Skip to content
Learnijoy

Class 10 · Maths · Chapter 6 · NCERT Class 10 Mathematics

Triangles Class 10 Notes

Free here: the full mind map and the first 3 of 6 parts of the notes. The rest is free with an account.

Chapter mind map

The whole chapter at a glance: the big idea, then each branch and what sits under it.

Similarity of Triangles

The study of figures with the same shape but not necessarily the same size, focusing on proportional sides and equal angles.

  1. Fundamentals of Similarity

    Criteria for general polygons and the distinction between congruence and similarity.

    • Polygon Conditions — Two polygons are similar if all corresponding angles are equal and all corresponding sides are in the same ratio.
    • Scale Factor — Also known as the Representative Fraction, it is the constant ratio of corresponding sides.
    • Congruence vs Similarity — All congruent figures are similar, but similar figures are only congruent if they have the same size.

  2. Basic Proportionality Theorem

    Also known as Thales Theorem, it relates parallel lines to side ratios within a triangle.

    • Theorem 6.1 (BPT) — A line parallel to one side of a triangle divides the other two sides in the same ratio.
    • Theorem 6.2 (Converse) — If a line divides two sides of a triangle in the same ratio, it must be parallel to the third side.
    • Area-Based Proof — Uses the ratio of areas of triangles with common altitudes and bases between parallel lines.

  3. Similarity Criteria

    Specific conditions that guarantee two triangles are similar without checking all angles and sides.

    • AA / AAA Criterion — If two angles of one triangle equal two angles of another, the triangles are similar.
    • SSS Criterion — If all three pairs of corresponding sides are proportional, the corresponding angles are equal.
    • SAS Criterion — Requires one pair of equal angles and that the sides including these angles are proportional.
    • RHS Similarity — Two right triangles are similar if their hypotenuses and one pair of sides are proportional.

  4. Practical Applications

    Using similarity for indirect measurement and solving geometric properties.

    • Indirect Measurement — Calculating heights of towers or mountains using shadows and similar triangle ratios.
    • Median Ratios — In similar triangles, the ratio of corresponding medians is equal to the ratio of corresponding sides.
    • Vertex Correspondence — Critical for writing similarity statements like ΔABC ~ ΔDEF to ensure correct side pairing.

Chapter notes

An exploration of the concept of similarity in triangles, covering the Basic Proportionality Theorem, criteria for similarity (AAA, SSS, SAS), and their practical applications in indirect measurements.

01 · Explore

Introduction to Similarity

In geometry, similarity describes figures that share the same shape but are not necessarily the same size.

You have previously studied congruence, where two figures must have exactly the same shape and the same size. Similarity is a broader concept: all congruent figures are similar, but similar figures are not necessarily congruent. For example, all circles are similar because they are always round, but they are only congruent if they have the same radius.

The principle of similarity allows for indirect measurement. This is how mathematicians and scientists calculate the heights of mountains like Mount Everest or the distance to the moon without using a physical measuring tape. By using the properties of similar triangles, we can relate known small-scale measurements to unknown large-scale distances.

Two polygons with the same number of sides are similar if two conditions are met: first, all corresponding angles must be equal; second, all corresponding sides must be in the same ratio. This ratio is known as the scale factor or Representative Fraction.

ABCPQRAB/PQ = BC/QR = AC/PR
Two triangles with the same shape. Corresponding sides have one common scale factor; corresponding angles are equal. Learning sketch; use the labels and stated dimensions, not measurements from the picture.

Pause & Try

Think it through first. Writing and checking your answer is free with an account.

Question

Are all squares similar? Why or why not?

Sign in to see the answer

Write your own answer and compare it with ours. It’s free.

Sign inNew here? Sign up free

Question

If two polygons have equal corresponding angles, are they definitely similar?

Sign in to see the answer

Write your own answer and compare it with ours. It’s free.

Sign inNew here? Sign up free

NCERT reference: chapter PDF pages 1, 2, 3, 4.

02 · Explore

Basic Proportionality Theorem (Thales Theorem)

The Basic Proportionality Theorem (BPT) is a fundamental result relating parallel lines and side ratios in a triangle.

Theorem 6.1 states: 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. This is also known as Thales Theorem, named after the Greek mathematician Thales.

To prove it, let D be on AB and E on AC, with DE parallel to BC. Triangles ADE and BDE have bases AD and DB on the same line AB and share the altitude from E, so area(ADE)/area(BDE) = AD/DB. Similarly, triangles ADE and CDE share the altitude from D to AC, so area(ADE)/area(CDE) = AE/EC. Triangles BDE and CDE have the same base DE and lie between the same parallel lines DE and BC, so their areas are equal. Therefore AD/DB = AE/EC.

The converse (Theorem 6.2) is also true: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side. This is verified by assuming a line is not parallel and showing that the parallel line must coincide with the original line.

For the converse, suppose AD/DB = AE/EC. Draw through D a line parallel to BC, meeting AC at E′. The theorem gives AD/DB = AE′/E′C. Hence AE/EC = AE′/E′C. The same internal ratio selects the same point on AC, so E and E′ coincide. Thus DE is parallel to BC.

Applying BPT to find a missing segment

AD/DB = AE/EC

In triangle ABC, DE || BC. If AD = 1.5 cm, DB = 3 cm, and AE = 1 cm, find EC. Substituting the values: 1.5/3 = 1/EC. Simplifying the left side gives 1/2 = 1/EC. Therefore, EC = 2 cm.

Pause & Try

Think it through first. Writing and checking your answer is free with an account.

Question

In triangle PQR, a line ST is drawn such that S is on PQ and T is on PR. If PS/SQ = 3/4 and PT/TR = 3/4, what can you conclude about ST?

Sign in to see the answer

Write your own answer and compare it with ours. It’s free.

Sign inNew here? Sign up free

NCERT reference: chapter PDF pages 7, 8, 9, 10.

03 · Explore

AAA and AA Similarity Criteria

The Angle-Angle-Angle (AAA) criterion simplifies the process of proving two triangles are similar.

Theorem 6.3 (AAA Criterion) states that if the corresponding angles of two triangles are equal, then their corresponding sides are automatically in the same ratio, making the triangles similar. This is unique to triangles; in other polygons, equal angles do not guarantee proportional sides.

A more practical version is the AA similarity criterion. Since the sum of angles in a triangle is always 180 degrees, if two angles of one triangle are equal to two angles of another, the third angles must also be equal. Therefore, showing just two pairs of equal angles is sufficient to prove similarity.

When writing similarity statements, vertex correspondence is critical. If triangle ABC is similar to triangle DEF, we write ΔABC ~ ΔDEF. This implies ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F, as well as the ratio AB/DE = BC/EF = CA/FD.

Logic of AA Similarity

  1. 1

    Identify Angles

    Find two pairs of corresponding equal angles in two triangles.

  2. 2

    Angle Sum Property

    The third pair of angles must be equal because total sum is 180°.

  3. 3

    Establish Similarity

    Conclude the triangles are similar by AA criterion.

  4. 4

    Proportional Sides

    Conclude that all corresponding side ratios are now equal.

The sequence of reasoning that allows two angles to prove full similarity and side proportionality.

Pause & Try

Think it through first. Writing and checking your answer is free with an account.

Question

Triangle ABC has angles 60° and 80°. Triangle PQR has angles 80° and 40°. Are they similar?

Sign in to see the answer

Write your own answer and compare it with ours. It’s free.

Sign inNew here? Sign up free

NCERT reference: chapter PDF pages 13, 14, 15, 16.

The rest of this chapter

Keep reading Triangles, free

  1. Locked: 1. SSS Similarity Criterion
  2. Locked: 2. SAS Similarity Criterion
  3. Locked: 3. Practical Applications and Problem Solving

Create a free account and you will continue right here, at the next section. You also get Joy, your AI tutor, a practice quiz, chapter videos and the NCERT chapter itself.

Continue reading — freeAlready have an account? Log in

All Class 10 Maths chapters