Triangles (Class 10): Notes और MCQ Practice

आसान भाषा में

Triangles similar तब होते हैं जब उनका shape same हो, भले ही size different हो! यानी इनके corresponding angles बराबर होने चाहिए और corresponding sides का ratio same (proportional) होना चाहिए। याद रखें, सभी congruent figures similar होते हैं, लेकिन सभी similar figures congruent नहीं होते।

Important Points & Theorems

• Basic Proportionality Theorem (BPT / Thales Theorem):

यदि किसी triangle में एक side के parallel line खींची जाए जो बाकी दो sides को distinct points पर intersect करे, तो वह sides same ratio में divide होती हैं।

In ΔABC, if DE || BC, then: AD/DB = AE/EC

• Converse of BPT:

यदि कोई line triangle की दो sides को same ratio में divide करे, तो वह third side के parallel होती है।

• Criteria for Similarity of Triangles:

1. AAA / AA Criterion: यदि दो triangles के corresponding angles equal हों, तो वे similar होते हैं (दो angles equal होना भी काफी है)।
2. SSS Criterion: यदि corresponding sides का ratio equal हो (AB/DE = BC/EF = AC/DF), तो triangles similar होते हैं।
3. SAS Criterion: यदि एक angle equal हो और उसे include करने वाली sides proportional हों।

2 Memory Tricks

1. "P for Parallel, P for Proportion": BPT में जब भी line Parallel दिखे, समझ जाओ sides का Proportion equal करना है।
2. "AA is Enough": Triangles में तीसरा angle naturally equal हो ही जाता है (180° rule से), इसलिए 3 angles चेक करने का time waste मत करो, बस 2 angles ढूंढो!

2 Common Board-Exam Mistakes

1. Vertex Order / Lettering की गलती: ΔABC ~ ΔDEF लिखते समय vertices का order exactly match होना चाहिए। AB/DE = BC/EF = AC/DF। गलत order लिखने पर full marks कटते हैं!
2. SAS में Non-included Angle लेना: SAS similarity तभी valid है जब angle दोनों proportional sides के बिल्कुल बीच में (included angle) हो, किसी कोने वाले angle पर नहीं।

Triangles - Class 10 Revision Notes

1. Chapter in Short

In Class 9, you studied congruent figures (figures that have the same shape and the same size). In Class 10, the focus shifts to similar figures—figures that have the same shape, but not necessarily the same size.

This chapter teaches:

• What makes two geometric figures similar.
• Basic Proportionality Theorem (BPT) or Thales Theorem, and its converse.
• The three core criteria for similarity of triangles: AAA (or AA), SSS, and SAS.

2. Key Concepts Explained Simply

Congruent vs Similar

• Congruent figures: Exact copies of each other. Same shape AND same size.

Everyday example: Two brand-new 500-rupee notes from the bank.

• Similar figures: Same shape, but different sizes.

Everyday example: A photograph of you and its passport-size copy, or a big circle and a small coin.

• Crucial Rule: All congruent figures are similar, but all similar figures are NOT necessarily congruent.

Similarity of Polygons and Triangles

Two triangles are similar if:

1. Their corresponding angles are equal, AND
2. Their corresponding sides are in the same ratio (proportional).

If in Triangle ABC and Triangle DEF:

• Angle A = Angle D, Angle B = Angle E, Angle C = Angle F
• AB ÷ DE = BC ÷ EF = AC ÷ DF

Then, Triangle ABC ~ Triangle DEF (the symbol '~' means 'is similar to').

Basic Proportionality Theorem (BPT / Thales Theorem)

• Concept: If you draw a line parallel to one side of a triangle intersecting the other two sides, it divides those two sides in the exact same ratio.
• In Triangle ABC, if a line DE is parallel to BC (where D is on AB and E is on AC), then:

AD ÷ DB = AE ÷ EC

Converse of BPT

• If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.
• If AD ÷ DB = AE ÷ EC, then DE is parallel to BC.

Criteria for Similarity of Triangles

Instead of checking all 3 angles and all 3 pairs of sides, we can use these shortcuts:

1. AAA (or AA) Criterion:

• If two angles of one triangle are equal to two angles of another triangle, the third angles must also be equal (angle sum property), so the triangles are similar.

1. SSS Criterion:

• If all three pairs of corresponding sides are in the same ratio, the triangles are automatically similar.

1. SAS Criterion:

• If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the triangles are similar.

(Note: The angle must be between the two proportional sides).

3. Important Definitions, Formulas, and Facts

• Thales of Miletus: A famous Greek mathematician who formulated the Basic Proportionality Theorem.
• Equiangular Triangles: Two triangles are equiangular if their corresponding angles are equal. The ratio of any two corresponding sides in equiangular triangles is always the same.
• Symbol of Similarity: ~ (read as "is similar to").
• BPT Relation:

In Triangle ABC, if DE || BC:

• AD ÷ DB = AE ÷ EC
• Also useful: AD ÷ AB = AE ÷ AC, and DB ÷ AB = EC ÷ AC
• Corresponding Parts of Similar Triangles:

If Triangle ABC ~ Triangle PQR:

• Angle A = Angle P, Angle B = Angle Q, Angle C = Angle R
• AB ÷ PQ = BC ÷ QR = AC ÷ PR
• Order of Letters Matters: Writing Triangle ABC ~ Triangle DEF means A matches D, B matches E, and C matches F. Writing Triangle ABC ~ Triangle EDF means something entirely different and could be marked wrong.

4. Solved Examples (Board-Exam Style)

Example 1: Finding an unknown segment using BPT

Question: In Triangle ABC, DE || BC. If AD = 1.5 cm, DB = 3 cm, and AE = 1 cm, find EC.

Solution:

1. Since DE || BC, by the Basic Proportionality Theorem (BPT):

AD ÷ DB = AE ÷ EC

1. Substitute the given values:

1.5 ÷ 3 = 1 ÷ EC

1. Simplify the left side:

1 ÷ 2 = 1 ÷ EC

1. Cross-multiply:

EC = 2 cm

Answer: EC = 2 cm

Example 2: Finding an unknown variable using BPT

Question: In Triangle ABC, DE || BC. If AD = x, DB = x − 2, AE = x + 2, and EC = x − 1, find the value of x.

Solution:

1. By BPT, since DE || BC:

AD ÷ DB = AE ÷ EC

1. Substitute the expressions:

x ÷ (x − 2) = (x + 2) ÷ (x − 1)

1. Cross-multiply:

x × (x − 1) = (x + 2) × (x − 2)

1. Expand both sides:

x² − x = x² − 4

1. Subtract x² from both sides:

−x = −4

x = 4

Answer: x = 4

Example 3: Shadow problem using AA Similarity

Question: A vertical pole of length 6 m casts a shadow 4 m long on the ground. At the same time, a nearby tower casts a shadow 28 m long. Find the height of the tower.

Solution:

1. Let the pole be AB = 6 m and its shadow be BC = 4 m.

Let the tower be PQ = h m and its shadow be QR = 28 m.

1. In Triangle ABC and Triangle PQR:

• Angle B = Angle Q = 90° (both stand vertical to the ground)
• Angle C = Angle R (sun's elevation is the same for both at the same time)

1. Therefore, by AA similarity criterion:

Triangle ABC ~ Triangle PQR

1. Since corresponding sides are proportional:

AB ÷ PQ = BC ÷ QR

6 ÷ h = 4 ÷ 28

1. Simplify:

6 ÷ h = 1 ÷ 7

h = 6 × 7 = 42 m

Answer: The height of the tower is 42 m.

Example 4: Proving a Trapezium Property

Question: Diagonals of a trapezium ABCD with AB || DC intersect each other at point O. Show that AO ÷ BO = CO ÷ DO.

Solution:

1. Consider Triangle AOB and Triangle COD:

• Angle OAB = Angle OCD (alternate interior angles, as AB || DC and AC is a transversal)
• Angle OBA = Angle ODC (alternate interior angles, as AB || DC and BD is a transversal)

1. Therefore, by AA similarity criterion:

Triangle AOB ~ Triangle COD

1. Since corresponding sides of similar triangles are proportional:

AO ÷ CO = BO ÷ DO

1. Rearranging terms (cross-multiplying and dividing):

AO ÷ BO = CO ÷ DO

Hence Proved.

5. Common Mistakes and Quick Revision Points

Common Mistakes

• Wrong side ratio with BPT: Students sometimes write DE ÷ BC = AD ÷ DB. This is incorrect!

BPT only gives AD ÷ DB = AE ÷ EC.

To relate DE and BC, you must use similarity of Triangle ADE and Triangle ABC, which gives: DE ÷ BC = AD ÷ AB.

• Ignoring vertex correspondence: Saying Triangle ABC ~ Triangle PQR means Angle A = Angle P. Do not write names in random order.
• Confusing SAS criterion: The angle in SAS MUST be the included angle (the angle between the two proportional sides). If the angle is elsewhere, SAS cannot be applied.

Quick Revision Points

• Two circles are always similar.
• Two squares are always similar.
• Two equilateral triangles are always similar.
• Parallel lines in a triangle? Think BPT first.
• Comparing shadows or heights? Think AA similarity.