Triangle congruence theorems practice worksheet with examples and handwritten annotations.
Diagrams illustrating various triangle congruence theorems (SSS, SAS, ASA, AAS, HL) with labeled triangles and corresponding proofs.
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Step-by-step solution for: Geometry Unit 8 Congruent Triangles SSS SAS ASA AAS HL Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Unit 8 Congruent Triangles SSS SAS ASA AAS HL Worksheet
Let’s go through each of the 9 triangles one by one and determine if they are congruent — and if so, which theorem (SSS, SAS, ASA, AAS, HL) proves it.
We’ll look at what parts are marked as equal in each diagram:
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Problem 1:
Two triangles sharing a side. All three sides are marked with tick marks meaning they’re equal. So we have SSS → Congruent by SSS.
✔ Answer: SSS
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Problem 2:
Two right triangles. Both have a right angle, one leg marked equal, and hypotenuse marked equal. That’s HL (Hypotenuse-Leg for right triangles). Also could be seen as SAS if you consider the right angle between the leg and... wait, no — actually, since it’s right triangle + hypotenuse + leg = HL. But some might say SAS if they misread. Let’s check: The right angle is between the two legs? No — here, the marked side is the leg, and the other marked side is the hypotenuse. So yes — HL is correct.
But wait — looking again: In problem 2, both triangles share a vertical side (marked equal), and their horizontal bases are marked equal, and both have right angles. So that’s two sides and included angle → SAS.
Actually, let me re-express: If two sides and the INCLUDED angle are equal → SAS. Here, the right angle is BETWEEN the two marked sides? Yes — because the vertical side and base meet at the right angle. So it’s SAS.
Wait — but also, since it’s a right triangle, and we have leg and hypotenuse? Actually, no — the marked sides are the two legs! Because the diagonal is not marked. Looking carefully: The right angle is at the bottom left for left triangle, bottom right for right triangle. The vertical side is marked equal on both. The horizontal base is marked equal on both. And the right angle is between them. So that’s SAS.
✔ Answer: SAS
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Problem 3:
Two triangles sharing a vertex. Angles at top are marked equal. Side from top to middle point is shared → so that side is equal. Another side on each side is marked equal. So we have two sides and the INCLUDED angle → SAS.
Wait — actually, the angle is at the top, and the two sides going down from the top are marked equal? Or is the side along the base marked?
Looking again: The two outer sides (from top to bottom corners) are marked equal. The angle at the top is marked equal. And the side connecting the two bottom points is NOT necessarily equal — but wait, actually, the side from top to the middle point on the base is drawn — maybe that’s an altitude? Hmm.
Actually, standard interpretation: Two sides and the included angle → SAS.
✔ Answer: SAS
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Problem 4:
Right triangles. One has legs marked 3 and 4, hypotenuse should be 5 (but not labeled). Other triangle has same markings? Wait — actually, both triangles have all three sides marked with ticks — meaning all corresponding sides are equal. So SSS.
Even though one is flipped, if all sides match, it’s SSS.
✔ Answer: SSS
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Problem 5:
Two triangles sharing a side. Angles at the ends of the shared side are marked equal. And the shared side itself is common → so we have two angles and the included side → ASA.
Yes — Angle-Side-Angle.
✔ Answer: ASA
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Problem 6:
Triangle with an altitude drawn. We see two right triangles formed. They share the altitude (so one leg equal). The segments of the base are marked equal (so other leg equal). Right angles are present. So again, two legs and included right angle → SAS.
Alternatively, since it’s right triangles with two legs equal → SAS or even LL (which is same as SAS for right triangles). But standard is SAS.
✔ Answer: SAS
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Problem 7:
Already labeled “SAS” in the image — and indeed, two sides and included angle are marked equal. So confirmed.
✔ Answer: SAS
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Problem 8:
Labeled “HL”. It’s a kite shape made of two right triangles sharing the diagonal. Each triangle has hypotenuse (the outer side) marked equal, and one leg (half the diagonal) shared → so Hypotenuse-Leg → HL.
✔ Answer: HL
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Problem 9:
Labeled “SSS”. Three sides of each triangle are marked with matching ticks → all three sides equal → SSS.
✔ Answer: SSS
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Now, compiling all answers in order:
1. SSS
2. SAS
3. SAS
4. SSS
5. ASA
6. SAS
7. SAS
8. HL
9. SSS
All triangles are congruent, and we’ve assigned the correct theorem for each.
Final Answer:
1. SSS
2. SAS
3. SAS
4. SSS
5. ASA
6. SAS
7. SAS
8. HL
9. SSS
We’ll look at what parts are marked as equal in each diagram:
---
Problem 1:
Two triangles sharing a side. All three sides are marked with tick marks meaning they’re equal. So we have SSS → Congruent by SSS.
✔ Answer: SSS
---
Problem 2:
Two right triangles. Both have a right angle, one leg marked equal, and hypotenuse marked equal. That’s HL (Hypotenuse-Leg for right triangles). Also could be seen as SAS if you consider the right angle between the leg and... wait, no — actually, since it’s right triangle + hypotenuse + leg = HL. But some might say SAS if they misread. Let’s check: The right angle is between the two legs? No — here, the marked side is the leg, and the other marked side is the hypotenuse. So yes — HL is correct.
But wait — looking again: In problem 2, both triangles share a vertical side (marked equal), and their horizontal bases are marked equal, and both have right angles. So that’s two sides and included angle → SAS.
Actually, let me re-express: If two sides and the INCLUDED angle are equal → SAS. Here, the right angle is BETWEEN the two marked sides? Yes — because the vertical side and base meet at the right angle. So it’s SAS.
Wait — but also, since it’s a right triangle, and we have leg and hypotenuse? Actually, no — the marked sides are the two legs! Because the diagonal is not marked. Looking carefully: The right angle is at the bottom left for left triangle, bottom right for right triangle. The vertical side is marked equal on both. The horizontal base is marked equal on both. And the right angle is between them. So that’s SAS.
✔ Answer: SAS
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Problem 3:
Two triangles sharing a vertex. Angles at top are marked equal. Side from top to middle point is shared → so that side is equal. Another side on each side is marked equal. So we have two sides and the INCLUDED angle → SAS.
Wait — actually, the angle is at the top, and the two sides going down from the top are marked equal? Or is the side along the base marked?
Looking again: The two outer sides (from top to bottom corners) are marked equal. The angle at the top is marked equal. And the side connecting the two bottom points is NOT necessarily equal — but wait, actually, the side from top to the middle point on the base is drawn — maybe that’s an altitude? Hmm.
Actually, standard interpretation: Two sides and the included angle → SAS.
✔ Answer: SAS
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Problem 4:
Right triangles. One has legs marked 3 and 4, hypotenuse should be 5 (but not labeled). Other triangle has same markings? Wait — actually, both triangles have all three sides marked with ticks — meaning all corresponding sides are equal. So SSS.
Even though one is flipped, if all sides match, it’s SSS.
✔ Answer: SSS
---
Problem 5:
Two triangles sharing a side. Angles at the ends of the shared side are marked equal. And the shared side itself is common → so we have two angles and the included side → ASA.
Yes — Angle-Side-Angle.
✔ Answer: ASA
---
Problem 6:
Triangle with an altitude drawn. We see two right triangles formed. They share the altitude (so one leg equal). The segments of the base are marked equal (so other leg equal). Right angles are present. So again, two legs and included right angle → SAS.
Alternatively, since it’s right triangles with two legs equal → SAS or even LL (which is same as SAS for right triangles). But standard is SAS.
✔ Answer: SAS
---
Problem 7:
Already labeled “SAS” in the image — and indeed, two sides and included angle are marked equal. So confirmed.
✔ Answer: SAS
---
Problem 8:
Labeled “HL”. It’s a kite shape made of two right triangles sharing the diagonal. Each triangle has hypotenuse (the outer side) marked equal, and one leg (half the diagonal) shared → so Hypotenuse-Leg → HL.
✔ Answer: HL
---
Problem 9:
Labeled “SSS”. Three sides of each triangle are marked with matching ticks → all three sides equal → SSS.
✔ Answer: SSS
---
Now, compiling all answers in order:
1. SSS
2. SAS
3. SAS
4. SSS
5. ASA
6. SAS
7. SAS
8. HL
9. SSS
All triangles are congruent, and we’ve assigned the correct theorem for each.
Final Answer:
1. SSS
2. SAS
3. SAS
4. SSS
5. ASA
6. SAS
7. SAS
8. HL
9. SSS
Parent Tip: Review the logic above to help your child master the concept of asa and aas congruence worksheet.
