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Congruent Triangles - Worksheet 9 - 10th Grade

Hard

Prove triangle congruence using congruence criteria (Hard)

23 problems · 25-30 minutes · Skills: triangles, congruence, proofs

About This Worksheet

This hard-level worksheet contains 23 problems designed for 10th Grade students practicing congruent triangles. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Triangle congruence is the engine behind a huge portion of geometric proof. From proving that diagonals of a parallelogram bisect each other to establishing properties of isosceles triangles, CPCTC and the congruence criteria are used repeatedly throughout the course and in subsequent math courses.

  • Apply SSS, SAS, ASA, AAS, and HL congruence criteria to determine if triangles are congruent
  • Write two-column proofs to prove triangle congruence
  • Use CPCTC to conclude that corresponding parts of congruent triangles are congruent
  • Write correct congruence statements matching corresponding vertices in order
  • Identify which congruence criterion applies given marked diagrams or listed information

How to Solve: Worked Example

Example Problem: Is SSA enough to prove two triangles congruent?
Solution Steps:
  1. SSA gives two sides and a non-included angle.
  2. Two genuinely different triangles can satisfy the same SSA data — the ambiguous case.
  3. So SSA is not a valid congruence criterion.
Answer: No — SSA is not a valid congruence criterion
Tip: SSA and AAA both fail. AAA proves similarity only; SSA can produce two different triangles.
1
△ABC ≅ △DEF and ∠C = 22°. Find ∠F.
2
△ABC ≅ △DEF and ∠C = 106°. Find ∠F.
3
△ABC ≅ △DEF and ∠B = 37°. Find ∠E.
4
△ABC ≅ △DEF and ∠C = 91°. Find ∠F.
5
Is AAA enough to prove two triangles congruent?
6
△ABC ≅ △DEF and ∠B = 115°. Find ∠E.
7
Is SAS enough to prove two triangles congruent?
8
△ABC ≅ △DEF and the perimeter of △ABC is 19. Find the perimeter of △DEF.
9
△ABC ≅ △DEF and the perimeter of △ABC is 49. Find the perimeter of △DEF.
10
△ABC ≅ △DEF and AB = 23. Find DE.
11
△ABC ≅ △DEF and the perimeter of △ABC is 21. Find the perimeter of △DEF.
12
△ABC ≅ △DEF and the perimeter of △ABC is 18. Find the perimeter of △DEF.
13
Is SSA enough to prove two triangles congruent?
14
△ABC ≅ △DEF and AC = 13. Find DF.
15
△ABC ≅ △DEF and ∠C = 31°. Find ∠F.
16
△ABC ≅ △DEF and the perimeter of △ABC is 34. Find the perimeter of △DEF.
17
△ABC ≅ △DEF and the perimeter of △ABC is 41. Find the perimeter of △DEF.
18
△ABC ≅ △DEF and ∠A = 130°. Find ∠D.
19
△ABC ≅ △DEF and the perimeter of △ABC is 26. Find the perimeter of △DEF.
20
△ABC ≅ △DEF and the perimeter of △ABC is 60. Find the perimeter of △DEF.
21
△ABC ≅ △DEF and AB = 35. Find DE.
22
△ABC ≅ △DEF and the perimeter of △ABC is 48. Find the perimeter of △DEF.
23
△ABC ≅ △DEF and the perimeter of △ABC is 69. Find the perimeter of △DEF.

Answer Key

1. 22°2. 106°3. 37°4. 91°5. No — AAA proves similarity, not congruence6. 115°7. Yes — SAS is a valid congruence postulate8. 199. 4910. 2311. 2112. 1813. No — SSA is not a valid congruence criterion14. 1315. 31°16. 3417. 4118. 130°19. 2620. 6021. 3522. 4823. 69

Frequently Asked Questions

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What skills does this triangles worksheet cover?

This worksheet focuses on triangles, congruence, proofs. It is designed for 10th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 23 problems. It typically takes 25-30 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.