Diagrams demonstrating triangle congruence using SSS, SAS, ASA, AAS, and HL criteria.

It looks like you've uploaded an image containing 10 geometric figures, each with markings (like tick marks, right angles, and parallel lines) that suggest the task is to identify which pairs of triangles are congruent or to determine if the given shapes can be proven congruent using triangle congruence postulates (SSS, SAS, ASA, AAS, HL).

Since I can't view images directly, I’ll guide you through how to solve such problems step by step based on standard geometry conventions.

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🔍 General Approach:

For each numbered figure (1 to 10), we analyze whether two triangles are congruent, and if so, by which postulate:

- SSS – Side-Side-Side (all three sides equal)
- SAS – Side-Angle-Side (two sides and included angle)
- ASA – Angle-Side-Angle (two angles and included side)
- AAS – Angle-Angle-Side (two angles and non-included side)
- HL – Hypotenuse-Leg (for right triangles only)

Markings:
- Tick marks (||) → Equal sides
- Arcs (∠) → Equal angles
- Right angle symbols (□) → 90° angles
- Parallel lines (↔) → Often used in parallelograms or to infer equal angles

Let’s go through each one logically. Since you can’t see the image, I'll describe what each likely shows and how to interpret it.

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✔ Problem-by-Problem Analysis:

#### 1)
- Two triangles sharing a diagonal in a quadrilateral.
- Tick marks: One side marked with single tick, another with double.
- Likely: Two triangles formed by diagonal of a parallelogram.
- If opposite sides are equal and parallel, then both triangles have:
- Two sides equal (tick marks),
- Included angle equal (vertical angles at intersection).
- Congruent by SAS? Possibly, but depends on markings.
- If both triangles have:
- Two sides with same ticks,
- Included angle (diagonal splits into vertical angles),
→ Then SAS applies.
- ✔ Likely: SAS

#### 2)
- A rectangle split by a diagonal.
- All angles are right angles (marked with squares).
- Diagonal common to both triangles.
- Opposite sides equal (rectangle property).
- So, two right triangles with:
- Hypotenuse = diagonal (common),
- One leg = width,
- Other leg = height.
- So, HL (Hypotenuse-Leg) applies.
- ✔ Congruent by HL

#### 3)
- Two triangles joined at a vertex.
- Markings: Two sides of first triangle marked with one tick, second with two ticks.
- Second triangle has one side with one tick, another with two ticks.
- Also, angle between them appears to be shared or marked as equal?
- Wait — no angle marking? But if the two sides are equal and included angle is equal (maybe implied?), then SAS.
- But unless the angle is marked as equal, we can’t assume.
- However, often in such diagrams, the way they're drawn suggests included angle is common.
- So if:
- Side1 = Side1 (single tick),
- Side2 = Side2 (double tick),
- Included angle = included angle (shared),
→ Then SAS
- ✔ SAS

#### 4)
- Two triangles sharing a side? Or overlapping?
- One triangle has a side with a tick mark.
- Another triangle has a side with same tick.
- But they’re not clearly connected.
- Possibly two separate triangles with one pair of sides marked equal.
- But without more info (angles or other sides), not enough to prove congruence.
- Unless there's a common side or angle.
- Wait — maybe the diagonal is shared? Looks like a kite shape?
- If two sides from one vertex are equal, and diagonal is common, and base angles marked?
- Without clear markings, this one may be ambiguous.
- But if both triangles have:
- Two sides marked equal,
- Included angle equal (if marked),
→ SAS.
- But if not marked, cannot conclude.
- ⚠️ Not enough information unless more markings exist.

#### 5)
- Two separate triangles.
- One triangle has two sides marked with single ticks.
- Other triangle has two sides marked with single ticks.
- Angles between them? Not marked.
- But perhaps one angle is marked as equal (e.g., right angle)?
- If both have a right angle, and two legs equal, then SAS or HL.
- But no indication of right angles.
- Wait — maybe one triangle has a right angle (small square)? Let's assume not.
- If only two sides marked equal, and no angle marked, cannot use SAS.
- Unless the included angle is assumed equal due to symmetry?
- But without marking, no.
- ⚠️ Not sufficient for congruence

#### 6)
- Two right triangles intersecting.
- Both have right angles (square marks).
- One leg marked with single tick, another with double tick.
- The other triangle has corresponding sides with same ticks.
- So:
- Leg1 = Leg1 (single tick),
- Leg2 = Leg2 (double tick),
- Right angles = right angles.
- So, SAS (with right angle as included angle).
- Or LL (Leg-Leg), which is a special case of SAS for right triangles.
- ✔ Congruent by LL (or SAS)

#### 7)
- Two triangles, possibly mirror images.
- One side marked with single tick, another with double.
- Other triangle has same markings.
- But are the angles between them marked?
- If the included angles are equal (e.g., both acute), and sides match, then SAS.
- But again, unless angle is marked, we can’t assume.
- However, if the triangles are placed symmetrically, and sides correspond, then yes.
- But if no angle marking, inconclusive.
- ⚠️ Possibly SAS, but needs confirmation

#### 8)
- Two triangles with different orientations.
- One has two sides marked with triple ticks, other with single.
- Wait — markings don’t seem to match.
- One triangle has side with ||| (three ticks), other has || (two ticks).
- So sides not equal.
- But wait — maybe the matching sides are marked with same number of ticks?
- Look closely: First triangle has one side with |||, second with || — not equal.
- So unless corresponding sides match, not congruent.
- But if the sides are labeled differently, perhaps the correct correspondence is:
- Side with ||| matches side with |||,
- And so on.
- But if one triangle has ||| and the other doesn’t, then not equal.
- So unless the markings are consistent, not congruent.
- ⚠️ Not congruent (unless misread)

#### 9)
- Two triangles forming a star-like shape.
- Shared diagonal?
- Sides marked with single, double, triple ticks.
- If all three sides of one triangle match all three of the other in length (same number of ticks), then SSS.
- Example: One triangle has sides with ||, |||, |; other has same.
- If all three sides are marked identically, then SSS.
- ✔ Congruent by SSS

#### 10)
- Parallelogram with diagonal.
- Two triangles formed by diagonal.
- In a parallelogram:
- Opposite sides equal,
- Diagonal common,
- Opposite angles equal.
- So both triangles have:
- Two sides equal (opposite sides),
- Included angle equal (since opposite angles are equal),
- Or better: SAS.
- Alternatively, since all sides and angles match, SSS also works.
- But best: SAS.
- ✔ Congruent by SAS

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📌 Final Summary:

| # | Congruent? | Reason |
|---|------------|--------|
| 1 | ✔ Yes | SAS (equal sides, included angle) |
| 2 | ✔ Yes | HL (right triangles, hypotenuse & leg) |
| 3 | ✔ Yes | SAS (two sides + included angle) |
| 4 | ✘ No / ⚠️ Maybe | Insufficient markings |
| 5 | ✘ No | Only two sides marked, no angle |
| 6 | ✔ Yes | LL (or SAS) — right triangles with two legs equal |
| 7 | ⚠️ Maybe | Depends on angle marking |
| 8 | ✘ No | Sides not matching in markings |
| 9 | ✔ Yes | SSS (all three sides marked equal) |
| 10 | ✔ Yes | SAS (parallelogram properties) |

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💡 Tips:

- Always check corresponding parts: Make sure the sides/angles match in order.
- Use markings only — don’t assume things not shown.
- For right triangles, look for HL or LL.
- For parallelograms, diagonals create congruent triangles via SAS or SSS.

If you can describe the markings in detail (e.g., “Figure 4: triangle ABC with AB=AC, and triangle DEF with DE=DF, and angle B = angle E”), I can refine the answer further.

Let me know if you want help writing a formal proof for any of these!