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

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

$ \frac{AD}{DB}=\frac{AE}{EC}=\frac{DE}{BC} $

$ \frac{AD}{AE}=\frac{AE}{DB}=\frac{DE}{BC} $

Congruent Triangles ADE and ABC: Geometric Properties Analysis

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

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

Q: How do I know which sides correspond to each other?

Look at the triangle names! In triangles ADE and ABC, the order of vertices tells you the correspondence: A↔A, D↔B, E↔C. So side AD corresponds to side AB.

Q: Why are all the ratios equal in congruent triangles?

Since congruent triangles are identical, when one triangle sits inside the other (like ADE inside ABC), the ratio of corresponding sides is the same constant value - it's the scale factor!

Q: Can the ratio be greater than 1?

Yes! If triangle ABC is smaller than ADE, then $ \frac{AD}{AB} > 1 $. The ratio just tells you how many times bigger one triangle is compared to the other.

Q: What if I write the ratios in different order?

The ratios must be written consistently! If you write $ \frac{AD}{AB} $, then you must write $ \frac{AE}{AC} $ and $ \frac{DE}{BC} $ - all smaller over larger or all larger over smaller.

Triangle Similarity with Congruent 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

00:05 Let's find the right answer together.

00:08 The triangles are similar, based on the given data.

00:13 We'll start at point A. Let's identify the matching sides. Ready? Here we go!

00:28 Now, let's repeat this process for points D and B. You're doing great!

00:34 And that's how we find the answer to this question. Well done!

Step-by-step written solution

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Understand the problem

Triangles ADE and ABC are congruent.

Choose the correct answer.

Step-by-step solution

Triangles ADE and ABC are given as congruent. When two triangles are congruent, their corresponding sides are equal in proportion. Therefore, we need to find the correct proportional relationship between the sides of these triangles.

Let's recall that for two congruent triangles, their corresponding sides are in equal ratios. Specifically, for triangles ADE and ABC, the sides AD, AE, and DE correspond to the sides AB, AC, and BC, respectively.

Thus, the correct proportional relationship between the corresponding sides of triangles ADE and ABC is:

$\frac{AD}{AB}$ corresponding to the first set of sides.
$\frac{AE}{AC}$ corresponding to the second set of sides.
$\frac{DE}{BC}$ corresponding to the third set of sides.

Therefore, the choice that correctly represents the proportional relationship of the sides of triangles ADE and ABC is:

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Congruent Triangles: Triangles ADE and ABC are congruent, meaning identical in shape and size
Correspondence: Match vertices correctly - A to A, D to B, E to C
Proportion Check: All ratios equal same value: $\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$ ✓

Common Mistakes

Avoid these frequent errors

Confusing similarity with congruence in proportions
Don't write ratios like $\frac{AB}{AD}$ for congruent triangles = backwards thinking! This creates ratios greater than 1 for smaller triangle to larger triangle. Always write smaller triangle measurements over corresponding larger triangle measurements for proper proportional relationships.

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

How do I know which sides correspond to each other?

Look at the triangle names! In triangles ADE and ABC, the order of vertices tells you the correspondence: A↔A, D↔B, E↔C. So side AD corresponds to side AB.

Why are all the ratios equal in congruent triangles?

Since congruent triangles are identical, when one triangle sits inside the other (like ADE inside ABC), the ratio of corresponding sides is the same constant value - it's the scale factor!

Can the ratio be greater than 1?

Yes! If triangle ABC is smaller than ADE, then $\frac{AD}{AB} > 1$ . The ratio just tells you how many times bigger one triangle is compared to the other.

What if I write the ratios in different order?

The ratios must be written consistently! If you write $\frac{AD}{AB}$ , then you must write $\frac{AE}{AC}$ and $\frac{DE}{BC}$ - all smaller over larger or all larger over smaller.

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