Triangles | Maths Grade 10 CBSE

Play with it

The Thales theorem, live

Drag the line DE up and down — it stays parallel to the base BC. No matter where it sits, AD/DB = AE/EC. That's the Basic Proportionality Theorem.

△ABC with DE ∥ BCdrag the slider

position of DE 0.45

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The three ideas in this chapter

Two figures are similar (written A ∼ B) if they have the same shape — even if their sizes differ. Photos enlarged, maps, and scale models are all similar to the original.

Two triangles are similar when both conditions hold:

Corresponding angles are equal, and
Corresponding sides are in the same ratio (proportional).

Similar vs congruent

Congruent = same shape AND size (ratio 1). Similar = same shape, any size. Every congruent pair is also similar.

Watch out: equal angles alone make triangles similar, but for other polygons you need both equal angles and proportional sides (e.g. a square and a rectangle have equal angles but aren't similar).

You don't need to check every angle and side. Any one of these is enough:

AA — two angles of one triangle equal two angles of the other (the third is then automatic).
SSS — all three pairs of corresponding sides are proportional.
SAS — one pair of equal angles, with the two sides containing that angle proportional.

Worked example · find a missing angle

△ABC ∼ △DEF with ∠A = 50° and ∠B = 60°. Find ∠F.

Angles of △ABC sum to 180°: ∠C = 180 − 50 − 60 = 70°.
In similar triangles, ∠F corresponds to ∠C.
So ∠F = 70°.

The star theorem of this chapter:

If DE ∥ BC, then AD/DB = AE/EC

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio. The converse is also true: if a line divides two sides in the same ratio, it must be parallel to the third side.

Worked example · find EC

In △ABC, DE ∥ BC with AD = 2 cm, DB = 3 cm and AE = 4 cm. Find EC.

By BPT: AD/DB = AE/EC → 2/3 = 4/EC.
Cross-multiply: 2 × EC = 3 × 4 = 12.
EC = 6 cm.

Common mistake: pairing the wrong segments. AD goes with AE (both from vertex A) and DB with EC — keep the "top with top, bottom with bottom" order.

Why this matters

Where you'll actually use this

Similar triangles let you measure things you can't reach — the height of a tower, the width of a river — using only a shadow or a ruler. Surveyors, architects and photographers rely on it daily.

Scaling without distortion

Resize a photo, a logo or a building plan and every length grows by the same scale factor while the angles stay fixed — that's similarity. Get the scale factor wrong on one side and the image looks stretched. Designers and architects keep figures similar so nothing warps.

Scale factor

Measure a tower from its shadow

A stick and a tower in the same sunlight cast shadows that form similar triangles — the sun's rays hit both at the same angle. So height ÷ shadow is the same for both: 2/3 = h/30 → the tower is 20 m tall. No ladder required.

Indirect measurement

🗺️ Maps & scale models

Every map is a similar, scaled-down copy of the land — the scale is the ratio of similitude.

📷 Cameras & projectors

A lens forms a similar image; object and image triangles share the same ratios.

📐 Surveying rivers

Surveyors find an unreachable width by building a similar triangle on the near bank.

🔒 More real-world applications

Maps, cameras, surveying and more — each explained with a diagram. Free to unlock.

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Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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