Triangle congruence interactive worksheet - Free Printable

Let's solve each problem on the Triangle Congruence Worksheet by identifying which congruence postulate or theorem applies to each pair of triangles. The options are:

- SSS (Side-Side-Side) – All three sides are equal.
- SAS (Side-Angle-Side) – Two sides and the included angle are equal.
- 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.
- Info not enough (5) – Not enough information to determine congruence.

We’ll go through each problem one by one, based on the markings in the diagrams.

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1. Triangle FGI and HGI

- Markings:
- FG = HG (marked with one tick)
- FI = HI (marked with one tick)
- GI is common side
- So, all three sides are equal: SSS
✔ Answer: 1 (SSS)

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2. Triangles NOP and QOP

- Diagram shows:
- NO = QP (one tick)
- NP = QO (one tick)
- OP is common
- But note: This looks like two triangles sharing side OP, but we don't have info about angles.
- However, if it’s a kite or rhombus-like shape, maybe symmetry?
- Wait: Let's look closely — the figure appears to be a quadrilateral with diagonals? Actually, it's two triangles sharing side OP.
- Markings:
- NO = QP
- NP = QO
- OP = OP (common)
- So, three sides: SSS
✔ Answer: 1 (SSS)

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3. Triangles ABC and EDC

- Diagonals AC and BD intersect at C.
- Markings:
- AB = ED (not marked directly, but angles at B and D are right angles)
- Angles at B and D are both right angles (marked)
- BC = DC (one tick)
- AC and EC are parts of same line?
- Wait: Points A-B-C and E-D-C, with diagonals crossing at C.
- We see:
- ∠B = ∠D = 90°
- BC = DC (one tick)
- AC = EC? Not marked.
- But also, vertical angles at C: ∠ACB = ∠ECD (vertical angles)
- So we have:
- ∠B = ∠D (right angles)
- ∠ACB = ∠ECD (vertical angles)
- BC = DC (side)
- So: Two angles and included side → ASA
✔ Answer: 3 (ASA)

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4. Triangles RSU and TSU

- Square or rectangle? Right angles at S and U.
- Markings:
- ∠S = ∠U = 90°
- SU is common
- RU = TU? No marking.
- But wait: RT is diagonal, so triangle RSU and TSU share SU.
- From diagram:
- ∠R = ∠T = 90°? Wait, only S and U are marked as right angles.
- Actually, ∠S and ∠U are both 90°
- SU is common
- But what about other sides?
- Look: RS and TU are opposite sides; if it’s a rectangle, they are equal.
- But no tick marks on RS and TU.
- However, we can see:
- ∠S = ∠U = 90°
- SU = SU (common)
- ∠RSU = ∠TSU? No, not marked.
- Wait: There's a diagonal RU and TU?
- Actually, triangle RSU and TSU:
- SU is common
- ∠S = ∠U = 90°
- But no side between them?
- Alternatively: Maybe we need to consider the diagonal ST?
- Wait: It's a square/rectangle with diagonal from R to U and T to S?
- Actually, points: R-S-T-U form a rectangle? Then diagonal RT and SU?
- But here, we have triangle RSU and TSU.
- So:
- RS and TU: likely equal (opposite sides), but not marked
- SU common
- ∠S and ∠U: both 90°
- But we don’t know if RS = TU or if angles at R and T are equal
- But we do have:
- ∠S = ∠U = 90°
- SU = SU (common)
- And RU and TU? No.
- Wait: In this figure, we have:
- ∠S = ∠U = 90°
- SU = SU (common)
- But unless we know another side or angle, we can't conclude.
- But notice: RU and TU are not marked, but RS and TU might be equal.
- Actually, looking again: Only SU is shared, and two right angles.
- But we need more.
- Wait: Is there a mark on RU and TU? No.
- But perhaps the triangles are RST and TUS? No, labeled as RSU and TSU.
- Let’s reconsider: Triangle RSU and triangle TSU:
- Both have SU
- ∠S = ∠U = 90°
- But no indication that RS = TU or RU = TS
- Unless it's a square, we can't assume.
- But wait: the figure has RS and TU as horizontal sides — likely equal.
- But without tick marks, we can’t assume.
- However, in many such problems, when you have a rectangle with diagonal, the two triangles formed are congruent by SAS or SSS.
- But here, diagonal is SU? Wait, no — diagonal should be RT or SU?
- Actually, if it's a rectangle RSTU, then diagonal is RU or ST?
- Wait: Points: R-S-T-U, so order is R→S→T→U→R.
- So diagonal would be RT or SU.
- Here, SU is drawn, so triangle RSU and TSU.
- So:
- SU is common
- ∠S = ∠U = 90°
- RS and TU: opposite sides — in rectangle, they are equal
- But not marked
- Without tick marks, we can’t use SSS or SAS unless shown.
- But wait: the diagram shows tick marks on RS and TU? Let's check:
- In image: RS and TU both have one tick mark? Yes! Look carefully:
- RS has one tick
- TU has one tick
- So RS = TU
- Also, SU is common
- ∠S and ∠U are both 90°
- So: Two sides and included angle?
- In triangle RSU: RS, SU, ∠S
- In triangle TSU: TU, SU, ∠U
- RS = TU, SU = SU, ∠S = ∠U = 90°
- So: SAS (two sides and included angle)
✔ Answer: 2 (SAS)

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5. Triangles JKM and LMK

- Quadrilateral JKML with diagonal JM.
- Markings:
- ∠J = ∠L (both marked with arcs)
- ∠K = ∠M (both marked)
- JM is common
- So:
- ∠J = ∠L
- ∠K = ∠M
- JM = JM
- So: Two angles and a non-included side → AAS
✔ Answer: 4 (AAS)

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6. Triangles NQO and PQO

- Diamond-shaped figure, probably a rhombus.
- Markings:
- NQ = PQ (one tick)
- NO = PO (one tick)
- OQ is common
- So: Three sides equal → SSS
✔ Answer: 1 (SSS)

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7. Triangles ABC and EDC

- Similar to #3
- Intersecting diagonals at C
- Markings:
- AB = ED (one tick)
- BC = DC (one tick)
- ∠ABC = ∠EDC (both right angles)
- So:
- ∠B = ∠D = 90°
- BC = DC
- AB = ED
- So: Two sides and included angle? Yes: AB = ED, ∠B = ∠D, BC = DC → SAS
✔ Answer: 2 (SAS)

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8. Triangles FGI and HGI

- Same as #1
- FG = HG (tick)
- FI = HI (tick)
- GI common
- So: SSS
✔ Answer: 1 (SSS)

Wait — this is identical to #1. Probably a typo in labeling? Or different triangles?

Wait: #1 was FGI and HGI — same as this. So yes, same triangle pair.

But in #1, we already did this.

So #8 is same as #1 → SSS

✔ Answer: 1 (SSS)

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9. Triangles JKM and LMK

- Rectangle with diagonal KM
- Markings:
- JK = LM (one tick)
- JM = LK? Not marked
- But KM is common
- Also, ∠J = ∠L = 90° (assumed)
- But wait: only JK and LM are marked as equal
- Also, JM and LK not marked
- But in rectangle, opposite sides are equal, but not marked
- However, the diagram shows:
- JK = LM (one tick)
- JM = LK? Not marked
- KM = KM (common)
- But no angles marked except possibly at K and M?
- Wait: At K and M, we see angle marks? No.
- But in rectangle, angles are 90°, but not marked.
- But look: the figure has ticks on JK and LM (equal), and also on JM and LK? No — only JK and LM have ticks.
- So only one pair of sides equal.
- But we have:
- JK = LM
- KM = KM
- But no other sides or angles marked
- So insufficient?
- Wait: Is it a parallelogram? Yes, rectangle.
- But without more markings, we cannot conclude.
- But actually, in rectangle, opposite sides are equal, so:
- JK = LM (given)
- JM = LK (implied, but not marked)
- But since not marked, we can’t use it.
- However, if we assume it’s a rectangle, then:
- ∠J = ∠L = 90°
- JK = LM
- JM = LK
- KM = KM
- But again, no tick marks on JM and LK.
- But look: are there any angle markings?
- In the diagram, at J and L, are there right angles? Yes — both marked with squares.
- So ∠J = ∠L = 90°
- Also, JK = LM (one tick)
- KM = KM (common)
- But we don’t have the third side or angle.
- So: ∠J = ∠L, JK = LM, KM = KM → SAS? Only if the included angle is between JK and KM, and LM and KM.
- Yes: in triangle JKM: sides JK and KM, angle at K
- In triangle LMK: sides LM and KM, angle at M
- But we don’t know if ∠K = ∠M
- We know ∠J = ∠L, but not necessarily ∠K = ∠M
- So we have:
- ∠J = ∠L
- JK = LM
- KM = KM
- This is AAS? Because two angles and a non-included side?
- Wait: we only have one pair of angles (∠J = ∠L), and one side (JK = LM), and KM = KM
- But we don’t know if ∠K = ∠M
- So unless we know another angle, we can't use AAS.
- But in rectangle, adjacent angles are supplementary, but not necessarily equal.
- However, in rectangle, all angles are 90°, so ∠K = ∠M = 90°?
- Wait: at point K and M, are they right angles? Not marked.
- Only J and L are marked as right angles.
- So we cannot assume.
- Therefore, only known:
- ∠J = ∠L = 90°
- JK = LM
- KM = KM
- So: SAS? No — because the angle at J is not between JK and KM? Wait:
- In triangle JKM: sides JK and KM, angle at K
- But we don’t know angle at K
- We know angle at J
- So we have:
- Side JK
- Angle at J
- Side JM? Not given
- So it’s ASA? No — we don’t have two angles
- Wait: we have:
- ∠J = ∠L
- JK = LM
- KM = KM
- This is SAS only if the included angle is between JK and KM, but we don’t know angle at K
- So we cannot apply SAS or ASA
- But if it’s a rectangle, then:
- ∠K = ∠M = 90°? But not marked
- So unless the markings show it, we can’t assume
- But in the diagram, only ∠J and ∠L are marked as right angles
- So we have:
- One pair of equal sides: JK = LM
- One pair of equal angles: ∠J = ∠L
- Common side KM
- But not enough to prove congruence
- Wait: actually, in rectangle, opposite sides are equal, and angles are 90°, so:
- ∠J = ∠L = 90°
- JK = LM
- JM = LK (but not marked)
- KM = KM
- But still, we don’t have enough marked elements
- But perhaps the diagram implies symmetry
- But let's look: is there a tick on JM and LK? No
- So only JK = LM, and ∠J = ∠L, KM common
- This is AAS? Only if we had two angles and a side
- We have one angle, one side, and a common side
- But the side KM is not opposite to the angle
- So we have:
- ∠J = ∠L
- JK = LM
- KM = KM
- This is SAS only if the angle is between the two sides
- But angle at J is between JK and JM, not between JK and KM
- So not SAS
- So we cannot conclude
- But wait: perhaps it's HL for right triangles?
- But HL is not listed — only SSS, SAS, ASA, AAS, Info not enough
- Since both triangles are right triangles (if ∠J and ∠L are right angles), and hypotenuse KM is common, and leg JK = LM, then HL applies
- But HL is not an option — closest is AAS or SAS
- But HL is equivalent to AAS in right triangles
- So we can say AAS?
- But AAS requires two angles and a side
- In right triangle JKM: ∠J = 90°, ∠K, ∠M
- In LMK: ∠L = 90°, ∠M, ∠K
- But we don’t know if ∠K = ∠M
- But in rectangle, opposite angles are equal, so ∠K = ∠M
- But not marked
- So unless we can deduce, we can't
- Given the markings, only ∠J = ∠L and JK = LM, and KM = KM
- This is AAS if we consider:
- ∠J = ∠L
- ∠K = ∠M (from geometry)
- JK = LM
- But since ∠K and ∠M are not marked, we can't use it
- So Info not enough
✘ Answer: 5 (Info not enough)

Wait — but in many textbooks, if it's a rectangle with diagonal, the two triangles are congruent by SAS or AAS, but here the markings are minimal.

But let's recheck: are there any angle marks at K or M? No.

Only ∠J and ∠L are marked as right angles.

And only JK and LM are marked equal.

So we have:
- ∠J = ∠L (90°)
- JK = LM
- KM = KM

But this is not sufficient for any standard postulate unless we know the angle at K or M.

So Info not enough

✔ Answer: 5 (Info not enough)

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10. Triangles NOP and QOP

- Quadrilateral NOPQ with diagonal OP
- Markings:
- ∠N = ∠Q (arcs)
- ∠P = ∠P (common)
- OP = OP (common)
- So:
- ∠N = ∠Q
- ∠P = ∠P
- OP = OP
- So: Two angles and a non-included side → AAS
✔ Answer: 4 (AAS)

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11. Triangles JKL and MNL

- Rectangle with diagonal JL
- Markings:
- JK = ML (one tick)
- KL = NL (one tick)
- JL = JL (common)
- So: three sides equal → SSS
✔ Answer: 1 (SSS)

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12. Triangles SRU and TSU

- Rectangle with diagonal SU
- Markings:
- RS = TU (one tick)
- RU = TS (one tick)
- SU = SU (common)
- So: three sides equal → SSS
✔ Answer: 1 (SSS)

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Final Answers:

| # | Answer |
|---|--------|
| 1 | 1 (SSS) |
| 2 | 1 (SSS) |
| 3 | 3 (ASA) |
| 4 | 2 (SAS) |
| 5 | 4 (AAS) |
| 6 | 1 (SSS) |
| 7 | 2 (SAS) |
| 8 | 1 (SSS) |
| 9 | 5 (Info not enough) |
|10 | 4 (AAS) |
|11 | 1 (SSS) |
|12 | 1 (SSS) |

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Summary of Answers:

1. 1
2. 1
3. 3
4. 2
5. 4
6. 1
7. 2
8. 1
9. 5
10. 4
11. 1
12. 1

Let me know if you'd like a printed version or explanation for any specific one!