Triangles ADE and ABC are congruent.Choose the correct answer.AAABBBCCCDDDEEE

$ \frac{AB}{AD}=\frac{AC}{AE}=\frac{DE}{BC} $

$ \frac{BC}{DB}=\frac{AB}{BE}=\frac{AC}{ED} $

$ \frac{AD}{A}=\frac{AE}{AC}=\frac{DE}{BA} $

Congruent Triangles ADE and ABC: Analyzing Geometric Properties

Q: What's the difference between congruent and similar triangles?

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

Q: How do I identify corresponding sides in congruent triangles?

Match the vertex letters in the same positions! In triangles ADE and ABC, vertex A corresponds to A, D corresponds to B, and E corresponds to C. So AD↔AB, AE↔AC, and DE↔BC.

Q: Why are all the ratios equal to each other?

Since the triangles are congruent, corresponding sides are equal: AD=AB, AE=AC, DE=BC. When you divide equal quantities, you get: $ \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} = 1 $

Q: What if the triangles were just similar instead of congruent?

For similar triangles, the ratios would still be equal to each other, but they wouldn't equal 1. For example: $ \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} = \frac{1}{2} $ if triangle ADE is half the size of ABC.

Q: How can I remember which ratio goes in the numerator?

Think of the naming order! Triangle ADE is named first, so its sides (AD, AE, DE) go in the numerator. Triangle ABC sides (AB, AC, BC) go in the denominator.

Congruent Triangles with Proportional Relationships

Triangles ADE and ABC are congruent.

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 appropriate answer

00:03 The triangles are similar according to the given data

00:09 According to the similarity ratio, we have pairs of corresponding sides

00:14 We take corresponding pairs of sides from each triangle

00:23 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Triangles ADE and ABC are congruent.

Choose the correct answer.

Step-by-step solution

To find the correct proportionality relationship between the triangles ADE and ABC, we need to identify the corresponding sides:

In triangle ADE, the sides $AD$ , $AE$ , and $DE$ correspond to triangle ABC’s sides $AB$ , $AC$ , and $BC$ , respectively.

Because these triangles are congruent, the ratios of their corresponding sides are equal:

$\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$

This relationship matches with choice 1:

$\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$

Therefore, the correct answer is choice 1.

Final Answer

$\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$

Key Points to Remember

Essential concepts to master this topic

Congruent Triangles: All corresponding sides and angles are equal
Corresponding Sides: Match vertices: AD↔AB, AE↔AC, DE↔BC
Verification: Check that ratios equal 1 for congruent triangles ✓

Common Mistakes

Avoid these frequent errors

Confusing congruent with similar triangles
Don't write ratios like AD/AB = 1/2 for congruent triangles = wrong proportions! Congruent means identical size and shape, so corresponding sides are equal. Always remember that for congruent triangles, the ratio of corresponding sides equals 1.

Practice Quiz

Test your knowledge with interactive questions

Is the similarity ratio between the three triangles equal to one?

FAQ

Everything you need to know about this question

What's the difference between congruent and similar triangles?

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

How do I identify corresponding sides in congruent triangles?

Match the vertex letters in the same positions! In triangles ADE and ABC, vertex A corresponds to A, D corresponds to B, and E corresponds to C. So AD↔AB, AE↔AC, and DE↔BC.

Why are all the ratios equal to each other?

Since the triangles are congruent, corresponding sides are equal: AD=AB, AE=AC, DE=BC. When you divide equal quantities, you get: $\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} = 1$

What if the triangles were just similar instead of congruent?

For similar triangles, the ratios would still be equal to each other, but they wouldn't equal 1. For example: $\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} = \frac{1}{2}$ if triangle ADE is half the size of ABC.

How can I remember which ratio goes in the numerator?

Think of the naming order! Triangle ADE is named first, so its sides (AD, AE, DE) go in the numerator. Triangle ABC sides (AB, AC, BC) go in the denominator.

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