All ________ triangles are similar.
- (a)isosceles
- (b)equilateral
- (c)right-angled
- (d)scalene
Triangles Class 10: notes and important questions
Revision notes, 40 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.
- 40 questions
- 12 multiple choice, 3 assertion–reason, 9 very short answer, 8 short answer, 5 long answer, 3 case study
- About 14 hours to master
Geometry unit: 15 of 80 theory marks (Triangles and Circles).
Revision notes
Triangles — revision notes
1. Similar figures
- Two polygons with the same number of sides are similar if (i) corresponding angles are equal and (ii) corresponding sides are in the same ratio. Both conditions are needed for polygons (a square and a rhombus fail (i)).
- All circles, all squares and all equilateral triangles are similar. Congruent shapes are similar (ratio \(1:1\)), but not conversely.
2. Basic Proportionality Theorem (BPT / Thales)
If \(DE \parallel BC\) in \(\triangle ABC\) (\(D\) on \(AB\), \(E\) on \(AC\)), then \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\). Consequences: \(\dfrac{AD}{AB} = \dfrac{AE}{AC}\), \(\dfrac{DB}{AB} = \dfrac{EC}{AC}\).
Proof idea (examinable): compare areas. \(\dfrac{\text{ar}(ADE)}{\text{ar}(BDE)} = \dfrac{AD}{DB}\) and \(\dfrac{\text{ar}(ADE)}{\text{ar}(DEC)} = \dfrac{AE}{EC}\); and \(\text{ar}(BDE) = \text{ar}(DEC)\) (same base \(DE\), same parallels).
Converse: if \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\), then \(DE \parallel BC\).
Note: \(DE\) itself is not in the ratio \(AD : DB\); use \(\dfrac{DE}{BC} = \dfrac{AD}{AB}\) (from similarity).
3. Criteria for similarity of triangles
- AAA / AA: two pairs of equal angles (the third then matches).
- SSS: all three pairs of sides in the same ratio.
- SAS: two pairs of sides in the same ratio and the included angles equal.
Write the correspondence in order: \(\triangle ABC \sim \triangle PQR\) means \(A \leftrightarrow P\), \(B \leftrightarrow Q\), \(C \leftrightarrow R\).
4. Useful consequences
- Ratio of perimeters = ratio of corresponding sides = ratio of corresponding altitudes/medians.
- Trapezium \(ABCD\) (\(AB \parallel DC\)) with diagonals meeting at \(O\): \(\triangle AOB \sim \triangle COD\), so \(\dfrac{OA}{OC} = \dfrac{OB}{OD} = \dfrac{AB}{CD}\).
- Shadow problems: vertical objects and parallel sun rays give similar triangles (AA).
Worked example 1
\(DE \parallel BC\), \(AD = 5\), \(DB = 3\), \(AC = 12\). Then \(\dfrac{AE}{AC} = \dfrac{AD}{AB} = \dfrac58 \Rightarrow AE = 7.5\).
Worked example 2
Sides \(5, 12, 13\) and \(7.5, 18, 19.5\): ratios all \(\dfrac23\), so similar by SSS.
Worked example 3
A \(1.8\) m man casts a \(2.4\) m shadow; a pole casts a \(10\) m shadow at the same time: height \(= \dfrac{1.8}{2.4} \times 10 = 7.5\) m.
Common errors
- Writing \(\dfrac{AD}{DB} = \dfrac{DE}{BC}\) — wrong; use \(\dfrac{AD}{AB} = \dfrac{DE}{BC}\).
- Matching vertices in the wrong order, leading to wrong side ratios.
- Using SAS similarity with an angle that is not between the two sides.
- Claiming two rhombuses/rectangles are similar from sides alone.
Board-exam tips
- In proofs, state the reason for every equality (given, common, alternate angles, corresponding angles, BPT, criterion).
- Draw and label your own rough sketch for every triangle question, even when none is printed.
- The BPT proof is a frequently asked 3-mark item: learn it with the construction.
Topics in this chapter: Similar figures · Basic Proportionality Theorem and its converse · Criteria for similarity of triangles · Applications of similarity.
Route to 95: four steps
Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.
Step 1
Secure the basics
You know when figures are similar and can use the Basic Proportionality Theorem to find missing lengths.
Step 2
Board standard
You can prove triangles similar using AA, SAS or SSS with a reason for every step, and use the converse of BPT.
Step 3
Full marks on long answers
You can write the BPT proof and the long 'prove that' answers the board asks, with figure, given, to prove, construction and proof.
Step 4
95+ stretch (HOTS)
You can handle the hard proofs: altitudes, parallelograms with extended sides, and algebraic BPT problems.
Practice questions
Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 6 of the 40 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.
Which of the following pairs of shapes are always similar?
- (a)Two rectangles
- (b)Two rhombuses
- (c)Two isosceles triangles
- (d)Two squares
Which of the following statements is true?
- (a)Any two similar triangles are congruent.
- (b)Any two right triangles are similar.
- (c)Any two congruent triangles are similar.
- (d)Any two isosceles triangles are similar.
Where marks are lost in Triangles
- Writing AD/DB = DE/BC. Fix: DE/BC matches AD/AB (whole side), not AD/DB; write the similarity statement first and read ratios from it.
- Similarity statement with vertices in the wrong order (△ABC ~ △QPR when angle A matches angle P). Fix: match equal angles first, then write the letters in that order.
- Steps in a proof with no reasons. Fix: every equality needs a bracket: (given), (common), (corresponding angles), (BPT), (AA criterion). R marks depend on it.
- Using SAS with an angle that is not between the two sides. Fix: check that the angle is the included angle before quoting SAS.
- BPT proof without the construction (join BE, CD; draw EN ⟂ AB, DM ⟂ AC). Fix: learn the proof with the construction written as a separate line.
- No figure in a geometry proof. Fix: draw and label a rough figure even when none is printed; it usually carries a mark in 3- and 5-mark proofs.
Triangles in our sample papers
Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).
- Sample paper 1 (Board standard): questions 8, 23, 28, 33
- Sample paper 2 (Board standard): questions 9, 23, 29, 33
- Sample paper 3 (Above board): questions 10, 12, 19, 28, 33
- Sample paper 4 (Above board): questions 9, 29, 33
- Sample paper 5 (95+ challenge): questions 9, 10, 28, 33