Triangle congruence worksheet for SSS, SAS, ASA, AAS, and HL criteria.

Let’s solve each problem step by step, using the triangle congruence postulates:

- SSS (Side-Side-Side): All three sides of one triangle are congruent to all three sides of another.
- SAS (Side-Angle-Side): Two sides and the included angle of one triangle are congruent to two sides and the included angle of another.
- ASA (Angle-Side-Angle): Two angles and the included side of one triangle are congruent to two angles and the included side of another.
- AAS (Angle-Angle-Side): Two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another.
- HL (Hypotenuse-Leg): Only for right triangles — hypotenuse and one leg of one triangle are congruent to hypotenuse and one leg of another.

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Problem 1:

Two triangles share a common side (the vertical line). The base angles are marked congruent (arc marks), and the shared side is between them. So we have two angles and the included side → ASA

✔ Answer: ASA

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Problem 2:

Parallelogram split into two triangles by a diagonal. Opposite sides of a parallelogram are equal, so both triangles have:
- One pair of sides equal (opposite sides),
- Another pair of sides equal (other opposite sides),
- Shared diagonal = third side.

So three sides equal → SSS

✔ Answer: SSS

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Problem 3:

Two triangles with two sides marked congruent and one angle marked congruent. But note: the angle is not between the two sides — it’s opposite one of them. That’s SSA, which is NOT a valid congruence postulate (unless it’s a right triangle — HL).

Here, no right angle is shown, so cannot be proven congruent.

⚠️ Answer: Not Congruent (or “None” — depending on worksheet instructions)

*(Note: Some worksheets may accept "SSA" as an answer, but technically it's not sufficient unless HL applies. Since no right angle, we say NOT congruent.)*

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Problem 4:

Rectangle split by diagonal → two right triangles. Both have:
- Right angle (marked),
- Shared hypotenuse (diagonal),
- One leg equal (side of rectangle).

So: Right triangle, hypotenuse + leg → HL

✔ Answer: HL

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Problem 5:

Two triangles sharing a vertex, with two sides marked congruent on each, and the vertical angles at the intersection are congruent (vertically opposite angles).

We have:
- Two sides congruent,
- Included angle (vertical angle) congruent.

→ SAS

✔ Answer: SAS

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Problem 6:

Two triangles with two angles marked congruent (arcs) and a side marked congruent. The side is not between the two angles — it’s adjacent to one angle. So this is AAS.

✔ Answer: AAS

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Problem 7:

Isosceles triangle split into two by altitude to base. We see:
- Two legs equal (isosceles),
- Base split into two equal parts (markings),
- Shared altitude (common side).

So all three sides of each small triangle are equal → SSS

✔ Answer: SSS

*(Also could be SAS if you consider the right angle and legs, but since all sides are marked, SSS is direct.)*

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Problem 8:

Quadrilateral split into two triangles. Each triangle has:
- Two sides marked equal (one side shared),
- And the included angle marked congruent (at the top).

So: Two sides + included angle → SAS

✔ Answer: SAS

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Problem 9:

Two right triangles. Both have:
- Right angle (marked),
- One leg equal (marked),
- Hypotenuse equal (marked).

→ HL

✔ Answer: HL

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Problem 10:

Two triangles with two angles marked congruent and one side marked congruent. The side is not between the angles → AAS

✔ Answer: AAS

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Problem 11:

Two triangles sharing a common vertex. Markings show:
- Two sides equal (on each triangle),
- Included angle (at shared vertex) is marked congruent.

→ SAS

✔ Answer: SAS

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Problem 12:

Two triangles intersecting, with two sides marked congruent on each, and vertical angles at intersection are congruent.

So: Two sides + included angle (vertical angle) → SAS

✔ Answer: SAS

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## ✔ Final Answers:

1. ASA
2. SSS
3. Not Congruent *(or SSA — but not valid)*
4. HL
5. SAS
6. AAS
7. SSS
8. SAS
9. HL
10. AAS
11. SAS
12. SAS

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📌 *Note: For #3, if your curriculum accepts SSA in some contexts, check with your teacher. But standard geometry says SSA is not sufficient unless it’s HL (right triangle). Since no right angle, safest answer is “Not Congruent.”*

Let me know if you want diagrams or explanations for any specific one!