Triangle Similarity Ratio: Finding the Scale Factor Between Two Intersecting Triangles

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

Match by equal angles! Since ∠A = ∠E and ∠B = ∠C, vertex A corresponds to E, and B corresponds to C. The shared vertex D corresponds to itself.

Q: Why are both angles at D equal to 90 degrees?

The diagram shows perpendicular lines intersecting at point D. When two lines are perpendicular, they form four right angles, so both ∠ADB and ∠EDC equal 90°.

Q: Can I write the similarity ratio in any order?

No! The order matters. You must match corresponding sides: if triangle ABC ~ triangle ECD, then write $ \frac{AB}{EC} = \frac{BC}{CD} = \frac{AC}{ED} $.

Q: What if I can't see all the angle measurements?

Look for visual clues like right angle markers (squares) or parallel lines. In this problem, the square at D shows 90° angles, and the diagram suggests ∠A = ∠E.

Q: How do I remember which triangle is which?

Write the vertices in corresponding order. Since A↔E, B↔C, and D↔D, triangle ABC corresponds to triangle ECD, giving us the ratio pattern.

Similar Triangles with Angle-Angle Correspondence

What is the ratio of similarity between the triangles shown in the diagram below?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find out which triangles are similar and what their similarity ratio is.

00:11 First, look for equal angles given in the problem. Remember, equal angles help us identify similar triangles.

00:18 Also, spot any perpendicular lines in the data. This is important for our comparison.

00:24 If two angles are equal between the triangles, then we know the third angle must be equal too!

00:33 This means the triangles are similar by the Angle-Angle, or AA, rule.

00:44 Next, remember that corresponding sides are opposite to those equal angles.

00:54 And that's how we find our similarity ratio, by comparing those sides.

01:01 Great job! That's the solution to the question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the ratio of similarity between the triangles shown in the diagram below?

Step-by-step solution

From the drawing it appears that angle E equals angle A

Since angle D equals 90 degrees, its adjacent angle also equals 90 degrees.

In other words, angle D1 equals angle D2 and both equal 90 degrees.

Since we have two pairs of equal angles, the triangles are similar.

Also angle B equals angle C

Now let's write the similar triangles according to their corresponding angle letters:

$ABC=ECD$

Let's write the ratio of sides according to the corresponding letters of the similar triangles:

$\frac{AB}{EC}=\frac{AD}{ED}=\frac{BD}{CD}$

Final Answer

$\frac{AB}{EC}=\frac{AD}{ED}=\frac{BD}{CD}$

Key Points to Remember

Essential concepts to master this topic

Similarity Rule: Two triangles are similar if two angles are equal
Correspondence: Match vertices by equal angles: ∠A = ∠E, ∠B = ∠C
Check: Verify ratios are equal: $\frac{AB}{EC} = \frac{AD}{ED} = \frac{BD}{CD}$ ✓

Common Mistakes

Avoid these frequent errors

Writing ratios without matching corresponding vertices
Don't write ratios like AB/CD when A doesn't correspond to C = wrong proportions! This mixes up which sides actually match between similar triangles. Always match vertices by their equal angles first, then write ratios using corresponding 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 vertices correspond to each other?

Match by equal angles! Since ∠A = ∠E and ∠B = ∠C, vertex A corresponds to E, and B corresponds to C. The shared vertex D corresponds to itself.

Why are both angles at D equal to 90 degrees?

The diagram shows perpendicular lines intersecting at point D. When two lines are perpendicular, they form four right angles, so both ∠ADB and ∠EDC equal 90°.

Can I write the similarity ratio in any order?

No! The order matters. You must match corresponding sides: if triangle ABC ~ triangle ECD, then write $\frac{AB}{EC} = \frac{BC}{CD} = \frac{AC}{ED}$ .

What if I can't see all the angle measurements?

Look for visual clues like right angle markers (squares) or parallel lines. In this problem, the square at D shows 90° angles, and the diagram suggests ∠A = ∠E.

How do I remember which triangle is which?

Write the vertices in corresponding order. Since A↔E, B↔C, and D↔D, triangle ABC corresponds to triangle ECD, giving us the ratio pattern.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Similar Triangles and Polygons questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime