📐 Concept diagram

02-02 - Triangles

Phase: 2 | Subject: 02-02 Prerequisites: 02-01-angles-and-lines.md Next subject: 02-03-polygons-and-circles.md

Learning Objectives

By the end of this subject, you will be able to:

1. Classify triangles by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse)
2. Apply the angle sum property (180°) and exterior angle theorem
3. Understand congruence criteria (SSS, SAS, ASA, AAS, RHS)
4. Understand similarity and the similarity ratio
5. Calculate area of triangles using multiple methods

Core Content

Triangle Classifications

By Sides

• Scalene: All sides different, all angles different
• Isosceles: Two sides equal, two angles equal (base angles equal)
• Equilateral: All sides equal, all angles 60°

By Angles

• Acute: All angles < 90°
• Right: One angle = 90°
• Obtuse: One angle > 90°

A triangle must be at least one of each type (e.g., an isosceles right triangle).

The Angle Sum Property

⚠️ THIS IS CRITICAL — the fact that triangle angles sum to 180° is used in virtually every geometry problem. Master this now.

The sum of interior angles in ANY triangle is exactly 180°.

This is fundamental — it's why we can find missing angles.

Proof sketch: Draw a line through one vertex parallel to the opposite side. The alternate angles formed are equal to the other two interior angles. Together with the angle at the vertex, they form a straight line (180°).

Example: Find the missing angle if two angles are 50° and 60°.

180° - 50° - 60° = 70°

The Exterior Angle Theorem

An exterior angle equals the sum of the two opposite interior angles.

$      /\
     /  \
    /    \
   /______\
  A   B    C
$

Exterior angle at C = ∠A + ∠B

Why: The exterior angle and interior angle at C form a straight line (180°). Since ∠A + ∠B + ∠C = 180°, the exterior angle must equal ∠A + ∠B.

Congruent Triangles

Two triangles are congruent if they have exactly the same shape AND size. All corresponding sides and angles are equal.

Congruence Criteria

Important: SSA (two sides and a non-included angle) is NOT a valid congruence criterion (except for RHS).

Similar Triangles

Two triangles are similar if they have the same shape but possibly different sizes. Corresponding angles are equal, and corresponding sides are proportional.

Similarity Criteria

Similarity Ratio

If triangle ABC ~ triangle DEF (similar), then:

AB/DE = BC/EF = AC/DF = k

k is the similarity ratio (or scale factor).

Example: Two similar triangles have sides 3, 4, 5 and 6, 8, 10. Ratio k = 6/3 = 8/4 = 10/5 = 2. The second triangle is twice as large.

Triangle Area

Basic Formula

$Area = (1/2) × base × height
A = (1/2)bh
$

The height must be perpendicular to the chosen base.

Heron's Formula (when you know all three sides)

$s = (a + b + c)/2  (semi-perimeter)
Area = √[s(s-a)(s-b)(s-c)]
$

Example: Sides 3, 4, 5 (right triangle!) s = (3 + 4 + 5)/2 = 6 Area = √[6(3)(2)(1)] = √36 = 6

Check with basic formula: base = 3, height = 4, Area = (1/2)(3)(4) = 6 ✓

Key Terms

• 02 02 Triangles
• Basic Formula
• By Angles
• By Sides
• Congruence Criteria
• Congruent Triangles
• Correct: A)
• Correct: B)
• Correct: C)
• Criterion
• Example 1: Classify and find missing angles
• Example 2: Congruence proof

Worked Examples

Example 1: Classify and find missing angles

Triangle ABC has angles: ∠A = 45°, ∠B = 65°. Find ∠C and classify.

1. ∠C = 180° - 45° - 65° = 70°
2. All angles < 90°, so it's an acute triangle
3. All angles different, so it's scalene

Triangle: Acute scalene

Example 2: Congruence proof

Given: AB = DE, BC = EF, AC = DF. Prove ΔABC ≅ ΔDEF.

1. We have three pairs of equal sides: AB = DE, BC = EF, AC = DF
2. By SSS criterion, ΔABC ≅ ΔDEF

Example 3: Similarity ratio and area

Two similar triangles have similarity ratio k = 3. If the smaller has area 12, what is the larger?

1. Area ratio = k² = 3² = 9
2. Larger area = 12 × 9 = 108

Key insight: Area scales with the SQUARE of the similarity ratio.

Quiz

Q1: What does the concept of Basic Formula primarily refer to in this subject?

A) A historical anecdote about Basic Formula B) The definition and application of Basic Formula C) A visual representation of Basic Formula D) A computational error related to Basic Formula

Correct: B)

• If you chose A: This is incorrect. Basic Formula is defined as: the definition and application of basic formula. The other options describe different aspects that are not the primary focus.
• If you chose B: Basic Formula is defined as: the definition and application of basic formula. The other options describe different aspects that are not the primary focus. Correct!
• If you chose C: This is incorrect. Basic Formula is defined as: the definition and application of basic formula. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Basic Formula is defined as: the definition and application of basic formula. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of By Angles?

A) It is used only in advanced research contexts B) It is used to by angles in mathematical analysis C) It replaces all other methods in this domain D) It is primarily a historical notation system

Correct: B)

• If you chose A: This is incorrect. By Angles serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: By Angles serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose C: This is incorrect. By Angles serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. By Angles serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about By Sides is TRUE?

A) By Sides is an advanced topic beyond this subject's scope B) By Sides is mentioned only as a historical footnote C) By Sides is a fundamental concept covered in this subject D) By Sides is not related to this subject

Correct: C)

• If you chose A: This is incorrect. By Sides is a fundamental concept covered in this subject. This subject covers By Sides as part of its core content.
• If you chose B: This is incorrect. By Sides is a fundamental concept covered in this subject. This subject covers By Sides as part of its core content.
• If you chose C: By Sides is a fundamental concept covered in this subject. This subject covers By Sides as part of its core content. Correct!
• If you chose D: This is incorrect. By Sides is a fundamental concept covered in this subject. This subject covers By Sides as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) The inverse of the correct answer B) A different result from a common mistake C) An unrelated numerical value D) 65°

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is 65°. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is 65°. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is 65°. The other options represent common errors.
• If you chose D: The worked examples show that the result is 65°. The other options represent common errors. Correct!

Q5: How are By Sides and Congruence Criteria related?

A) By Sides and Congruence Criteria are completely unrelated topics B) By Sides and Congruence Criteria are closely related concepts C) By Sides is a special case of Congruence Criteria D) By Sides is the inverse of Congruence Criteria

Correct: B)

• If you chose A: This is incorrect. Both By Sides and Congruence Criteria are covered in this subject as interconnected topics.
• If you chose B: Both By Sides and Congruence Criteria are covered in this subject as interconnected topics. Correct!
• If you chose C: This is incorrect. Both By Sides and Congruence Criteria are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both By Sides and Congruence Criteria are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Congruent Triangles?

A) A common mistake is confusing Congruent Triangles with a similar concept B) The main error with Congruent Triangles is using it when it is not needed C) Congruent Triangles has no common misconceptions D) Congruent Triangles is always computed the same way in all contexts

Correct: A)

• If you chose A: Students often confuse Congruent Triangles with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose B: This is incorrect. Students often confuse Congruent Triangles with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Congruent Triangles with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Congruent Triangles with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Criterion?

A) Apply Criterion to solve problems in this subject's domain B) Criterion is not practically useful C) Use Criterion only in pure mathematics contexts D) Avoid Criterion unless explicitly instructed

Correct: A)

• If you chose A: Criterion is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose B: This is incorrect. Criterion is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Criterion is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Criterion is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Find the third angle of a triangle with angles 40° and 75°. Answer: 180° - 40° - 75° = 65°

2. In isosceles triangle ABC with AB = AC, if ∠B = 50°, find ∠A and ∠C. Answer: ∠C = 50° (base angles equal). ∠A = 180° - 50° - 50° = 80°.

3. An exterior angle of a triangle is 100°. If one opposite interior angle is 35°, find the other. Answer: 100° - 35° = 65°

4. Are triangles with sides (3, 4, 5) and (6, 8, 10) congruent or similar? Answer: Similar (sides in ratio 2:1). Not congruent (different sizes).

5. Find area of triangle with base 8 and height 5. Answer: (1/2) × 8 × 5 = 20

6. Two similar triangles have areas 18 and 72. What is their similarity ratio? Answer: Area ratio = 72/18 = 4. Similarity ratio = √4 = 2.

7. In triangle ABC, AB = 5, BC = 7, AC = 8. Find area using Heron's formula. Answer: s = (5+7+8)/2 = 10. Area = √[10(5)(3)(2)] = √300 = 10√3 ≈ 17.32

Summary

Key takeaways:

• Triangle angle sum: always exactly 180°
• Exterior angle = sum of two opposite interior angles
• Isosceles: base angles equal; Equilateral: all angles 60°
• Congruent = same shape AND size (SSS, SAS, ASA, AAS, RHS)
• Similar = same shape, different size (AA, SAS, SSS)
• Area = (1/2) × base × height
• Area ratio of similar figures = (similarity ratio)²

Pitfalls

• Confusing SSA with valid congruence criteria. SSA (two sides and a non-included angle) is NOT a valid congruence criterion (except for RHS in right triangles). Two triangles can satisfy SSA and still be different shapes. Only SSS, SAS, ASA, AAS, and RHS are valid.
• Using the wrong height in the area formula. The height in A = (1/2)bh must be PERPENDICULAR to the chosen base. Using a slanted side as the height (instead of the altitude) gives an incorrect area. Always ensure the height is at a right angle to the base.
• Forgetting to square the similarity ratio for area. If the similarity ratio is k = 3, the area ratio is k² = 9. Using k instead of k² for area comparisons is a persistent error. Area scales with the SQUARE of the linear scale factor.
• Misapplying the exterior angle theorem. The exterior angle equals the sum of the TWO OPPOSITE interior angles, not the adjacent interior angle. For triangle with angles A, B, C, the exterior angle at vertex C equals A + B, not 180° - C (which is the same thing, but students often confuse the relationship).
• Assuming an isosceles triangle tells you which sides are equal. "Isosceles triangle ABC" doesn't specify which sides are equal — you need additional information (like AB = AC) or a diagram. Without it, you cannot determine which angles are the base angles.

Next Steps

Next up: 02-03-polygons-and-circles.md