Similar Triangles Construction: Finding Lengths with ∢A=∢D

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

Look at the angles! Since ∠A = ∠D, these vertices correspond. The sides extending from these equal angles are corresponding sides. So AB corresponds to DE, and AC corresponds to DF.

Q: Why do all the ratios need to be the same number?

For triangles to be similar, they must have the same shape but different size. This means all corresponding sides must be scaled by the same factor. If ratios are different, the triangles have different shapes!

Q: What if I get different ratios for the sides?

Then the triangles are not similar! Even if one pair of angles is equal, you need all corresponding sides to have the same ratio for true similarity.

Q: How do I calculate the ratio correctly?

Always put corresponding sides in the same position: $ \frac{\text{Triangle 1 side}}{\text{Triangle 2 side}} $. For example: $ \frac{AB}{DE} $ and $ \frac{AC}{DF} $, keeping the same triangle on top.

Similar Triangles with Proportional Side Verification

Given two triangles. Choose appropriate lengths to obtain similar triangles if given $∢A=∢D$

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Step-by-step video solution

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00:07 Let's find the side lengths so that the triangles are similar.

00:11 First, we need to find the ratio of the sides.

00:15 Next, substitute the correct values based on the given information.

00:20 Now, find the ratio between the first pair of sides and the second pair.

00:25 Again, use the given data to substitute the right values.

00:30 If the side ratios match, then the triangles are similar.

00:34 And that's how we solve this problem!

Step-by-step written solution

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Understand the problem

Given two triangles. Choose appropriate lengths to obtain similar triangles if given $∢A=∢D$

Step-by-step solution

The correct answer is D. Since angle A is equal to angle D, then BAC is similar to FDE

Therefore, we calculate according to the data from answer D as follows:

$\frac{AB}{DE}=\frac{40}{8}=5$

$\frac{AC}{DF}=\frac{45}{9}=5$

Final Answer

AB=40 DF=9

AC=45 DE=8

Key Points to Remember

Essential concepts to master this topic

Similarity Rule: Equal angles require proportional corresponding sides
Technique: Calculate ratios: $\frac{AB}{DE} = \frac{40}{8} = 5$ and $\frac{AC}{DF} = \frac{45}{9} = 5$
Check: All corresponding side ratios must be equal for similarity ✓

Common Mistakes

Avoid these frequent errors

Matching sides incorrectly based on position
Don't assume sides in the same visual position correspond = wrong ratios! The diagram position doesn't determine correspondence. Always match sides based on the angles they form: if ∠A = ∠D, then sides AB and DE are corresponding because they both extend from the equal angles.

Practice Quiz

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FAQ

Everything you need to know about this question

How do I know which sides correspond to each other?

Look at the angles! Since ∠A = ∠D, these vertices correspond. The sides extending from these equal angles are corresponding sides. So AB corresponds to DE, and AC corresponds to DF.

Why do all the ratios need to be the same number?

For triangles to be similar, they must have the same shape but different size. This means all corresponding sides must be scaled by the same factor. If ratios are different, the triangles have different shapes!

What if I get different ratios for the sides?

Then the triangles are not similar! Even if one pair of angles is equal, you need all corresponding sides to have the same ratio for true similarity.

Can I use any two sides to check similarity?

Yes! If you know one pair of angles is equal, checking that two pairs of corresponding sides have the same ratio is enough to prove similarity by SAS similarity theorem.

How do I calculate the ratio correctly?

Always put corresponding sides in the same position: $\frac{\text{Triangle 1 side}}{\text{Triangle 2 side}}$ . For example: $\frac{AB}{DE}$ and $\frac{AC}{DF}$ , keeping the same triangle on top.

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