Geometry worksheet on triangle congruence using criteria like ASA, SSS, and SAS.

Let's go through each problem carefully and analyze whether the triangles are congruent, and if so, by which congruence criterion (SSS, SAS, ASA, AAS, or HL). Then we'll address the last two questions about additional information needed for ASA and SAS.

---

Part 1: Determine if the triangles are congruent

We’ll go one by one:

---

1)
- Two sides marked with single tick marks, and a non-included angle marked.
- But the included angles are not necessarily the same; it’s SSA-like, which is not a valid congruence criterion.
- Also, the orientation suggests that the sides may not correspond properly.
- Answer: Not congruent. (SSA is not sufficient.)

> ✔ Your answer "Not congruent" is correct.

---

2)
- Two angles marked as equal (with arcs), and the included side between them is common (shared).
- This is ASA (Angle-Side-Angle).
- The shared side is between the two marked angles.
- Answer: Congruent by ASA.

> ✔ Your answer "ASA" is correct.

---

3)
- All three sides of both triangles have matching tick marks (three pairs of equal sides).
- This is SSS (Side-Side-Side).
- Answer: Congruent by SSS.

> ✔ Your answer "SSS" is correct.

---

4)
- Two angles marked with arcs, and the side between them is shared (common side).
- So, two angles and the included side — this is ASA.
- Answer: Congruent by ASA.

> ✔ Your answer "ASA" is correct.

---

5)
- One side and one angle are marked, but the corresponding parts don’t seem to match in position.
- The triangles appear rotated or reflected, but only one side and one angle are indicated.
- No clear correspondence of two sides and an angle or two angles and a side.
- Answer: Not congruent. (Insufficient information.)

> ✔ Your answer "Not congruent" is correct.

---

6)
- Both triangles are right triangles (right angles marked).
- One leg is marked equal (single tick), and the hypotenuse is also marked equal (double tick).
- This matches the HL (Hypotenuse-Leg) theorem for right triangles.
- Answer: Congruent by HL.

> ✘ You wrote "ASA", but this is actually HL.
> - Note: ASA would require two angles and the included side. Here, we have a right angle, one leg, and hypotenuse — that’s HL.
> ✔ Correct answer: HL

---

7)
- Two sides marked with double ticks, and one angle marked with arc.
- But the angle is not between the two marked sides — it's opposite one of them.
- This is SSA, which does not guarantee congruence.
- Answer: Not congruent.

> ✔ Your answer "Not congruent" is correct.

---

8)
- All three sides are marked with corresponding tick marks (three pairs).
- This is SSS.
- Answer: Congruent by SSS.

> ✔ Your answer "SSS" is correct.

---

9)
- Two sides marked with ticks, and the included angle is marked (angle between them).
- This is SAS (Side-Angle-Side).
- Answer: Congruent by SAS.

> ✘ You wrote "SAS", but the diagram shows:
> - Two sides marked (one with single tick, one with double tick),
> - And the included angle is marked with an arc.
> Yes, this is SAS.
> ✔ Your answer "SAS" is correct.

---

10)
- Two triangles share a diagonal in a parallelogram-like figure.
- All three sides are marked with triple ticks (same markings on both).
- So all three sides are equal → SSS.
- Answer: Congruent by SSS.

> ✔ Your answer "SSS" is correct.

---

✔ Summary of Part 1 (Correct Answers):

| # | Congruent? | Reason |
|---|------------|--------|
| 1 | No | SSA not valid |
| 2 | Yes | ASA |
| 3 | Yes | SSS |
| 4 | Yes | ASA |
| 5 | No | Insufficient info |
| 6 | Yes | HL (not ASA) |
| 7 | No | SSA not valid |
| 8 | Yes | SSS |
| 9 | Yes | SAS |
| 10| Yes | SSS |

> 🔴 Only #6 was incorrect — you said ASA, but it's HL.

---

Part 2: Additional Information Required

Now, let's look at the bottom section:

#### 11) ASA
Given: Triangle $ \triangle SUT $ and $ \triangle DUT $
- Shared side $ UT $
- Right angles at $ U $ and $ T $? Wait — looking at diagram:
- $ \angle SUT $ and $ \angle DUT $ are marked as equal (arcs)
- $ UT $ is common
- Need another angle or side?

Wait — the diagram shows:
- Point $ U $ has a right angle mark
- $ \angle SUT $ and $ \angle DUT $ are marked with arcs — so they are equal?
But both angles are at point $ U $, sharing side $ UT $.

Actually, from the diagram:
- $ \angle SUT $ and $ \angle DUT $ are adjacent angles at point $ U $
- $ UT $ is common
- We already have $ \angle SUT = \angle DUT $ (marked)
- $ UT $ is common
- To use ASA, we need:
- Two angles and the included side

So currently, we have:
- $ \angle SUT = \angle DUT $
- Side $ UT $ is common
- But we need another angle adjacent to $ UT $

For ASA, we need:
- $ \angle SUT = \angle DUT $
- $ UT = UT $
- $ \angle STU = \angle DTU $ (the other angle at $ T $)

But wait — the diagram shows $ \angle STU $ and $ \angle DTU $ are not marked.

Alternatively, perhaps the goal is to make the two triangles congruent via ASA.

So to apply ASA, we need:
- Two angles and the included side

Currently:
- $ \angle SUT = \angle DUT $ (given)
- $ UT $ is common (side between the two angles)
- We need the other angle at $ T $, i.e., $ \angle STU = \angle DTU $

So additional information needed:
👉 $ \angle STU \cong \angle DTU $

Or equivalently: $ \angle STU = \angle DTU $

> ✔ Answer: $ \angle STU \cong \angle DTU $

You wrote: $ \angle SUT \cong \angle DUT $ — but that’s already given (marked in diagram). So that’s not additional.

So your answer is incorrect.

✔ Correct answer: $ \angle STU \cong \angle DTU $

---

#### 12) SAS
Given: Triangles $ \triangle VWX $ and $ \triangle XKY $
- $ VX $ and $ XK $ are marked with tick marks — but not necessarily equal unless labeled
- $ \angle V $ and $ \angle K $ are right angles (marked)
- $ XW $ and $ XY $ are vertical? Or something else?

Wait — from the diagram:
- $ \angle VXW $ and $ \angle KXY $ are right angles
- $ VX $ and $ XK $ are legs
- $ XW $ and $ XY $ are other legs

But for SAS, we need:
- Two sides and the included angle

Currently:
- $ \angle VXW $ and $ \angle KXY $ are both right angles — so equal
- $ VX $ and $ XK $ are marked with single ticks — so $ VX = XK $
- $ XW $ and $ XY $ — are these marked? Not clearly.

But to apply SAS, we need:
- Two sides and the included angle

Suppose we want to use SAS:
- $ VX = XK $ (given)
- $ \angle VXW = \angle KXY $ (both 90°)
- Need: $ XW = XY $ — the other leg

So the additional information needed is:
👉 $ XW = XY $

Or $ \overline{XW} \cong \overline{XY} $

You wrote: $ VW \cong XK $ — but $ VW $ is not even a side in triangle $ XKY $. That doesn't make sense.

Also, $ XK $ is already marked equal to $ VX $, but $ VW $ is not connected.

So your answer is incorrect.

✔ Correct answer: $ XW = XY $ (or $ \overline{XW} \cong \overline{XY} $)

---

✔ Final Answers:

#### Part 1: Are the triangles congruent?

| # | Congruent? | Reason |
|---|------------|--------|
| 1 | No | SSA not valid |
| 2 | Yes | ASA |
| 3 | Yes | SSS |
| 4 | Yes | ASA |
| 5 | No | Insufficient info |
| 6 | Yes | HL (not ASA) |
| 7 | No | SSA not valid |
| 8 | Yes | SSS |
| 9 | Yes | SAS |
| 10| Yes | SSS |

#### Part 2: Additional Information

11) ASA
To use ASA, we need two angles and the included side.
We have:
- $ \angle SUT = \angle DUT $
- $ UT $ common
- Missing: $ \angle STU = \angle DTU $

✔ Additional info: $ \angle STU \cong \angle DTU $

12) SAS
We have:
- $ VX = XK $
- $ \angle VXW = \angle KXY $ (both right angles)
- Need: $ XW = XY $

✔ Additional info: $ \overline{XW} \cong \overline{XY} $

---

✔ Corrections to Your Work:

- #6: Should be HL, not ASA.
- #11: Should be $ \angle STU \cong \angle DTU $, not $ \angle SUT \cong \angle DUT $ (that's already given).
- #12: Should be $ XW = XY $, not $ VW = XK $ (invalid side).

---

Let me know if you'd like a visual explanation or a diagram breakdown!