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How did Thales of Miletus measure the height of the Great Pyramid of Giza 2,600 years ago using only his walking stick and sunlight shadows? The secre...
How did Thales of Miletus measure the height of the Great Pyramid of Giza 2,600 years ago using only his walking stick and sunlight shadows? The secret is Triangle Similarity and the Basic Proportionality Theorem.
Why This Chapter Matters
In Class 10 Mathematics, "Triangles" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
Before You Begin (Prerequisites)
- Congruence of triangles from Class 9.
- Ratios and proportions.
- Angles and parallel lines.
What You Will Learn (Core Objectives)
- Distinguish between Congruent figures (same shape & size) and Similar figures (same shape, proportional size).
- State and prove the Basic Proportionality Theorem (BPT / Thales Theorem).
- State and apply the Converse of the Basic Proportionality Theorem.
- Apply Similarity Criteria: AAA (or AA), SSS, and SAS similarity.
- Solve complex board examination proofs using BPT and triangle similarity.
Chapter Roadmap & Progression
1
1. Similarity vs. Congruence
2
2. Basic Proportionality Theorem (T...
3
3. Criteria for Triangle Similarity
Complete Concept Guide (100% Curriculum Coverage)
1. Similarity vs. Congruence
Two polygons are Similar ($\sim$) if their corresponding angles are equal and their corresponding sides are in the same ratio (proportional). All circles, all squares, and all equilateral triangles are similar!
2. Basic Proportionality Theorem (Thales Theorem)
BPT Theorem: If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, the other two sides are divided in the same ratio: $$\mathbf{DE \parallel BC \implies \frac{AD}{DB} = \frac{AE}{EC}}$$
Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side!
3. Criteria for Triangle Similarity
- AAA / AA Criterion: If two angles of one triangle are equal to two angles of another, the triangles are similar.
- SSS Criterion: If corresponding sides are proportional: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$.
- SAS Criterion: One angle equal and including sides proportional.
Visual Learning & Conceptual Map
Triangles Master Matrix
Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture
1. Similarity vs. Congruence • 2. Basic Proportionality Theorem (Thales Theorem)
Chapter Summary & 10 Key Takeaways
Takeaway 1
Similarity: Same shape, proportional sides ($\sim$).
Takeaway 2
BPT (Thales): Parallel line divides two sides in equal ratio ($\frac{AD}{DB} = \frac{AE}{EC}$).
Takeaway 3
AA Criterion: Two equal angles guarantee triangle similarity.
Takeaway 4
SAS Similarity: Included angle equal with proportional adjacent sides.
Takeaway 5
High-Yield Proof: BPT proof is a permanent 5-mark CBSE Board favorite.
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
In $\triangle ABC$, $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If $AD = 1.5\text{ cm}, DB = 3\text{ cm},$ and $AE = 1\text{ cm}$, find $EC$.
Reveal Answer & Explanation
Answer: By BPT: $\frac{AD}{DB} = \frac{AE}{EC} \implies \frac{1.5}{3} = \frac{1}{EC} \implies \frac{1}{2} = \frac{1}{EC} \implies EC = 2\text{ cm}$.
BPT formula.
2
State the Basic Proportionality Theorem.
Reveal Answer & Explanation
Answer: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
BPT definition.
3
A vertical pole of length 6 m casts a shadow 4 m long on the ground, and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
Reveal Answer & Explanation
Answer: By AA similarity: $\frac{\text{Height of Tower}}{\text{Height of Pole}} = \frac{\text{Shadow of Tower}}{\text{Shadow of Pole}} \implies \frac{h}{6} = \frac{28}{4} = 7 \implies h = 42\text{ meters}$.
Height = 42 m.
4
If $\triangle ABC \sim \triangle DEF$, $AB = 4\text{ cm}, DE = 6\text{ cm}$, and perimeter of $\triangle ABC = 16\text{ cm}$, find the perimeter of $\triangle DEF$.
Reveal Answer & Explanation
Answer: Ratio of perimeters equals ratio of corresponding sides: $\frac{\text{Perimeter } ABC}{\text{Perimeter } DEF} = \frac{AB}{DE} = \frac{4}{6} = \frac{2}{3} \implies \text{Perimeter } DEF = 16 \times \frac{3}{2} = 24\text{ cm}$.
Perimeter = 24 cm.
5
Are all congruent triangles similar? Are all similar triangles congruent?
Reveal Answer & Explanation
Answer: All congruent triangles are similar (ratio of sides is 1:1), but similar triangles are NOT necessarily congruent because their sizes can differ.
Congruent implies similar, not vice versa.
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