Solved CONGRUENT TRIANGLES? SSS -SAS-ASA-AAS - HL (Right | Chegg.com - Free Printable

Let's go through each of the given pairs of triangles and determine whether they are congruent and, if so, by which congruence criterion:
- SSS (Side-Side-Side)
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- HL (Hypotenuse-Leg – for right triangles only)
- Or None, if not enough information or not congruent.

We’ll analyze them one by one based on markings (tick marks for equal sides, arcs for equal angles, right angles, etc.).

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(1)

Two triangles with:
- One angle marked as equal (arc)
- Two sides marked with ticks (equal)

The angle is between the two marked sides → This is SAS.

✔ Answer: SAS

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(2)

Two triangles with:
- One side marked with tick
- One angle marked with arc
- Another side marked with tick

But the angle is not between the two sides. The marked sides are adjacent to the angle, but the configuration suggests two sides and a non-included angle.

This is not SAS — it’s SSA, which is not a valid congruence rule unless it's a right triangle.

Here, no right angle shown → Not sufficient.

✘ Answer: None

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(3)

Two triangles:
- One side marked (tick)
- One angle marked (arc)
- Another side marked (tick)

Again, the angle is between the two sides? Let's check:

- The angle is at the base, and the two sides are the ones forming it → yes, the included angle is between the two sides.

So this is SAS.

✔ Answer: SAS

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(4)

Two triangles:
- One angle marked (arc)
- One side marked (tick)
- Another side marked (tick)

But again, the angle is not between the two sides — it's opposite one of them.

So we have two sides and a non-included angle → SSA, which is not valid.

✘ Answer: None

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(5)

Not visible in image — skip.

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(8)

Two triangles:
- One angle marked (arc)
- One side marked (tick)
- But the second triangle has the same angle and side, but not corresponding?

Wait — let’s look closely:

First triangle: angle at top, side marked on left leg.

Second triangle: angle at bottom, side marked on right leg.

Are these matching?

They seem to have:
- One angle
- One side
- But not corresponding properly

Also, no indication that the other parts match.

But actually, both triangles have:
- A marked angle
- A marked side adjacent to it
- And another side?

Wait — the second triangle has a tick mark on a different side.

But looking more carefully: both have:
- One angle
- One side adjacent to it
- And another side marked?

No clear correspondence.

Actually, both have:
- One angle
- One side
- But the side is not between the angle and another known part.

It looks like one angle and one side, but not enough.

Alternatively, maybe AAS?

Wait — both triangles have:
- One angle (marked)
- One side (marked)
- And another angle?

No — only one angle is marked in each.

So only one angle and one side → insufficient.

✘ Answer: None

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(9)

Two triangles:
- Both have:
- One angle marked (at bottom-left)
- One side marked (adjacent to angle)
- Another angle marked (at top)

So: two angles and a side between them?

Yes! The side is between the two angles → ASA

✔ Answer: ASA

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(10)

Right triangles:
- Both have right angles (square symbol)
- One leg marked (tick)
- Hypotenuse marked (double tick?)

Wait: first triangle has:
- Right angle
- One leg marked (vertical side)
- Hypotenuse has a tick

Second triangle:
- Right angle
- Same leg marked (vertical)
- Hypotenuse also marked

So: right angle, one leg, and hypotenuse → this is HL (Hypotenuse-Leg)

✔ Answer: HL

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(11)

Right triangles:
- Both have right angles
- One leg marked (tick on vertical leg)
- Hypotenuse? Not marked.
- But the other leg is marked?

Wait — only one leg is marked in each, and it's the same leg.

But no hypotenuse marked, and only one side besides the right angle.

So we have: right angle + one leg → not enough for congruence.

Need either:
- HL (hypotenuse and leg), or
- SAS (leg and included angle), but here only one leg is marked.

So insufficient.

✘ Answer: None

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(15)

Parallelogram divided by diagonal:
- Opposite sides marked equal (ticks)
- Diagonal splits it into two triangles

We need to see if the two triangles formed are congruent.

In a parallelogram:
- Opposite sides are equal
- Diagonal is common to both triangles

So:
- Side AB = CD (given by ticks)
- Side AD = BC (given)
- Diagonal AC is common

So three sides equal → SSS

✔ Answer: SSS

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(16)

Parallelogram with diagonal and:
- One angle marked
- One side marked

Triangles formed:
- Shared diagonal
- One angle marked in each triangle
- One side marked

But not enough to conclude congruence.

Wait — the diagram shows:
- One angle marked in each triangle (same angle?)
- One side marked (tick)

But not clearly matching.

Actually, in parallelogram:
- Opposite sides equal
- Diagonal common
- Angles may be equal due to parallel lines

But here only one side and one angle marked.

Without more info, cannot confirm.

But wait — if it's a parallelogram, then:
- Triangles share diagonal
- Two sides of each triangle are sides of parallelogram → equal
- So SSS?

Wait — but the markings show only one side marked with a tick.

So perhaps only one side is marked.

But in a parallelogram, opposite sides are equal → so even without marking, we know.

But the problem relies on markings.

Since only one side is marked, and one angle, not enough.

But wait — the angle is at the vertex where two sides meet.

If we assume the parallelogram properties, then:
- Two sides are equal (opposite sides)
- Diagonal is common
- So SSS applies

But since markings don’t show all three, we must rely on what’s shown.

Only one side and one angle marked → not enough.

But wait — the diagonal is shared, and the sides are marked?

Look: one side is marked with tick, and the other triangle has same side marked.

But the other sides are not marked.

Still, in a parallelogram, opposite sides are equal, so we can infer.

But do we assume that?

In such problems, we use only the markings unless geometry rules apply.

But usually, in these diagrams, if it's a parallelogram, we can use properties.

But here, the figure is drawn as a parallelogram, and one side is marked.

Wait — both triangles share the diagonal.

And the parallelogram has:
- AB = CD (opposite sides)
- AD = BC
- Diagonal AC common

So triangles ABC and CDA?

Wait — actually, the diagonal divides it into two triangles: say ΔABC and ΔCDA.

Then:
- AB = CD (opposite sides)
- BC = DA
- AC = AC (common)

So SSS → congruent.

Even though only one side is marked, the shape implies it.

But in standard test questions, markings are key.

Here, only one side is marked with tick, and the angle is marked.

But the diagonal is not marked, but it's common.

But the other sides are not marked.

So unless we assume parallelogram properties, we can't say.

But typically, when a figure is drawn as a parallelogram with tick marks on one pair of opposite sides, we assume the rest.

But here, only one side is marked with a tick, and the other triangle has the same side marked.

Wait — the marking shows one side in each triangle — but are they corresponding?

Actually, the diagram shows:
- One side marked in left triangle
- Same side marked in right triangle

But not necessarily the same length.

But in a parallelogram, opposite sides are equal.

So if one side is marked, and the figure is a parallelogram, then the opposite side is equal.

But the markings are only on one side.

So we have:
- One side marked (say AB)
- Then CD is equal (by parallelogram)
- But CD is not marked
- Diagonal is common
- Other sides: AD and BC — not marked

But since it's a parallelogram, AD = BC

So all three sides are equal → SSS

So even though only one side is explicitly marked, the figure implies the rest.

Thus, SSS

✔ Answer: SSS

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(17)

Parallelogram with diagonal and arrows on the diagonal.

Arrows suggest direction — possibly indicating that the diagonal is bisected?

But arrows point in opposite directions — could mean that the diagonal is split into two equal parts?

So if the diagonal is bisected, then:
- Two segments of diagonal are equal
- Also, opposite sides of parallelogram are equal
- And the diagonal is common

But wait — the arrows are on the diagonal, pointing toward center — suggesting equal segments

So:
- Diagonal is split into two equal parts
- Opposite sides are equal (parallelogram)
- So triangles formed by diagonal have:
- Two sides from parallelogram (equal)
- Diagonal split equally

So:
- Side AB = CD
- Side AD = BC
- Diagonal AC split into AE = EC

But the two triangles are: ΔABE and ΔCDE? No — the diagonal connects opposite corners.

Say diagonal from A to C, and E is midpoint.

Then triangles: ΔABC and ΔADC?

Wait — no — the diagonal is AC, and it's bisected at E.

Then triangles are ΔABE and ΔCBE? Not matching.

Actually, the diagonal divides the parallelogram into two triangles: ΔABC and ΔADC.

But if the diagonal is bisected, that doesn’t help unless we have more.

Wait — the arrows are on the diagonal, showing equal segments.

So if the diagonal is split into two equal parts, then:

But the two triangles formed by the diagonal are:
- ΔABC and ΔADC

But they share the diagonal AC.

But now, if the diagonal is bisected, that means point E is midpoint, but the two triangles are still ΔABC and ΔADC — they include the full diagonal.

So the entire diagonal is common.

But the arrows indicate that the two halves are equal — so the diagonal is bisected.

But that doesn’t directly help unless we have symmetry.

Wait — actually, in a parallelogram, the diagonals bisect each other, but here only one diagonal is shown.

But if the diagonal is bisected, and we have the parallelogram, then:

But we need to see the triangles.

The two triangles are:
- Left triangle: formed by diagonal
- Right triangle: same

With diagonal bisected — but the triangles share the whole diagonal.

So unless we have more, it's unclear.

But wait — the diagram shows arrows on the diagonal, pointing toward the center — suggesting the two parts are equal.

But the two triangles are symmetric across the diagonal?

No — the diagonal is common.

But the two triangles are:
- Triangle 1: vertices A, B, C
- Triangle 2: vertices D, C, A

Wait — actually, the diagonal is from A to C.

So triangles are ΔABC and ΔADC.

Now, in parallelogram:
- AB = DC
- AD = BC
- AC = AC

So SSS → congruent.

But the arrows suggest the diagonal is bisected — but that's not needed.

So again, SSS applies.

But the arrows might be indicating something else.

Alternatively, maybe the arrows show that the diagonal is split equally, but the triangles are not necessarily congruent unless we know more.

But in a parallelogram, the two triangles formed by a diagonal are always congruent by SSS or ASA.

So regardless of arrows, they should be congruent.

But the arrows might be emphasizing that the diagonal is bisected — but that's not necessary for congruence.

So probably, SSS or ASA.

Let’s see:
- AB = DC (opposite sides)
- AD = BC
- AC = AC

→ SSS

✔ Answer: SSS

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(18)

Parallelogram with diagonal and arrows on the sides.

Arrows on the two opposite sides — pointing in same direction?

Wait — the arrows are on the top and bottom sides, pointing right.

So likely indicating parallel and equal.

But in a parallelogram, opposite sides are equal and parallel.

So again, diagonal divides into two triangles.

Same as above: triangles formed by diagonal are congruent by SSS or ASA.

So:
- AB = CD
- AD = BC
- AC = AC

→ SSS

✔ Answer: SSS

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Summary of Answers:

| Number | Congruence Criterion |
|--------|----------------------|
| (1) | SAS |
| (2) | None |
| (3) | SAS |
| (4) | None |
| (8) | None |
| (9) | ASA |
| (10) | HL |
| (11) | None |
| (15) | SSS |
| (16) | SSS |
| (17) | SSS |
| (18) | SSS |

Note: Some problems like (5), (6), (7), (12)-(14) are missing from image — so skipped.

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Final Answers:

1. SAS
2. None
3. SAS
4. None
8. None
9. ASA
10. HL
11. None
15. SSS
16. SSS
17. SSS
18. SSS

Let me know if you want explanations for any specific ones!