AAABBBCCCDDDEEE60°30°30°60°$ ΔACB∼ΔBED $Choose the correct answer.

$ \frac{CB}{BE}=\frac{AC}{ED} $

$ \frac{AB}{BD}=\frac{ED}{AC} $

Similar Triangles Proof: Analyzing ΔACB∼ΔBED with 30° and 60° Angles

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

Match the angles first! In similar triangles, sides opposite equal angles are corresponding. For example, if angle C = angle E = 30°, then the sides opposite these angles (AB and BD) correspond.

Q: Why are there multiple correct ratio equations?

In similar triangles, all corresponding sides are proportional. This means you can write three different ratio equations, and they'll all be equivalent. That's why both answers a and b can be correct!

Q: What does the ~ symbol mean in ΔACB∼ΔBED?

The symbol ~ means "is similar to". When triangles are similar, their corresponding angles are equal and their corresponding sides are proportional.

Q: How can I tell if triangles are similar just from angles?

If two angles in one triangle equal two angles in another triangle, the triangles are similar! The third angles must also be equal since angles in a triangle sum to 180°.

Q: Do I need to worry about the order of vertices in similar triangles?

Yes! The order matters. In ΔACB∼ΔBED, vertex A corresponds to B, vertex C corresponds to E, and vertex B corresponds to D. This tells you which sides and angles match up.

Similar Triangle Ratios with Angle Correspondence

$ΔACB∼ΔBED$

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 Find the side opposite to the 30-degree angle in both triangles

00:14 Use the same method for the 90-degree angle in both triangles

00:23 And the same for the 60-degree angle in both triangles

00:33 The similarity ratio is always the side opposite to the equal angle

00:53 Let's use the transition rule

00:57 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

$ΔACB∼ΔBED$

Choose the correct answer.

Step-by-step solution

First, let's look at angles C and E, which are equal to 30 degrees.

Angle C is opposite side AB and angle E is opposite side BD.

$\frac{AB}{DB}$

Now let's look at angle B, which is equal to 90 degrees in both triangles.

In triangle ABC the opposite side is AC and in triangle EBD the opposite side is ED.

$\frac{AC}{ED}$

Let's look at angles A and D, which are equal to 60 degrees.

Angle A is the opposite side of CB, angle D is the opposite side of EB

$\frac{CB}{EB}$

Therefore, from this it can be deduced that:

$\frac{AB}{BD}=\frac{AC}{ED}$

And also:

$\frac{CB}{ED}=\frac{AB}{BD}$

Final Answer

Answers a + b are correct.

Key Points to Remember

Essential concepts to master this topic

Correspondence: Similar triangles have proportional sides and equal corresponding angles
Technique: Match angles first: 30° to 30°, 60° to 60°, then write ratios
Check: All three ratios must be equal: $\frac{AB}{BD}=\frac{AC}{ED}=\frac{CB}{BE}$ ✓

Common Mistakes

Avoid these frequent errors

Writing ratios without matching corresponding angles
Don't randomly pair sides from similar triangles = wrong ratios! This gives incorrect proportions because sides opposite different angle measures aren't corresponding. Always identify which angles are equal first, then match the opposite sides.

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?

Match the angles first! In similar triangles, sides opposite equal angles are corresponding. For example, if angle C = angle E = 30°, then the sides opposite these angles (AB and BD) correspond.

Why are there multiple correct ratio equations?

In similar triangles, all corresponding sides are proportional. This means you can write three different ratio equations, and they'll all be equivalent. That's why both answers a and b can be correct!

What does the ~ symbol mean in ΔACB∼ΔBED?

The symbol ~ means "is similar to". When triangles are similar, their corresponding angles are equal and their corresponding sides are proportional.

How can I tell if triangles are similar just from angles?

If two angles in one triangle equal two angles in another triangle, the triangles are similar! The third angles must also be equal since angles in a triangle sum to 180°.

Do I need to worry about the order of vertices in similar triangles?

Yes! The order matters. In ΔACB∼ΔBED, vertex A corresponds to B, vertex C corresponds to E, and vertex B corresponds to D. This tells you which sides and angles match up.

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