Class 10 Maths Triangles Notes

1. Similar Figures

Two figures are similar when they have the same shape, though their sizes may be different.

For similar polygons:

Corresponding angles are equal.
Corresponding sides are proportional.

The common ratio between corresponding sides is called the scale factor.

Important:

Every pair of congruent figures is similar.
Similar figures need not be congruent.

2. Similarity of Triangles

Two triangles are similar when their corresponding angles are equal and their corresponding sides have the same ratio.

△ABC ~ △DEF

then the correspondence is:

A ↔ D
B ↔ E
C ↔ F

Therefore, $\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$

Correct vertex order is important when writing a similarity statement.

3. Basic Proportionality Theorem (BPT)

Theorem

If a line is drawn parallel to one side of a triangle and cuts the other two sides, it divides those two sides in the same ratio.

In △ABC, if DE ∥ BC, where D lies on AB and E lies on AC, then $\frac{AD}{DB}=\frac{AE}{EC}$

Why it matters

BPT is one of the main tools for finding unknown lengths in triangles containing parallel lines.

4. Converse of BPT

The reverse is also useful:

If a line divides two sides of a triangle in the same ratio, then that line is parallel to the third side.

Thus, in △ABC, $\frac{AD}{DB}=\frac{AE}{EC}$

implies $DE\parallel BC$

Quick recognition

Parallel line given → use BPT.
Equal side ratios given → use Converse of BPT.

5. Criteria for Similarity of Triangles

There are three main criteria.

AA — Angle–Angle

If two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar. $\angle A=\angle D,\qquad \angle B=\angle E$

Therefore, $\triangle ABC\sim\triangle DEF$

The third pair of angles is automatically equal because the angles of a triangle add up to $180^\circ$ .

SSS — Side–Side–Side

If the three corresponding sides of two triangles are proportional, the triangles are similar. $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$

Therefore, $\triangle ABC\sim\triangle DEF$

Exam tip: Match corresponding sides carefully before comparing ratios.

SAS — Side–Angle–Side

If:

one angle of one triangle equals the corresponding angle of another triangle, and
the two sides containing those angles are proportional,

then the triangles are similar.

For example, $\frac{AB}{DE}=\frac{AC}{DF}$

and $\angle A=\angle D$

then $\triangle ABC\sim\triangle DEF$

Important: The equal angle must be the angle between the two proportional sides.

6. How to Prove Two Triangles Similar

A reliable method:

Step 1: Identify the two triangles.

Step 2: Look for:

equal angles → AA
three proportional sides → SSS
two proportional sides + included equal angle → SAS

Step 3: Establish the correspondence of vertices.

Step 4: Write the similarity statement in the correct order.

Step 5: Use corresponding sides or angles to obtain the required result.

7. Similarity and Indirect Measurement

Similarity allows us to find quantities that are difficult to measure directly.

Examples include:

heights of tall objects,
lengths of shadows,
distances that cannot be measured directly.

Typical shadow problem

If a tall object and a smaller object form right triangles with their shadows, the triangles are usually similar by AA.

Then corresponding lengths can be compared: $\frac{\text{height of tall object}} {\text{height of small object}} = \frac{\text{shadow of tall object}} {\text{shadow of small object}}$

The key is to identify corresponding sides correctly.

8. Useful Results from Similar Triangles

If two triangles are similar, then:

Corresponding sides are proportional

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$

Corresponding angles are equal

$\angle A=\angle D,\quad \angle B=\angle E,\quad \angle C=\angle F$

Medians correspond proportionally

When corresponding triangles are similar, their corresponding medians have the same ratio as their corresponding sides.

For example, $\frac{CM}{RN}=\frac{AB}{PQ}$

when the relevant triangles and medians correspond as established in the chapter.

9. Midpoint–Parallel-Line Results

Two important results connect with BPT:

Result 1

A line through the midpoint of one side of a triangle, parallel to another side, bisects the third side.

Result 2

The line joining the midpoints of two sides of a triangle is parallel to the third side.

These are frequent applications of BPT and its converse.

10. RHS Similarity

For two right-angled triangles, if the hypotenuse and one corresponding side are proportional, the triangles are similar.

This is called the RHS Similarity Criterion.

11. Most Important Theorems to Remember

Concept	What to remember
Similar figures	Same shape, size may differ
Polygon similarity	Equal corresponding angles + proportional corresponding sides
BPT	Parallel line → same ratio on two sides
Converse BPT	Same ratio → parallel line
AA	Two corresponding angles equal
SSS	Three corresponding sides proportional
SAS	Two proportional sides + included equal angle
RHS	Right triangles: proportional hypotenuse + one side

12. Common Mistakes to Avoid

Do not assume figures are similar just because they look alike.
In BPT, make sure the line is actually parallel to the required side.
In SAS, use the angle included between the proportional sides.
Do not mix corresponding sides from different vertex pairings.
Write similarity statements in the correct vertex order.
Before using a ratio, clearly identify which lengths correspond.

13. One-Page Revision Formula Sheet

BPT

$DE\parallel BC \quad\Rightarrow\quad \frac{AD}{DB}=\frac{AE}{EC}$

Converse BPT

$\frac{AD}{DB}=\frac{AE}{EC} \quad\Rightarrow\quad DE\parallel BC$

Similar triangles

$\triangle ABC\sim\triangle DEF$

implies $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$

and $\angle A=\angle D,\quad \angle B=\angle E,\quad \angle C=\angle F$

Similarity tests

AA: 2 equal corresponding angles
SSS: 3 proportional corresponding sides
SAS: 2 proportional sides + included equal angle
RHS: Right triangles + proportional hypotenuse and one side