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SparkEd Math offers 60+ practice questions for Triangles across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

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Yes! All Triangles content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Triangles differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Triangles problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Triangles content aligned with the CBSE syllabus?

Yes, all Triangles questions and lessons on SparkEd Math are fully aligned with the Central Board of Secondary Education (CBSE) syllabus for Class 10. The content follows the official curriculum structure and exam pattern.

Triangles — Class 10 CBSE (2025-26)

About Triangles — Class 10 CBSE

Prove and apply BPT, similarity criteria, and Pythagoras theorem with proofs. This topic is part of the CBSE Class 10 mathematics syllabus (chapter: Chapter 6). On this page you can practice 62 questions across three difficulty levels — 20 easy, 20 medium, and 22 hard — each with a visual step-by-step solution, plus a timed 37-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Triangles

Introduction to Similar Triangles
Basic Proportionality Theorem (BPT)
Criteria for Similarity of Triangles
Pythagoras Theorem and its Converse
Summary and Practice Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Triangles — solved examples for Class 10 CBSE

Example 1easy

Consider a triangle ABC. A line DE is drawn parallel to BC, intersecting AB at D and AC at E. Which of the following statements correctly represents the Basic Proportionality Theorem (BPT)?

A)AD/AB = AE/AC
B)AD/DB = AE/EC
C)AD/DE = AB/BC
D)DB/AB = EC/AC

Step-by-step solution

The Basic Proportionality Theorem (BPT), also known as Thales Theorem, states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
In ΔABC, if DE || BC, then according to BPT, the ratio of the segments on AB is equal to the ratio of the segments on AC.
$AD/DB = AE/EC$

Answer: AD/DB = AE/EC

Example 2medium

Which of the following statements is TRUE regarding similar figures?

A)A. All congruent figures are similar.
B)B. All similar figures are congruent.
C)C. Similar figures always have the same size.
D)D. Similar figures must have corresponding sides equal.

Step-by-step solution

Congruent figures have both the same shape and the same size. Similar figures have the same shape but not necessarily the same size.
Since congruent figures have the same shape, they meet the criterion for similarity. Thus, all congruent figures are similar.
However, similar figures only require the same shape, so they are not necessarily congruent (they might have different sizes).

Answer: A. All congruent figures are similar.

Example 3hard

In a triangle ABC, a line DE is drawn parallel to BC, intersecting AB at D and AC at E. If AD/DB = 3/2, and the area of ΔADE is 27 cm², what is the area of the trapezium DBCE?

A)18 cm²
B)27 cm²
C)48 cm²
D)75 cm²

Step-by-step solution

Given AD/DB = 3/2. This means AD = 3k and DB = 2k for some constant k. So, AB = AD + DB = 3k + 2k = 5k.
Since DE || BC, by AA similarity criterion (∠ADE = ∠ABC and ∠AED = ∠ACB, common ∠A), ΔADE ~ ΔABC.
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area(ΔADE) / Area(ΔABC) = (AD/AB)² = (3k/5k)² = (3/5)² = 9/25.
Given Area(ΔADE) = 27 cm². So, 27 / Area(ΔABC) = 9/25. This gives Area(ΔABC) = 27 × (25/9) = 3 × 25 = 75 cm². The area of trapezium DBCE = Area(ΔABC) - Area(ΔADE) = 75 cm² - 27 cm² = 48 cm².

Answer: 48 cm²

Practice questions on Triangles

Q1.easy
In triangle PQR, points S and T are on PQ and PR respectively. If PS = 3 cm, SQ = 6 cm, PT = 4 cm, and TR = 8 cm, which of the following statements is true?
1. A)ST || QR
2. B)ST is perpendicular to QR
3. C)ST bisects ∠P
4. D)ST is twice QR
Show answer
Answer: ST || QR
Hint: Check if the ratios of the segments on the sides are equal.
Q2.easy
Consider two triangles, ΔABC and ΔDEF. If ∠A = 50°, ∠B = 60°, ∠D = 50°, and ∠F = 70°, which similarity criterion would prove ΔABC ~ ΔDEF?
1. A)SSS Similarity
2. B)AA Similarity
3. C)SAS Similarity
4. D)None of these
Show answer
Answer: AA Similarity
Hint: First, find the missing angles in both triangles using the angle sum property. Then compare the angles.
Q3.easy
Two triangles, ΔPQR and ΔXYZ, have sides such that PQ = 6 cm, QR = 8 cm, PR = 10 cm and XY = 3 cm, YZ = 4 cm, XZ = 5 cm. Which of the following statements is true about these triangles?
1. A)They are congruent.
2. B)They are similar by SAS criterion.
3. C)They are similar by SSS criterion.
4. D)They are not similar.
Show answer
Answer: They are similar by SSS criterion.
Hint: Check the ratios of corresponding sides. Ensure you match the smallest side with the smallest, medium with medium, and largest with largest.
Q4.medium
In ΔABC, a line DE is drawn parallel to BC, intersecting AB at D and AC at E. If AD = 3 cm, DB = 4 cm, and AE = 4.5 cm, what is the length of EC?
1. A)A. 6 cm
2. B)B. 4.5 cm
3. C)C. 7.5 cm
4. D)D. 9 cm
Show answer
Answer: A. 6 cm
Hint: Apply the Basic Proportionality Theorem (Thales Theorem) which relates the ratios of segments formed by a parallel line.
Q5.medium
In ΔPQR, points S and T are on sides PQ and PR respectively. If PS = 3.9 cm, SQ = 3.6 cm, PT = 3 cm, and TR = 2.8 cm, then which of the following is true?
1. A)A. ST || QR
2. B)B. ST is perpendicular to QR
3. C)C. ST intersects QR at midpoint
4. D)D. ST is not parallel to QR
Show answer
Answer: D. ST is not parallel to QR
Hint: To check for parallelism, use the converse of the Basic Proportionality Theorem. Compare the ratios of the segments.
Q6.medium
In ΔABC and ΔPQR, it is given that ∠A = ∠P and ∠B = ∠Q. If AB = 6 cm, BC = 8 cm, AC = 10 cm, and PQ = 9 cm, find the length of QR.
1. A)A. 10 cm
2. B)B. 12 cm
3. C)C. 15 cm
4. D)D. 13.5 cm
Show answer
Answer: B. 12 cm
Hint: Since two angles are equal, the triangles are similar by AA criterion. Then, the ratio of corresponding sides will be equal.
Q7.hard
Consider a quadrilateral ABCD. E and F are points on AD and BC respectively such that EF is parallel to AB. If AE/ED = BF/FC, which of the following statements must be true?
1. A)AB || DC
2. B)AD || BC
3. C)AC and BD bisect each other
4. D)Triangle ABE is similar to Triangle CDF
Show answer
Answer: AB || DC
Hint: Recall the converse of the Basic Proportionality Theorem (BPT) and how it applies when a line divides two sides of a triangle proportionally. Consider constructing a diagonal.
Q8.hard
Two poles of heights 'a' metres and 'b' metres are 'p' metres apart. The poles are perpendicular to the ground. The lines joining the top of each pole to the foot of the other pole intersect at a height 'h' metres from the ground. Which of the following expressions correctly represents 'h'?
1. A)h = (a + b) / 2
2. B)h = ab / (a + b)
3. C)h = √(ab)
4. D)h = (a² + b²) / (a + b)
Show answer
Answer: h = ab / (a + b)
Hint: Draw a diagram. Identify similar triangles formed by the intersecting lines and the poles. Use ratios of corresponding sides.
Q9.hard
Given two triangles, ΔABC and ΔXYZ. If AB/XY = BC/YZ = CA/ZX = k, and k = 1, then the triangles are congruent. If k ≠ 1, the triangles are similar. Which of the following statements about SSS similarity is always true?
1. A)If the perimeters of ΔABC and ΔXYZ are equal, then k must be 1.
2. B)If the areas of ΔABC and ΔXYZ are equal, then k must be 1.
3. C)If k = 2, then the altitude from A to BC is twice the altitude from X to YZ.
4. D)All of the above.
Show answer
Answer: All of the above.
Hint: Recall the properties of similar triangles related to perimeter, area, and corresponding altitudes. The ratio of perimeters is the same as the ratio of sides, and the ratio of areas is the square of the ratio of sides.

These are 9 of the 62 questions available for Triangles. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.