Analyzing Similar Triangles: Compare 34° Angles and 4-5 Unit Sides

Triangle Congruence with SAS Theorem

Look at the triangles in the diagram.

Determine which of the statements is correct.

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the triangles in the diagram.

Determine which of the statements is correct.

343434343434555444444555AAABBBCCCDDDEEEFFF

2

Step-by-step solution

Let's consider that:

AC=EF=4

DF=AB=5

Since 5 is greater than 4 and the angle equal to 34 is opposite the larger side in both triangles, the angle ACB must be equal to the angle DEF

Therefore, the triangles are congruent according to the SAS theorem, as a result of this all angles and sides are congruent, and all answers are correct.

3

Final Answer

All of the above.

Key Points to Remember

Essential concepts to master this topic
  • SAS Rule: Two sides and included angle determine congruence
  • Technique: Match sides AC = EF = 4, AB = DF = 5, angle C = angle E = 34°
  • Check: When triangles are congruent by SAS, all corresponding parts are equal ✓

Common Mistakes

Avoid these frequent errors
  • Not identifying the included angle correctly
    Don't assume any angle works with any two sides = wrong congruence! The 34° angle must be between the two known sides (4 and 5). Always verify the angle is included between the matching sides.

Practice Quiz

Test your knowledge with interactive questions

Determine whether the triangles DCE and ABE congruent?

If so, according to which congruence theorem?

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FAQ

Everything you need to know about this question

How do I know which angle is the included angle?

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The included angle is the angle that sits between the two sides you're comparing. In triangle ABC, the 34° angle at C is between sides AC (4) and CB. Match this pattern in the other triangle.

Why does SAS prove all parts are congruent?

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When two triangles are congruent by SAS, they are identical in shape and size. This means all corresponding sides and angles must be equal, not just the ones used to prove congruence.

What if the sides are in different positions?

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That's okay! You need to identify corresponding parts. AC corresponds to EF (both equal 4), AB corresponds to DF (both equal 5), and angle C corresponds to angle E (both 34°).

Could these triangles be similar instead of congruent?

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Since the actual side lengths are equal (4 = 4 and 5 = 5), not just proportional, the triangles are congruent, which is stronger than just being similar.

How do I remember what SAS stands for?

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S-A-S means Side-Angle-Side. The angle must be the included angle between the two sides. Think of it as a sandwich: the angle is the filling between two side pieces!

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