CBSE Class 10 Mathematics (2026–27)
Chapter 6: Triangles
20 Important Questions and Answers
Q1. State the Basic Proportionality Theorem (BPT).
Answer:
The Basic Proportionality Theorem (BPT), also known as Thales’ Theorem, states that if a line is drawn parallel to one side of a triangle and intersects the other two sides at distinct points, then it divides those sides in the same ratio. In triangle ABC, if DE is drawn parallel to BC such that D lies on AB and E lies on AC, then:
[\frac{AD}{DB}=\frac{AE}{EC}]
This theorem is useful in proving proportional relationships in triangles and solving geometrical problems. It forms the basis for many concepts related to similarity and is widely used in coordinate geometry, construction, and practical applications involving measurements.
Q2. Write the converse of the Basic Proportionality Theorem.
Answer:
The converse of the Basic Proportionality Theorem states that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. Suppose in triangle ABC, points D and E lie on sides AB and AC respectively. If
[\frac{AD}{DB}=\frac{AE}{EC}]
then DE is parallel to BC. This theorem helps determine whether two lines are parallel based on proportional segments. It is frequently used in geometric proofs and construction problems. The converse theorem is important because it establishes a direct relationship between proportional division and parallel lines in triangles.
Q3. What are similar triangles?
Answer:
Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional. Similar triangles have the same shape but may differ in size. If triangles ABC and DEF are similar, then:
[\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}]
and
[\angle A=\angle D,\quad \angle B=\angle E,\quad \angle C=\angle F]
Similarity is represented by the symbol “~”. Similar triangles are useful in finding unknown lengths, heights, and distances without direct measurement. Many real-life applications such as map-making, architecture, and surveying use the concept of similar triangles.
Q4. Explain the AAA criterion of similarity.
Answer:
The AAA (Angle-Angle-Angle) similarity criterion states that if the three angles of one triangle are equal to the corresponding three angles of another triangle, then the two triangles are similar. Since the sum of angles in every triangle is 180°, equality of all corresponding angles ensures that the triangles have the same shape. Their corresponding sides are proportional, although not necessarily equal. This criterion is commonly used when angle measurements are known. AAA similarity helps in proving relationships between figures and solving geometry problems involving proportional sides and corresponding angles.
Q5. Explain the AA similarity criterion.
Answer:
The AA (Angle-Angle) similarity criterion states that if any two angles of one triangle are equal to any two corresponding angles of another triangle, then the triangles are similar. Since the third angle automatically becomes equal due to the angle sum property, all corresponding angles are equal. Therefore, the corresponding sides are proportional. This criterion is one of the simplest methods to prove similarity. It is useful in many geometric proofs and numerical problems. AA similarity is widely applied in finding unknown lengths and establishing proportional relationships between different triangles.
Q6. Explain the SAS similarity criterion.
Answer:
The SAS (Side-Angle-Side) similarity criterion states that two triangles are similar if two sides of one triangle are proportional to the corresponding two sides of another triangle and the included angles between those sides are equal. For example, if
[\frac{AB}{DE}=\frac{AC}{DF}]
and
[\angle A=\angle D]
then ΔABC ~ ΔDEF. This criterion is useful when complete side information is not available. It helps establish similarity by comparing only two sides and the included angle. SAS similarity is commonly used in geometry problems involving proportional measurements.
Q7. Explain the SSS similarity criterion.
Answer:
The SSS (Side-Side-Side) similarity criterion states that if the corresponding sides of two triangles are proportional, then the triangles are similar. For example, if:
[\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}]
then ΔABC ~ ΔDEF. This criterion does not require angle measurements. It is especially useful when all side lengths are known. Once similarity is established, many properties such as proportional sides, equal corresponding angles, and area relationships can be derived. SSS similarity plays an important role in geometry and practical applications involving scale drawings and models.
Q8. State the relationship between areas of similar triangles.
Answer:
The areas of two similar triangles are proportional to the squares of their corresponding sides. If two triangles are similar and the ratio of their corresponding sides is:
[\frac{a}{b}]
then the ratio of their areas is:
[\frac{\text{Area of first triangle}}{\text{Area of second triangle}}=\frac{a^2}{b^2}]
This theorem is useful for calculating areas without directly measuring dimensions. It is applied in geometry, engineering, architecture, and map scaling. The theorem shows that a small increase in side lengths can produce a much larger increase in area because area depends on the square of dimensions.
Q9. What is the significance of similar triangles in mathematics?
Answer:
Similar triangles are important because they help determine unknown lengths and distances without direct measurement. They provide a basis for proportional reasoning and geometric proofs. Similar triangles are widely used in trigonometry, surveying, architecture, navigation, and engineering. For example, the height of a building can be calculated using shadows and similar triangles. They also help in understanding scale models, maps, and photographs. Since corresponding angles are equal and sides are proportional, many complex geometric problems become easier to solve using similarity concepts.
Q10. State the Pythagoras Theorem.
Answer:
The Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If a right triangle has sides a and b and hypotenuse c, then:
[c^2=a^2+b^2]
This theorem is one of the most important results in geometry. It is used to calculate unknown sides of right triangles and has applications in construction, engineering, navigation, and physics. The theorem forms the basis of many advanced mathematical concepts and practical calculations involving distances.
Q11. State the converse of Pythagoras Theorem.
Answer:
The converse of Pythagoras Theorem states that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled. If a triangle has sides a, b, and c such that:
[c^2=a^2+b^2]
then the angle opposite side c is 90°. This theorem helps determine whether a given triangle is right-angled without measuring angles. It is widely used in geometry problems, construction work, and practical applications involving measurements and layouts.
Q12. How can similarity be used to find the height of a building?
Answer:
Similarity can be used to determine the height of a building by comparing it with another object of known height and measuring their shadows. If the sun’s rays create equal angles, the triangles formed by the objects and their shadows are similar. By using proportional sides, the unknown height can be calculated. For example, if a 2 m pole casts a 1 m shadow and a building casts a 10 m shadow, then:
[\frac{2}{1}=\frac{H}{10}]
Thus, H = 20 m. This method is commonly used in surveying and engineering.
Q13. Why are corresponding angles equal in similar triangles?
Answer:
Corresponding angles are equal in similar triangles because similarity is defined by having the same shape. Even if the size changes, the angle measures remain unchanged. Equal corresponding angles ensure that the sides maintain a constant ratio. This property helps in proving similarity and solving geometric problems. When triangles are similar, every angle in one triangle has an equal counterpart in the other. This feature is essential in applications involving maps, scale drawings, and architectural designs where shape preservation is important.
Q14. What happens to the ratio of sides in similar triangles?
Answer:
In similar triangles, the ratio of corresponding sides remains constant. This constant ratio is called the scale factor. If one triangle is enlarged or reduced, all corresponding sides change proportionally. For example, if the scale factor is 2, every side of the larger triangle is twice the corresponding side of the smaller triangle. This property helps determine unknown lengths and establish relationships between figures. The constant ratio of sides is one of the most important characteristics of similar triangles and forms the basis of many geometrical calculations.
Q15. Define the scale factor in similar triangles.
Answer:
The scale factor is the ratio of corresponding sides of two similar triangles. It indicates how much one triangle has been enlarged or reduced compared to the other. If:
[\frac{AB}{DE}=k]
then k is called the scale factor. When k > 1, the figure is enlarged, and when k < 1, it is reduced. The scale factor is useful in maps, blueprints, engineering drawings, and model construction. It helps maintain the same shape while changing the size proportionally. Understanding scale factors is essential for solving problems related to similarity.
Q16. How is BPT useful in geometry?
Answer:
The Basic Proportionality Theorem is useful because it helps establish proportional relationships between line segments in a triangle. It is often used to prove similarity of triangles, calculate unknown lengths, and verify parallel lines. BPT simplifies many geometrical constructions and proofs. It also provides the foundation for advanced concepts involving proportional division. Engineers, architects, and surveyors frequently apply similar principles when dealing with scaled representations and measurements. The theorem is considered one of the fundamental tools in Euclidean geometry.
Q17. What is the importance of the Pythagoras Theorem in daily life?
Answer:
The Pythagoras Theorem has many practical applications in everyday life. It is used in construction to ensure right angles in buildings and roads. Surveyors use it to calculate distances and heights. Navigation systems apply the theorem to determine the shortest paths between locations. Engineers use it in designing structures and machines. The theorem is also important in computer graphics, physics, and architecture. Its ability to relate side lengths in right triangles makes it one of the most widely used mathematical principles in practical situations.
Q18. How can similar triangles help in map-making?
Answer:
Map-making relies heavily on the concept of similarity. Maps are scaled representations of real locations, meaning all distances are reduced proportionally while preserving shape. Similar triangles help maintain accurate proportions between different features. Surveyors use triangulation methods based on similarity to determine distances and positions. Because corresponding sides remain proportional, measurements taken from maps can be converted into actual distances. This concept is also used in geographic information systems and engineering surveys, ensuring accurate representation of large areas on a smaller scale.
Q19. What conditions are necessary for triangles to be similar?
Answer:
Triangles are similar when they satisfy any one of the three similarity criteria: AA, SAS, or SSS. In the AA criterion, two corresponding angles are equal. In the SAS criterion, two sides are proportional and the included angle is equal. In the SSS criterion, all corresponding sides are proportional. Once similarity is established, corresponding angles become equal and corresponding sides remain proportional. These conditions ensure that the triangles have the same shape, although their sizes may differ. Similarity is a key concept in geometry and practical applications.
Q20. Distinguish between congruent and similar triangles.
Answer:
Congruent triangles are exactly equal in both shape and size. Their corresponding sides and angles are equal. Similar triangles, however, have the same shape but may differ in size. In similar triangles, corresponding angles are equal, but corresponding sides are proportional rather than equal. Congruence is represented by the symbol “≅”, while similarity is represented by “~”. Every congruent triangle is similar, but every similar triangle is not necessarily congruent. Understanding this distinction is important for solving geometric problems involving measurements, scaling, and transformations.