Congruent Triangles Worksheet: Determine if each pair of triangles is congruent and justify your answer.

Let's go through each pair of triangles in the Congruent Triangles Worksheet and determine whether they are congruent. We'll use the standard triangle congruence criteria:

- SSS (Side-Side-Side): All three sides of one triangle are equal to the corresponding sides of another.
- SAS (Side-Angle-Side): Two sides and the included angle of one triangle are equal to the corresponding parts of another.
- ASA (Angle-Side-Angle): Two angles and the included side are equal.
- AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
- HL (Hypotenuse-Leg): For right triangles, if the hypotenuse and one leg are equal.

We’ll analyze each problem step by step.

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1. Triangle PQR and Triangle ABC

- Both have a right angle: ∠Q = ∠B = 90°
- One leg marked with a single tick: PQ = AB
- Hypotenuse marked with double tick: PR = AC

This is HL (Hypotenuse-Leg) congruence for right triangles.

✔ Congruent by HL

> Answer: Yes, congruent by HL

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2. Triangle LMP and Triangle STQ

- ∠P = ∠T (marked)
- Side LP = ST (single tick)
- Side PM = TQ (double tick)

But wait — we need to check if these are corresponding sides.

In triangle LMP:
- ∠P is between sides LP and PM

In triangle STQ:
- ∠T is between sides ST and TQ

So we have:
- ∠P = ∠T
- LP = ST
- PM = TQ

→ This is SAS (Side-Angle-Side), since the angle is between the two sides.

✔ Congruent by SAS

> Answer: Yes, congruent by SAS

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3. Triangle ABC and Triangle PQR

- AB = PQ (double tick)
- BC = QR (single tick)
- ∠C = ∠R (angle marked)

But note: In triangle ABC, the angle at C is between sides BC and AC. But AC is not marked as equal to any side in PQR.

Wait: The given equal sides are:
- AB = PQ
- BC = QR
- ∠C = ∠R

But ∠C is not between AB and BC — it's between BC and AC. So this is not SAS, because the angle is not included.

Also, we don’t know about AC or PR.

So we only have two sides and a non-included angle — that’s SSA, which is not a valid congruence criterion unless it's a right triangle.

But here, no right angle is shown.

✘ Not necessarily congruent

> Answer: No, not congruent (SSA not valid)

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4. Triangle SRT and Triangle VRW

- The figure looks like an "X" shape with point R in the middle.
- SR = VR (single tick)
- TR = WR (double tick)
- ∠SRT = ∠VRW (vertical angles, so equal)

So:
- Two sides and the included angle: SR = VR, TR = WR, ∠SRT = ∠VRW

→ This is SAS

✔ Congruent by SAS

> Answer: Yes, congruent by SAS

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5. Triangle STV and Triangle AED

- ∠T = ∠A = 90° (right angles)
- One leg: ST = AE (single tick)
- Hypotenuse: SV = AD (double tick)

Wait: Is SV = AD? Let's see:

In triangle STV:
- Right angle at T
- ST = AE (given)
- SV = AD?

But in triangle AED:
- Right angle at A
- AE = ST
- AD = ? → marked with double tick, same as SV

Yes: SV = AD (both double tick)

So we have:
- Hypotenuse: SV = AD
- Leg: ST = AE

→ HL congruence (for right triangles)

✔ Congruent by HL

> Answer: Yes, congruent by HL

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6. Triangle EFG and Triangle ABC

- ∠E = ∠A (both marked with arc)
- ∠F = ∠B (both marked with arc)
- EF = AB (single tick)

So:
- Two angles and a non-included side: ∠E = ∠A, ∠F = ∠B, EF = AB

→ This is AAS (since the side is not between the two angles, but opposite one of them)

Note: AAS is valid.

✔ Congruent by AAS

> Answer: Yes, congruent by AAS

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7. Triangle LMN and Triangle PQR

- Markings:
- LN = PR (triple tick)
- MN = QR (double tick)
- LM = PQ (single tick)

So all three sides are marked equal:
- LM = PQ
- MN = QR
- LN = PR

→ SSS

✔ Congruent by SSS

> Answer: Yes, congruent by SSS

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8. Triangle ABC and Triangle CDB

- Quadrilateral ABCD has diagonals drawn.
- AB = CD (triple tick)
- BC = DA (double tick)
- AC = BD (single tick) — wait, no, actually both diagonals are marked with triple tick?

Wait: Look carefully.

- AB = CD (three ticks)
- BC = DA (two ticks)
- Diagonal AC and BD: both have three ticks — so AC = BD?

But the question is whether triangles ABC and CDB are congruent.

Triangle ABC and triangle CDB:

- AB = CD (three ticks)
- BC = CB (common side)
- AC = DB? — both diagonals have three ticks → yes

So:
- AB = CD
- BC = CB (same side)
- AC = DB

→ All three sides equal → SSS

✔ Congruent by SSS

> Answer: Yes, congruent by SSS

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✔ Final Answers:

| Problem | Congruent? | Reason |
|--------|------------|--------|
| 1 | Yes | HL |
| 2 | Yes | SAS |
| 3 | No | SSA not valid |
| 4 | Yes | SAS |
| 5 | Yes | HL |
| 6 | Yes | AAS |
| 7 | Yes | SSS |
| 8 | Yes | SSS |

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✍️ Written Answers (as per worksheet):

1. Yes, congruent by HL
2. Yes, congruent by SAS
3. No, not congruent (SSA not sufficient)
4. Yes, congruent by SAS
5. Yes, congruent by HL
6. Yes, congruent by AAS
7. Yes, congruent by SSS
8. Yes, congruent by SSS

Let me know if you'd like a printable version or explanations with diagrams!