CH – 6 TRIANGLES

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Chapter Overview

In this chapter, you will learn how to work with shapes that have the same form but different sizes. You will cover:

• Similar figures: Understanding shapes that look the same regardless of size.
• Similar triangles: Comparing two triangles based on their angles and side ratios.
• Criteria for similarity: Testing triangle similarity using AA, SAS, and SSS rules.
• Basic Proportionality Theorem (BPT): Finding side ratios when a line is drawn parallel to one side of a triangle.
• Converse of BPT: Checking if a line is parallel to a side based on side ratios.
• Areas of similar triangles: Understanding how area ratios relate to the squares of corresponding sides.
• Pythagoras Theorem: Working with side relationships in right-angled triangles.
• Converse of Pythagoras Theorem: Checking whether a given triangle contains a right angle.

🎯 Board Exam Focus

Practice the following core skills carefully for board exam preparation:

• Identifying whether two given triangles are similar.
• Applying AA similarity to solve for missing angles or sides.
• Using SAS similarity when two side ratios and an included angle are given.
• Checking side ratios using SSS similarity.
• Applying the Basic Proportionality Theorem to find unknown segment lengths.
• Writing down and using ratios of corresponding sides accurately.
• Finding areas of similar triangles using corresponding side ratios.
• Applying the Pythagoras Theorem in right-angled triangles.
• Using the converse of Pythagoras Theorem to identify right-angled triangles.

Most Important Concepts

Similar Figures

Similar figures are figures that have the exact same shape, but they can have different sizes.

Daily Life Examples:

• A full-size photograph and its smaller passport-size version.
• A large globe and a small desktop globe.
• A real car and a toy scale model of the same car.

Difference Between Same Shape and Same Size:

• Same shape: The figures look identical in form, but one can be a scaled-up or scaled-down version of the other.
• Same size: Both figures cover the exact same amount of space and match perfectly when placed over each other.

Figures that have both the same shape and the same size are called congruent. Figures that have the same shape (regardless of size) are called similar.

Similar Triangles

Two triangles are said to be similar if:

1. All their corresponding angles are equal.
2. All their corresponding sides are in the same ratio (proportional).

What does “Corresponding” mean?

The word corresponding means matching positions. The first angle of the first triangle matches the first angle of the second triangle, and the side between two vertices in the first triangle matches the side between the matching vertices in the second triangle.

Simple Example:

Consider triangle ABC and triangle DEF.

If:

• Angle A = Angle D
• Angle B = Angle E
• Angle C = Angle F

And the side ratios are equal:

AB/DE = BC/EF = AC/DF

Then, triangle ABC is similar to triangle DEF (written as ΔABC ~ ΔDEF).

Criteria for Similarity of Triangles

1. AA Similarity (Angle-Angle)

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. (Because the third angles will automatically be equal by the angle sum property).

• Example: In ΔABC and ΔDEF, if Angle A = 50° and Angle B = 70°, and in ΔDEF, Angle D = 50° and Angle E = 70°, then ΔABC ~ ΔDEF by AA similarity.

2. SAS Similarity (Side-Angle-Side)

If two pairs of corresponding sides of two triangles are in the same ratio, and the included angle (the angle formed directly between those two sides) is equal, then the two triangles are similar.

• What is an Included Angle? It is the angle located directly between two specified sides. For sides AB and AC, the included angle is Angle A.
• Example: In ΔABC and ΔDEF, if AB/DE = 1/2 and AC/DF = 1/2, and the included Angle A = Angle D = 60°, then ΔABC ~ ΔDEF by SAS similarity.

3. SSS Similarity (Side-Side-Side)

If all three pairs of corresponding sides of two triangles are in the same ratio, then their corresponding angles are equal, and the triangles are similar.

• Example: In ΔABC, side lengths are AB = 2 cm, BC = 3 cm, AC = 4 cm. In ΔDEF, side lengths are DE = 4 cm, EF = 6 cm, DF = 8 cm.Here, AB/DE = 2/4 = 1/2, BC/EF = 3/6 = 1/2, and AC/DF = 4/8 = 1/2.Since all three ratios are equal to 1/2, ΔABC ~ ΔDEF by SSS similarity.

🧠 Easy Way to Remember Similarity Criteria

• AA (Angle – Angle):“Double Match” — Just match 2 angles, and the triangles are similar.
• SAS (Side – Angle – Side):“Sandwich Rule” — The equal angle must be trapped like a sandwich between the two proportional sides.
• SSS (Side – Side – Side):“Three Ratios” — All 3 side fraction ratios must equal the exact same number.

Basic Proportionality Theorem (BPT / Thales Theorem)

Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then it divides the other two sides in the same ratio.

Step-by-Step Understanding:

1. Take any triangle ABC.
2. Draw a straight line DE inside the triangle such that DE is parallel to side BC (DE || BC).
3. Let line DE touch side AB at point D, and touch side AC at point E.
4. The theorem states that the ratio of segment AD to DB equals the ratio of segment AE to EC.

Formula:

AD/DB = AE/EC

Example:

In ΔABC, DE is parallel to BC. If AD = 2 cm, DB = 4 cm, and AE = 3 cm, find EC.

• Apply BPT formula: AD/DB = AE/EC
• Substitute values: 2/4 = 3/EC
• 1/2 = 3/EC
• Cross-multiply: EC = 3 × 2 = 6 cm.

Remember Point: BPT can ONLY be used when you already know that a line inside the triangle is parallel to one of the sides.

Converse of Basic Proportionality Theorem

Statement: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

• Explanation: This is the exact reverse of BPT. If you calculate the ratios of the split sides and find that AD/DB = AE/EC, you can immediately conclude that line DE is parallel to line BC (DE || BC).

Areas of Similar Triangles

Statement: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Explanation:

If ΔABC ~ ΔDEF, then:

Area(ΔABC) / Area(ΔDEF) = (AB/DE)² = (BC/EF)² = (AC/DF)²

Example:

Given ΔABC ~ ΔDEF. Side AB = 3 cm and side DE = 5 cm. If Area(ΔABC) = 18 cm², find Area(ΔDEF).

• Apply the formula: Area(ΔABC) / Area(ΔDEF) = (AB/DE)²
• 18 / Area(ΔDEF) = (3/5)²
• 18 / Area(ΔDEF) = 9/25
• Cross-multiply: 9 × Area(ΔDEF) = 18 × 25
• 9 × Area(ΔDEF) = 450
• Area(ΔDEF) = 450 ÷ 9 = 50 cm².

Pythagoras Theorem

Statement: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Understanding the Right-Angled Triangle:

• Right-Angled Triangle: A triangle in which one angle measures exactly 90°.
• Hypotenuse: The longest side of a right-angled triangle, located directly opposite the 90° angle.
• Base and Perpendicular: The two shorter sides that form the 90° angle.

Formula:

(Hypotenuse)² = (Base)² + (Perpendicular)²

In ΔABC, if Angle B = 90°:

AC² = AB² + BC²

Example:

In a right triangle, Base = 3 cm and Perpendicular = 4 cm. Find the Hypotenuse.

• Formula: Hypotenuse² = Base² + Perpendicular²
• Hypotenuse² = 3² + 4²
• Hypotenuse² = 9 + 16 = 25
• Hypotenuse = √25 = 5 cm.

Remember Point: Always identify the 90° angle first. The side opposite to it is ALWAYS the hypotenuse.

Converse of Pythagoras Theorem

Statement: In a triangle, if the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, then the angle opposite the first side is a right angle (90°).

How to Check if a Triangle is Right-Angled:

1. Identify the longest side.
2. Square the longest side.
3. Square the other two smaller sides and add them together.
4. If (Longest Side)² = (Side 1)² + (Side 2)², then the triangle is right-angled.

Example:

A triangle has sides 6 cm, 8 cm, and 10 cm. Is it a right-angled triangle?

• Longest side = 10 cm. Square = 10² = 100.
• Other sides = 6 cm and 8 cm. Sum of squares = 6² + 8² = 36 + 64 = 100.
• Since 10² = 6² + 8², the triangle is a right-angled triangle.

📌 Important Theorems & Results

• Condition for Similarity:ΔABC ~ ΔDEF if Angle A = Angle D, Angle B = Angle E, Angle C = Angle F and AB/DE = BC/EF = AC/DF.
• Basic Proportionality Theorem (BPT): If DE || BC, then AD/DB = AE/EC.
• Converse of BPT: If AD/DB = AE/EC, then DE || BC.
• Area Relationship: If ΔABC ~ ΔDEF, then Area(ΔABC) / Area(ΔDEF) = (AB/DE)² = (BC/EF)² = (AC/DF)².
• Pythagoras Theorem: In ΔABC where Angle B = 90°, AC² = AB² + BC².
• Converse of Pythagoras Theorem: If AC² = AB² + BC², then Angle B = 90°.

🧠 How to Know Which Concept to Use?

• Use Similarity Criteria (AA, SAS, SSS) when you need to relate two different triangles with given angles or side lengths, or when proving ratios across nested triangles.
• Use Basic Proportionality Theorem (BPT) when you see a triangle with a line explicitly given as parallel to one of its sides.
• Use Converse of BPT when a question asks you to prove that a line is parallel to a side of a triangle using given segment measurements.
• Use Area Relationship when the question mentions the areas of two triangles alongside side measurements or side ratios.
• Use Pythagoras Theorem when working with a 90° angle and you need to calculate an unknown side length.
• Use Converse of Pythagoras Theorem when given three side lengths and asked to verify if an angle is 90° or if the triangle is right-angled.

🧠 Concepts Students Find Difficult

• Matching corresponding sides correctly: Students often pair sides based on how the figure looks rather than following the order of letters in the similarity statement (e.g., in ΔABC ~ ΔPQR, AB corresponds to PQ, NOT QR).
• Writing ratios in the correct order: Mixing numerator and denominator sources (e.g., writing AB/DE = EF/BC instead of AB/DE = BC/EF).
• Identifying parallel lines in BPT: Overlooking which side the line is parallel to when triangles are rotated or flipped.
• Identifying the hypotenuse: Choosing the vertical or horizontal side as hypotenuse instead of looking directly opposite the 90° angle.
• Knowing when to use Pythagoras Theorem: Attempting to use the formula a² + b² = c² on triangles that do not have a 90° angle.

⚠️ Common Mistakes Students Make

1. Matching Wrong Corresponding Sides
   • Mistake: Assuming horizontal sides always match horizontal sides.
   • Correction: Always read the similarity statement name. If ΔABC ~ ΔQRP, then side AB corresponds to side QR.
2. Writing Ratios in Mixed Orders
   • Mistake: Putting small triangle side on top in the first ratio, but putting big triangle side on top in the second ratio.
   • Correction: Keep sides of the same triangle consistently in the numerator across all ratios.
3. Using Similarity Results Without Proving Similarity
   • Mistake: Directly taking side ratios AB/DE = BC/EF without first stating or proving why the triangles are similar.
   • Correction: Always write the similarity criterion (AA, SAS, or SSS) before using side ratios.
4. Forgetting the Square in Area Questions
   • Mistake: Writing Area(ΔABC) / Area(ΔDEF) = AB/DE instead of (AB/DE)².
   • Correction: Remember that area is measured in square units, so the side ratio MUST be squared.
5. Selecting the Wrong Side as Hypotenuse
   • Mistake: Assuming the side marked ‘c’ or the vertical side is always the hypotenuse.
   • Correction: Find the 90° angle box first. Draw an arrow pointing away from it to locate the hypotenuse.
6. Applying Pythagoras Theorem on Non-Right Triangles
   • Mistake: Using AC² = AB² + BC² on acute or obtuse triangles.
   • Correction: Only apply Pythagoras Theorem when a 90° angle is given or proven.
7. Confusing a Theorem with Its Converse
   • Mistake: Using BPT when asked to prove lines are parallel.
   • Correction: If parallel line is given → use BPT. If parallel line needs to be proven → use Converse of BPT.

🔥 Priority Revision

⭐⭐⭐ HIGH-PRIORITY PRACTICE

• Basic Proportionality Theorem (BPT) problems and proofs.
• Finding lengths using AA similarity criterion.
• Pythagoras Theorem calculations in right-angled triangles.

⭐⭐ IMPORTANT PRACTICE

• Area ratio calculations of similar triangles using corresponding sides.
• Using SAS and SSS criteria to prove triangle similarity.
• Converse of BPT application problems.

⭐ ADDITIONAL REVISION

• Identifying similar daily life figures and basic definitions.
• Using the Converse of Pythagoras Theorem to verify right angles.

⚡ Triangles Quick Revision Sheet

• Similar Triangles: Same shape, corresponding angles are equal, corresponding sides are in equal ratios.
• AA Criterion: 2 matching equal angles = similar triangles.
• SAS Criterion: 2 proportional side ratios + 1 equal included angle = similar triangles.
• SSS Criterion: 3 proportional side ratios = similar triangles.
• BPT: If DE || BC, then AD/DB = AE/EC.
• Area Ratio: Area1 / Area2 = (Side1 / Side2)².
• Pythagoras Theorem: In 90° triangle, Hypotenuse² = Base² + Perpendicular².
• Important Reminder: Always write triangle letters in exact corresponding order (e.g., ΔABC ~ ΔDEF).

📝 Test Yourself

1. In ΔABC, a line DE is drawn parallel to BC such that D lies on AB and E lies on AC. If AD = 3 cm, DB = 5 cm, and AE = 4.5 cm, calculate the length of EC.
2. Two triangles ΔXYZ and ΔPQR are given such that Angle X = 60°, Angle Y = 70°, Angle P = 60°, and Angle R = 50°. Determine whether ΔXYZ is similar to ΔPQR and state the criterion used.
3. In ΔABC and ΔDEF, AB = 4 cm, BC = 6 cm, DE = 8 cm, and EF = 12 cm. If Angle B = 45° and Angle E = 45°, prove that ΔABC is similar to ΔDEF.
4. The side lengths of ΔABC are AB = 3 cm, BC = 4 cm, and AC = 5 cm. The side lengths of ΔLMN are LM = 9 cm, MN = 12 cm, and LN = 15 cm. Show that ΔABC ~ ΔLMN.
5. In ΔPQR, points S and T lie on sides PQ and PR respectively. If PS = 2 cm, SQ = 6 cm, PT = 3 cm, and TR = 9 cm, use a theorem to check whether ST is parallel to QR.
6. ΔABC is similar to ΔDEF. If the side AB = 4 cm and the corresponding side DE = 7 cm, find the ratio of the area of ΔABC to the area of ΔDEF.
7. The areas of two similar triangles ΔABC and ΔPQR are 36 cm² and 81 cm² respectively. If the length of side PQ is 12 cm, find the length of the corresponding side AB.
8. A right-angled triangle has a hypotenuse of length 13 cm. If one of the remaining two sides is 12 cm, calculate the length of the third side.
9. A triangle has side lengths measuring 7 cm, 24 cm, and 25 cm. Determine whether this triangle is a right-angled triangle.
10. A vertical pole of length 5 m casts a shadow of 2 m on the ground. At the same time, a nearby tower casts a shadow of 10 m on the ground. Find the height of the tower using similarity of triangles.

🎯 Before Your Exam

☐ I understand similar figures and how they differ from congruent figures.

☐ I can identify whether two given triangles are similar.

☐ I know how to apply AA, SAS, and SSS similarity criteria correctly.

☐ I can match corresponding sides and angles without making order errors.

☐ I can state and apply the Basic Proportionality Theorem (BPT) to find unknown lengths.

☐ I understand how to calculate area ratios of similar triangles using side lengths.

☐ I can apply the Pythagoras Theorem to find missing sides in right-angled triangles.

☐ I can check whether a triangle is right-angled using the converse of Pythagoras Theorem.

Believe in your preparation, stay calm, and approach every question step by step!