Triangle Congruence Analysis: Comparing Three 45-Degree Triangles

Triangle Congruence with Insufficient Information

Which of the triangles are congruent?

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Which of the triangles are congruent?

454545454545454545IIIIII

2

Step-by-step solution

Let's observe the angle in each of the triangles and note that each time it is opposite to the length of a different side.

Therefore, none of the triangles are congruent since it is impossible to know from the data.

3

Final Answer

It is not possible to know based on the data.

Key Points to Remember

Essential concepts to master this topic
  • Congruence Rule: Need matching sides and angles in same position
  • Analysis Method: Check if 45° angles are opposite same relative sides
  • Verification: Identify when given information is insufficient for congruence ✓

Common Mistakes

Avoid these frequent errors
  • Assuming triangles are congruent based on one matching angle
    Don't conclude triangles are congruent just because they each have a 45° angle = wrong assumption! The angle could be in different positions relative to the sides. Always check if corresponding parts (sides and angles in the same relative positions) are equal.

Practice Quiz

Test your knowledge with interactive questions

Look at the triangles in the diagram.

Which of the statements is true?

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FAQ

Everything you need to know about this question

Why aren't these triangles congruent if they all have 45° angles?

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Having the same angle measure doesn't guarantee congruence! The 45° angle might be opposite different sides in each triangle. For congruence, we need corresponding parts to be equal - angles and sides in the same relative positions.

What information would I need to prove these triangles congruent?

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You'd need to know which sides the 45° angles are opposite, plus additional matching sides or angles. For example: SAS (two sides and included angle), ASA (two angles and included side), or SSS (all three sides).

How do I know when there's insufficient information?

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Check if you can apply any congruence theorem (SAS, ASA, AAS, SSS, or HL). If you can't match the required corresponding parts, then the information is insufficient to prove congruence.

Could these triangles still be congruent even though we can't prove it?

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Possibly, but in mathematics we can only conclude what we can prove. Without sufficient given information, the answer is always 'impossible to determine' rather than making assumptions.

What does 'corresponding parts' mean exactly?

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Corresponding parts are sides and angles in the same relative position. For example, if angle A in triangle 1 is the largest angle, it corresponds to the largest angle in triangle 2, not just any 45° angle.

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