle MPQ:
Angles: ∠M = 78°, ∠P = 52°, ∠Q = 50°
Sides: MP = 39, PQ = 50, QM = 40

Let’s match corresponding parts:

- Angle L (50°) ↔ Angle Q (50°)
- Angle J (52°) ↔ Angle P (52°)
- Angle K (78°) ↔ Angle M (78°)

Sides:
- Side LJ (50) ↔ Side QP (50) → but QP is same as PQ = 50
- Side JK (39) ↔ Side PM (39) → PM is same as MP = 39
- Side KL (40) ↔ Side MQ (40) → MQ is same as QM = 40

So, if we write the correspondence:

△LJK ≅ △QPM
*(Order matters! L→Q, J→P, K→M)*

Check:
- ∠L = ∠Q = 50°
- ∠J = ∠P = 52°
- ∠K = ∠M = 78°
- LJ = QP = 50
- JK = PM = 39
- KL = MQ = 40

✔ All corresponding angles and sides match → Congruent

Answer for A: LJK = QPM

Fill in blanks:

Angles
L = Q
J = P
K = M

Sides
LJ = QP
JM = ? — Wait, JM is not a side of triangle LJK. Probably typo — should be JK = PM
KM = ? — Should be KL = MQ

Better to use correct sides:

Sides
LJ = QP
JK = PM
KL = MQ

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B. Triangle XYZ and Triangle RSZ

Looking at the diagram:

It appears these two triangles share side ZY and ZX? Actually, looking at the image:

Triangle XYZ has points X, Y, Z.
Triangle RSZ has points R, S, Z.

But from the markings:

- In triangle XYZ:
∠X = marked with single arc → so ∠X = ∠R (in triangle RSZ)
∠Y = marked with double arc → ∠Y = ∠S
∠Z = common vertex, marked with triple arc → ∠Z = ∠Z (same angle)

Also, sides:
- XY = RS (both have one tick mark)
- YZ = SZ (both have two tick marks)
- XZ = RZ (both have three tick marks)

So all sides and angles match.

Therefore: △XYZ ≅ △RSZ

Check order:
X ↔ R
Y ↔ S
Z ↔ Z

So: XYZ = RSZ

Fill in:

Angles
Y = S
X = R
Z = Z

Sides
YX = SR
XZ = RZ
ZY = ZS

*(Note: YX is same as XY, SR is same as RS, etc.)*

✔ Congruent by SSS and AAA (actually ASA or SAS also works).

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C. Triangle XYZ and Triangle ABC

Given:

Triangle XYZ:
- Right angle at X → ∠X = 90°
- ∠Y = 60°, so ∠Z = 30° (since 90+60+30=180)
- Sides: XY = 2.5, XZ = 4.3, YZ = 5

Triangle ABC:
- Right angle at A → ∠A = 90°
- ∠B = 60°, so ∠C = 30°
- Sides: AB = 2.5, AC = 4.3, BC = 5

So:

- ∠X = ∠A = 90°
- ∠Y = ∠B = 60°
- ∠Z = ∠C = 30°

Sides:
- XY = AB = 2.5
- XZ = AC = 4.3
- YZ = BC = 5

Perfect match!

So: △XYZ ≅ △ABC

Order: X↔A, Y↔B, Z↔C

Answer for C: ABC = XYZ
*(Wait — the question says “ABC = ______”, so we need to say which triangle it equals. Since ABC corresponds to XYZ, then ABC = XYZ)*

Actually, the format is “ABC = _____”, meaning which triangle is congruent to ABC? It’s XYZ.

So: ABC = XYZ

Fill in:

Angles
X = A
Y = B
Z = C

Sides
YX = BA → or XY = AB = 2.5
ZY = CB → or YZ = BC = 5
XZ = AC = 4.3

✔ All match → Congruent

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## Final Answers Summary

Definitions:

- Corresponding: matching parts
- Congruence Statement: congruent

CPCTC:

- C: Corresponding
- P: Parts in
- C: Congruent
- T: Triangles are
- C: Congruent

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A. LJK = QPM

Angles
L = Q
J = P
K = M

Sides
LJ = QP
JK = PM
KL = MQ

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B. RSZ = XYZ

Angles
Y = S
X = R
Z = Z

Sides
YX = SR
XZ = RZ
ZY = ZS

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C. ABC = XYZ

Angles
X = A
Y = B
Z = C

Sides
YX = BA (or XY = AB)
ZY = CB (or YZ = BC)
XZ = AC

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✔ All triangles are congruent with matching corresponding parts.

Remember: Order matters! When writing congruence statements, the vertices must be listed in corresponding order.

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