Solve the Proportion: Finding the Missing Term in AD/_ = AE/AC

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

Look at the vertex order in the triangle names! In triangles ADE and ABC, vertex A is common, so AD corresponds to AB and AE corresponds to AC.

Q: Why does BC being parallel to DE create similar triangles?

When lines are parallel, they create corresponding angles that are equal. Since triangles ADE and ABC share angle A and have equal corresponding angles, they must be similar!

Q: What if I can't see the triangle clearly in the diagram?

Focus on the three vertices mentioned in the proportion. Triangle ADE has vertices A, D, and E. Triangle ABC has vertices A, B, and C. The parallel line DE creates the similar triangles.

Q: Can I flip the fractions in the proportion?

Yes! You can write $ \frac{AB}{AD} = \frac{AC}{AE} $ instead, but make sure to flip both fractions consistently. Don't mix flipped and unflipped ratios!

Q: How do I remember the correct order for proportions?

Use the triangle names as your guide: ADE ~ ABC means A↔A, D↔B, E↔C. So AD/AB = AE/AC = DE/BC. Keep the same vertex order in both ratios!

Similar Triangles with Parallel Lines

BC is parallel to DE.

Fill in the gap:

$\frac{AD}{}=\frac{AE}{AC}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's fill in what's missing.

00:10 Notice, the lines are parallel, just as given.

00:15 For parallel lines, matching, or corresponding angles, are equal. We call this angle A.

00:23 And here, another corresponding angle is equal. Let's call this angle A2.

00:29 So, the triangles are similar because of Angle-Angle, or AA.

00:35 Now, corresponding sides face these equal angles. This helps us understand their relationship.

00:55 And that's how we solve this question!

00:59 Now, let's move on to the next chapter.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

BC is parallel to DE.

Fill in the gap:

$\frac{AD}{}=\frac{AE}{AC}$

Step-by-step solution

Since we are given that line BC is parallel to DE

Angle E equals angle C and angle D equals angle B - corresponding angles between parallel lines are equal.

Now let's observe that angle D is opposite to side AE and angle B is opposite to side AC, meaning:

$\frac{AE}{AC}$

Now let's observe that angle E is opposite to side AD and angle C is opposite to side AB, meaning:

$\frac{AD}{AB}$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Parallel Lines: Create similar triangles with proportional corresponding sides
Technique: AD/AB = AE/AC since triangles ADE and ABC are similar
Check: Verify corresponding angles are equal and ratios match ✓

Common Mistakes

Avoid these frequent errors

Using wrong corresponding sides in proportion
Don't write AD/AC = AE/AB with mixed up sides = wrong proportion! This ignores the order of corresponding vertices in similar triangles. Always match sides from the same relative positions: AD corresponds to AB, and AE corresponds to AC.

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

How do I know which sides correspond to each other?

Look at the vertex order in the triangle names! In triangles ADE and ABC, vertex A is common, so AD corresponds to AB and AE corresponds to AC.

Why does BC being parallel to DE create similar triangles?

When lines are parallel, they create corresponding angles that are equal. Since triangles ADE and ABC share angle A and have equal corresponding angles, they must be similar!

What if I can't see the triangle clearly in the diagram?

Focus on the three vertices mentioned in the proportion. Triangle ADE has vertices A, D, and E. Triangle ABC has vertices A, B, and C. The parallel line DE creates the similar triangles.

Can I flip the fractions in the proportion?

Yes! You can write $\frac{AB}{AD} = \frac{AC}{AE}$ instead, but make sure to flip both fractions consistently. Don't mix flipped and unflipped ratios!

How do I remember the correct order for proportions?

Use the triangle names as your guide: ADE ~ ABC means A↔A, D↔B, E↔C. So AD/AB = AE/AC = DE/BC. Keep the same vertex order in both ratios!

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