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Similar triangles have the same shape but not necessarily the same size. That idea lets you find the height of a tower from its shadow, or a missing length in a figure, without measuring it. In Class 10 it comes down to one theorem you must prove and a few criteria you must use correctly. This post covers what is in the syllabus, sets out the proof, works three problems, and lists the slips that lose marks.
What does the Class 10 syllabus include now?
CBSE's Class X Mathematics syllabus for 2026-27 lists one proof under Triangles: "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." Everything else is to be stated without proof: the converse of that theorem and the three similarity criteria (AAA, SSS and SAS). Geometry as a unit carries 15 of the 80 theory marks, shared with circles.
The NCERT chapter, rationalised and reprinted for 2026-27, ends at the criteria for similarity. The theorems on areas of similar triangles and the Pythagoras theorem proof are no longer sections of this chapter, so do not prepare them for the board paper unless your current syllabus says so.
How do you prove the basic proportionality theorem?
The theorem is also called Thales' theorem in the NCERT book.
Given: In triangle ABC, a line DE parallel to BC meets AB at D and AC at E. To prove: AD/DB = AE/EC. Construction: Join BE and CD. Draw DM perpendicular to AC and EN perpendicular to AB.
Proof.
Area of ADE = ½ × AD × EN, and area of BDE = ½ × DB × EN. So ar(ADE)/ar(BDE) = AD/DB.
Area of ADE = ½ × AE × DM, and area of DEC = ½ × EC × DM. So ar(ADE)/ar(DEC) = AE/EC.
Triangles BDE and DEC are on the same base DE and between the same parallels DE and BC, so ar(BDE) = ar(DEC).
From 1, 2 and 3, AD/DB = AE/EC.
Write "Given", "To prove", "Construction" and "Proof" as headings, and draw the figure first. Board marking schemes give marks for this structure.
Worked examples
Example 1. Using the theorem. In triangle ABC, DE is parallel to BC, with D on AB and E on AC. AD = 3 cm, DB = 5 cm and AE = 4.5 cm. Find EC.
By the basic proportionality theorem, AD/DB = AE/EC.
3/5 = 4.5/EC, so EC = (4.5 × 5)/3 = 7.5 cm.
Example 2. Using the converse. In triangle PQR, S is on PQ and T is on PR, with PS = 2 cm, SQ = 3 cm, PT = 3 cm and TR = 4.5 cm. Is ST parallel to QR?
PS/SQ = 2/3. PT/TR = 3/4.5 = 2/3.
The ratios are equal, so by the converse of the theorem, ST is parallel to QR.
Example 3. Shadows. At the same time of day, a 2 m pole casts a 1.5 m shadow and a tower casts a 24 m shadow. Find the height of the tower.
The sun's rays make the same angle with the ground for both, and both objects are vertical. So the two triangles formed by object, shadow and ray have two pairs of equal angles, and are similar by the AA criterion.
Corresponding sides are proportional: height/shadow is the same. So h/24 = 2/1.5.
h = 24 × 2 ÷ 1.5 = 32 m.
Why explain each step to yourself?
Geometry questions reward the reason behind each line. In a classic study, Chi and colleagues (1989) recorded students talking aloud while they studied worked mechanics examples. The stronger students produced many self-explanations that linked each step to a principle and noticed when they did not understand; the weaker students produced few and leaned on the examples later. When you read a solved triangle problem, say why each ratio or angle is equal before moving on.
Which mistakes lose marks?
Mixing up AD/DB and AD/AB. Both are valid, but only if you use the matching ratio on the other side: AD/DB = AE/EC, or AD/AB = AE/AC. Never AD/DB = AE/AC.
Vertices out of order. See the callout.
Naming a criterion that does not exist. There is no SSA similarity criterion. Use AAA (or AA), SSS or SAS, where the angle is between the two sides.
Treating similar as congruent. Similar triangles have equal angles and proportional sides, not equal sides.
Not stating the reason. Write "(by BPT)" or "(AA similarity)" after the line that uses it.
No figure. Draw and label one, even if the question gives one, and mark the parallel lines.
If you understand the theorem but freeze on a new figure, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can see which pair of triangles to pick and why.
A one-week practice routine
Day 1: Write the BPT proof from memory with figure and headings. Check against NCERT.
Day 2: Five problems using the theorem to find a length.
Day 3: Three problems using the converse to test for parallel lines.
Day 4: Four similarity problems. For each, write which criterion you used and the matching vertex order.
Day 5: Two real-life problems, such as shadows or a lamp post, then redo the proof.
To make self-explanation a habit, read our guide to the self-explanation technique. For how geometry answers are marked in the board paper, see common Class 10 maths board mistakes.
Today, draw triangle ABC with DE parallel to BC and write the proof without looking, headings first.
