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Class 10·Math
Class 10 · Math

Triangles

Triangles are the fundamental building blocks of geometry.

Feynman Lens

Start with the simplest version: this lesson is about Triangles. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.

Triangles are the fundamental building blocks of geometry. This chapter moves beyond the congruence you learned earlier (same shape AND size) to explore similarity (same shape but different sizes). Understanding similar triangles is powerful: you can find heights of buildings without measuring them, determine distances to unreachable objects, and understand how scaling works in photography and design. Similar triangles reveal that geometry is about proportions—and these proportions remain constant regardless of size, which is why a building blueprint scaled 100 times still represents the same structure.

Congruence vs. Similarity: Two Levels of Likeness

Congruence (from Class IX):

Similarity (new in this chapter):

Real-world example: A 1:10 scale model of a bridge is similar to the actual bridge. All angles match, but all linear dimensions are scaled by 1/10.

The AA Criterion: Angle-Angle Similarity

The most practical way to prove triangles are similar:

If two angles of one triangle equal two angles of another triangle, the triangles are similar.

Why this works: In any triangle, the three angles must sum to 180°. If two angles match, the third automatically matches too. Matching angles guarantee the same shape, which is exactly what similarity requires.

Example:

This criterion is simple to use in practice—you just need two matching angles.

The SAS and SSS Criteria: Side-Angle-Side and Side-Side-Side

SAS Similarity: If one angle of a triangle equals an angle of another, and the sides including these angles are proportional, the triangles are similar.

SSS Similarity: If all three sides of one triangle are proportional to all three sides of another, the triangles are similar.

These criteria give you flexibility: sometimes you have angle measurements, sometimes side lengths, and these criteria cover both scenarios.

Properties of Similar Triangles: Proportionality

When △ABC ~ △DEF:

  1. Corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
  1. Corresponding sides are proportional:
   AB/DE = BC/EF = AC/DF = k (the constant of proportionality)
  1. Ratio of areas: If the sides are in ratio k, the areas are in ratio k²

Intuition for area ratio: Area is two-dimensional. If you scale every linear dimension by k, area scales by k × k = k².

The Basic Proportionality Theorem (Thales' Theorem)

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Statement: In △ABC, if DE ∥ BC (DE parallel to BC), then:

AD/DB = AE/EC

Application: This theorem is the foundation for many real-world measurements. If you can create similar triangles using parallel lines, you can use known ratios to find unknown lengths without direct measurement.

Real-World Application: Heights and Distances

The Classic Problem: Find the height of a building without climbing it.

Method:

  1. Stand at a known distance from the building
  2. Measure the angle of elevation to the roof
  3. Use similar triangles formed by your position, the building, and the angle
  4. Apply proportions to find the building's height

This technique works because the triangle formed by you, the ground, and the building is similar to a triangle you can measure or calculate. This connects naturally to chapter-08-introduction-to-trigonometry where you'll use trigonometric ratios for such problems.

Similar triangles bridge geometry and other topics:

Key Theorems and Criteria

Socratic Questions

  1. If two triangles have all three angles equal, are they always similar? Why or why not?
  1. You're given that AB/DE = BC/EF = 1/2 for two triangles. What can you conclude about their similarity? What additional information would you need?
  1. In the Basic Proportionality Theorem, what is the geometric significance of the parallel line? Why must it be parallel to the third side?
  1. If a scale model of a building has all linear dimensions 1/100th of the original, what is the ratio of their surface areas? Their volumes?
  1. How would you use similar triangles to estimate the height of a tree without cutting it down? Outline the steps.
🃏 Flashcards — Quick Recall
Term / Concept
What is Triangles?
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Triangles is the central idea of this lesson. Use the chapter examples to explain what it means and why it matters.
Term / Concept
What is Similarity?
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(new in this chapter):
Term / Concept
What is Real-world example?
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A 1:10 scale model of a bridge is similar to the actual bridge. All angles match, but all linear dimensions are scaled by 1/10.
Term / Concept
What is Why this works?
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In any triangle, the three angles must sum to 180°. If two angles match, the third automatically matches too. Matching angles guarantee the same shape, which is exactly what similarity requires.
Term / Concept
What is Example?
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- Triangle ABC has angles 60°, 80°, 40°
Term / Concept
What is SAS Similarity?
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If one angle of a triangle equals an angle of another, and the sides including these angles are proportional, the triangles are similar.
Term / Concept
What is SSS Similarity?
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If all three sides of one triangle are proportional to all three sides of another, the triangles are similar.
Term / Concept
What is Intuition for area ratio?
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Area is two-dimensional. If you scale every linear dimension by k, area scales by k × k = k².
Term / Concept
What is Statement?
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In △ABC, if DE ∥ BC (DE parallel to BC), then:
Term / Concept
What is Application?
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This theorem is the foundation for many real-world measurements. If you can create similar triangles using parallel lines, you can use known ratios to find unknown lengths without direct measurement.
Term / Concept
What is The Classic Problem?
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Find the height of a building without climbing it.
Term / Concept
What is Method?
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1. Stand at a known distance from the building
Term / Concept
What is AA Similarity?
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Two matching angles ensure similarity
Term / Concept
What is Basic Proportionality Theorem?
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Parallel line divides sides proportionally
Term / Concept
What is Area Ratio?
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(Linear ratio)² = Area ratio
Term / Concept
What is the core idea of Congruence vs. Similarity: Two Levels of Likeness?
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Congruence (from Class IX): - Two figures are congruent if they have the same shape AND the same size - Congruent triangles are identical—you can place one exactly on top of the other - Denoted: △ABC ≅ △DEF Similarity…
Term / Concept
What is the core idea of The AA Criterion: Angle-Angle Similarity?
tap to flip
The most practical way to prove triangles are similar: If two angles of one triangle equal two angles of another triangle, the triangles are similar. Why this works: In any triangle, the three angles must sum to 180°.
Term / Concept
What is the core idea of The SAS and SSS Criteria: Side-Angle-Side and Side-Side-Side?
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SAS Similarity: If one angle of a triangle equals an angle of another, and the sides including these angles are proportional, the triangles are similar.
Term / Concept
What is the core idea of Properties of Similar Triangles: Proportionality?
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When △ABC ~ △DEF: 1. Corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F 2. Corresponding sides are proportional: AB/DE = BC/EF = AC/DF = k (the constant of proportionality) 3.
Term / Concept
What is the core idea of The Basic Proportionality Theorem (Thales' Theorem)?
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If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
Term / Concept
What is the core idea of Real-World Application: Heights and Distances?
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The Classic Problem: Find the height of a building without climbing it. Method: 1. Stand at a known distance from the building 2. Measure the angle of elevation to the roof 3.
Term / Concept
What is the core idea of Connecting to Related Topics?
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Similar triangles bridge geometry and other topics: - chapter-07-coordinate-geometry: Triangles in coordinate planes use distance and similarity concepts - chapter-08-introduction-to-trigonometry: Angle ratios in…
Term / Concept
What is the core idea of Key Theorems and Criteria?
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- AA Similarity: Two matching angles ensure similarity - SAS Similarity: One matching angle + proportional including sides = similarity - SSS Similarity: All three sides proportional = similarity - Basic…
Term / Concept
What is Two figures are congruent if they have?
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Two figures are congruent if they have the same shape AND the same size
Term / Concept
What is Congruent triangles are identical—you can place one?
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Congruent triangles are identical—you can place one exactly on top of the other
Term / Concept
What is Two figures are similar if they have?
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Two figures are similar if they have the same shape but not necessarily the same size
Term / Concept
What is Similar triangles have matching angles and proportional?
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Similar triangles have matching angles and proportional sides
Term / Concept
What is A scaled photograph is similar to the original?
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same angles, proportional sizes
Term / Concept
What is Triangle ABC has angles 60°, 80°, 40°?
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Triangle ABC has angles 60°, 80°, 40°
Term / Concept
What is Triangle PQR has angles 60°, 80°, 40°?
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Triangle PQR has angles 60°, 80°, 40°
Term / Concept
What is Then △ABC ~ △PQR (even if they're?
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Then △ABC ~ △PQR (even if they're different sizes)
Term / Concept
What is chapter-07-coordinate-geometry?
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Triangles in coordinate planes use distance and similarity concepts
Term / Concept
What is chapter-08-introduction-to-trigonometry?
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Angle ratios in similar triangles lead to trigonometric functions
Term / Concept
What is chapter-09-some-applications-of-trigonometry?
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Real-world height/distance problems use similar triangles
Term / Concept
What is chapter-10-circles?
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Similar circles and proportional relationships emerge naturally
Term / Concept
What does this formula or relation mean: AB/DE = BC/EF = AC/DF = k (the constant of propo?
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This formula or symbolic relation appears in the chapter. Explain what each part represents before using it.
Term / Concept
What does this formula or relation mean: AD/DB = AE/EC?
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This formula or symbolic relation appears in the chapter. Explain what each part represents before using it.
Term / Concept
Why Congruence vs. Similarity: Two Levels of Likeness matters?
tap to flip
Congruence vs. Similarity: Two Levels of Likeness matters because it connects the chapter idea to a reason, pattern, or method you can apply in problems.
Term / Concept
Why The AA Criterion: Angle-Angle Similarity matters?
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The AA Criterion: Angle-Angle Similarity matters because it connects the chapter idea to a reason, pattern, or method you can apply in problems.
Term / Concept
Why The SAS and SSS Criteria: Side-Angle-Side and Side-Side-Side matters?
tap to flip
The SAS and SSS Criteria: Side-Angle-Side and Side-Side-Side matters because it connects the chapter idea to a reason, pattern, or method you can apply in problems.
40 cards — click any card to flip
📝 Quick Quiz — Test Yourself
If two triangles have all three angles equal, are they always similar? Why or why not?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
You're given that AB/DE = BC/EF = 1/2 for two triangles. What can you conclude about their similarity? What additional information would you need?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
In the Basic Proportionality Theorem, what is the geometric significance of the parallel line? Why must it be parallel to the third side?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
If a scale model of a building has all linear dimensions 1/100th of the original, what is the ratio of their surface areas? Their volumes?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
How would you use similar triangles to estimate the height of a tree without cutting it down? Outline the steps.
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
Which approach best shows that you understand Triangles?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Similarity?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real-world example?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Why this works?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Example?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand SAS Similarity?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand SSS Similarity?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Intuition for area ratio?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Statement?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Application?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The Classic Problem?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Method?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand AA Similarity?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Basic Proportionality Theorem?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Area Ratio?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Congruence vs. Similarity: Two Levels of Likeness?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The AA Criterion: Angle-Angle Similarity?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The SAS and SSS Criteria: Side-Angle-Side and Side-Side-Side?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Properties of Similar Triangles: Proportionality?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The Basic Proportionality Theorem (Thales' Theorem)?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real-World Application: Heights and Distances?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Connecting to Related Topics?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Key Theorems and Criteria?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Two figures are congruent if they have?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Congruent triangles are identical—you can place one?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Two figures are similar if they have?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Similar triangles have matching angles and proportional?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand A scaled photograph is similar to the original?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Triangle ABC has angles 60°, 80°, 40°?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Triangle PQR has angles 60°, 80°, 40°?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Then △ABC ~ △PQR (even if they're?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-07-coordinate-geometry?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-08-introduction-to-trigonometry?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-09-some-applications-of-trigonometry?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-10-circles?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
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