Congruent Triangles ABC and DEF: Proving Equilateral Properties

Congruent Triangles with Equilateral Properties

AAABBBCCCDDDEEEFFFΔABC≅ΔDEF ΔABC≅Δ\text{DEF}

The above triangles are equilateral.

Choose the correct answer:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Choose the correct answer
00:03 Sides are equal according to congruence
00:16 Congruent triangles are necessarily similar
00:24 This is the similarity ratio
00:39 Let's substitute the equality of sides in the ratio of sides to find the ratio
00:45 This is the similarity ratio between the triangles
00:50 Congruent triangles are similar triangles with a similarity ratio of 1
00:53 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

AAABBBCCCDDDEEEFFFΔABC≅ΔDEF ΔABC≅Δ\text{DEF}

The above triangles are equilateral.

Choose the correct answer:

2

Step-by-step solution

To solve this problem, let's go through the steps clearly:

  • Step 1: Since triangles △ABC \triangle ABC and △DEF \triangle DEF are equilateral, each side of these triangles is equal to the other sides of the same triangle.
  • Step 2: Moreover, given that △ABC≅△DEF \triangle ABC \cong \triangle DEF , these triangles are congruent, meaning that the corresponding sides and angles are equal. Therefore, side AB=DE AB = DE , BC=EF BC = EF , and CA=FD CA = FD .

Since every side of △ABC \triangle ABC is equal to every corresponding side of △DEF \triangle DEF , the ratio between any corresponding sides of these triangles is:

ABDE=BCEF=CAFD=1 \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = 1

Therefore, all sides have a ratio of 1 1 , which means that the correct statement is that "The ratio between the sides is equal to 1."

Therefore, the correct choice is choice 4.

Hence, the solution to this problem is: The ratio between the sides is equal to 1.

3

Final Answer

The ratio between the sides is equal to 1.

Key Points to Remember

Essential concepts to master this topic
  • Equilateral Rule: All sides equal means every ratio equals 1
  • Congruency: △ABC≅△DEF \triangle ABC ≅ \triangle DEF means corresponding sides are equal
  • Check: Verify ABDE=BCEF=CAFD=1 \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = 1 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing proportional ratios with equal ratios
    Don't assume congruent triangles have ratios other than 1! Many students think congruent means proportional with different ratios = wrong answer. Congruent means exactly the same size and shape. Always remember: congruent triangles have all corresponding parts equal, so ratios equal 1.

Practice Quiz

Test your knowledge with interactive questions

If it is known that both triangles are equilateral, are they therefore similar?

FAQ

Everything you need to know about this question

What's the difference between congruent and similar triangles?

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Congruent triangles are exactly the same size and shape - all corresponding sides and angles are equal. Similar triangles have the same shape but different sizes, with proportional sides.

Why do equilateral triangles make this problem easier?

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In equilateral triangles, all three sides are equal! So if AB=BC=CA AB = BC = CA and the triangles are congruent, then DE=EF=FD DE = EF = FD and all sides match perfectly.

How can I tell if two triangles are congruent?

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Look for these clues: SSS (all sides equal), SAS (two sides and included angle), ASA (two angles and included side), or the problem tells you directly like here!

What if the triangles looked different sizes in the diagram?

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Trust the mathematical information, not just the visual! Diagrams aren't always drawn to scale. The problem states they're congruent, so use that fact regardless of how they appear.

Why isn't the answer one of the fraction ratios?

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Those fraction ratios like ABAC=BCAC \frac{AB}{AC} = \frac{BC}{AC} are comparing sides within the same triangle. But the question asks about ratios between the two triangles, which must equal 1 for congruent figures.

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