Triangles · Class 10 Mathematics: quick revision, 50 bites

What similar means

Same shape, any size

Two figures are similar if they have the same shape, whether or not they are the same size.

Takeaway. Angles stay put; lengths all scale together.

The difference

Similar or congruent?

Takeaway. Congruent is similar with scale factor 1.

Congruent ⟹ similar

One way only

ALL congruent figures are similar.

Takeaway. This exact sentence is an exam favourite.

The two conditions for polygons

Both, not either

Takeaway. The word AND is the whole point.

Square vs rectangle

Angles alone fail

Takeaway. This is why the ratio test exists.

Square vs rhombus

Sides alone fail

Takeaway. This is why the angle test exists.

Circles, squares, equilateral triangles

Always similar

Any two circles are similar; so are any two squares, and any two equilateral triangles.

Takeaway. Their shape is fixed; only size varies.

A tempting false rule

Not all rectangles

A 2 × 3 rectangle and a 2 × 9 rectangle are both rectangles.

Takeaway. Rectangles vary in shape; squares do not.

Correspondence

The letters carry the pairing

$ABCD \sim PQRS$ says A matches P, B matches Q, and so on.

Takeaway. Read the pairing off the letters, never off the picture.

Setting up the ratio

Keep the same figure on top

Write every fraction with the first figure's side as the numerator.

Takeaway. One direction, all the way through.

Scale factor

The number that scales everything

Takeaway. $k = 1$ means the figures are congruent.

Three lines

Finding an unknown side

Takeaway. $\frac69 = \frac{8}{QR}$ gives $QR = 12$.

Quadrilaterals 4, 6, 8, 10

A worked scale factor

Against $6, 9, 12, x$: $\frac46 = \frac69 = \frac8{12} = \frac23$.

Takeaway. Check three ratios before trusting the fourth.

How to answer

Name the test that failed

State both conditions, test each with numbers, then give the verdict.

Takeaway. A bare 'not similar' is not an answer.

What is coming

Triangles are easier

Polygons need both conditions checked. For triangles, one forces the other.

Takeaway. That is why this chapter is about triangles.

Maps and screens

Scale factors you use

A 1 : 50,000 map is similar to the ground; a cinema screen is similar to the film frame.

Takeaway. Every length scales, every angle stays.

The Basic Proportionality Theorem

A parallel line splits both sides alike

If $DE \parallel BC$ in triangle $ABC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.

Takeaway. Also known as Thales' Theorem.

Part : part and part : whole

The two forms

Takeaway. Choose one form and use it on both sides.

The commonest error here

Never mix the forms

$\frac{AD}{DB} = \frac{AE}{EC}$ ✓ and $\frac{AD}{AB} = \frac{AE}{AC}$ ✓

Takeaway. Decide the form first, then write both sides.

How to set it out

The proof, in four parts

Takeaway. The construction line alone usually carries a mark.

The first ratio

Two areas, one height

Takeaway. Taking DM instead gives $\frac{ar(ADE)}{ar(DEC)} = \frac{AE}{EC}$.

Equal areas

The step that does the work

$BDE$ and $DEC$ stand on the same base $DE$ and lie between the same parallels $DE$ and $BC$.

Takeaway. Same base, same parallels, same area.

Where marks are lost

Give the reason, not just the fact

Writing $ar(BDE) = ar(DEC)$ without 'same base, between the same parallels' loses the mark.

Takeaway. One clause, one mark.

What the proof uses

No angles, no similarity

The proof uses only areas — no similar triangles at all.

Takeaway. Using them here would be circular.

AD = 4, DB = 6, AE = 6

Finding a length

Takeaway. Quote the theorem before substituting.

The converse

The theorem backwards

If a line cuts two sides of a triangle in equal proportion, it runs parallel to the remaining side.

Takeaway. This is what proves parallelism.

Read what you are given

Which theorem do I need?

Takeaway. Using BPT to prove parallelism assumes the answer.

Is DE parallel to BC?

A converse question

$AD = 3$, $DB = 5$, $AE = 6$, $EC = 10$.

Takeaway. Equal ratios, therefore parallel.

Unequal ratios

And when it fails

With $AD = 2$, $DB = 3$, $AE = 4$, $EC = 5$: $\frac23 \ne \frac45$.

Takeaway. The converse decides both ways.

A condition in the statement

Distinct points

The line must cut the other two sides in DISTINCT points.

Takeaway. Quote the condition when you state the theorem.

How to finish

Name the theorem

End a parallelism answer with 'by the converse of the Basic Proportionality Theorem'.

Takeaway. Compute, compare, then name.

Getting between the forms

Adding one to both sides

$\frac{AD}{DB} + 1 = \frac{AB}{DB}$, because $AD + DB = AB$.

Takeaway. That is why both forms are true together.

Where the name comes from

Measuring a pyramid

Thales is said to have found a pyramid's height from the length of its shadow, using exactly this ratio idea.

Takeaway. The same idea scales a projected image.

One condition is enough

Why triangles get their own chapter

For polygons you must check equal angles AND proportional sides.

Takeaway. That is what these three criteria say.

AAA and AA

Angles alone

Takeaway. AA is the version you will use most.

Why AA works

Two angles give the third

$\angle C = 180° - \angle A - \angle B$ and $\angle R = 180° - \angle P - \angle Q$.

Takeaway. The third angle is never new information.

SSS

Sides alone

Takeaway. Then the angles are equal automatically.

Checking SSS

All three, not two

$3, 4, 5$ and $6, 8, 11$ agree in the first two ratios.

Takeaway. Two matching ratios prove nothing.

SAS

One angle, two sides

Takeaway. The letters S-A-S record that order.

The SAS trap

'Including' is the whole condition

The equal angle must be the INCLUDED angle.

Takeaway. Check that the angle sits between the two sides.

Angle A equal, sides 4, 6 and 6, 9

A worked SAS

Takeaway. Both ratios and the included angle.

3, 4, 5 and 6, 8, 10

A worked SSS

$\frac36 = \frac48 = \frac5{10} = \frac12$.

Takeaway. Their angles must then be equal too.

Read what you are given

Which criterion?

Takeaway. The question chooses the criterion for you.

Writing the statement

The order of the letters

$\triangle ABC \sim \triangle PQR$ asserts A↔P, B↔Q, C↔R.

Takeaway. Follow the matching, not the alphabet.

After similarity is established

Finding a length

Takeaway. $\frac64 = \frac9{QR}$ gives $QR = 6$.

Not independent facts

They come from the BPT

All three criteria are proved from the Basic Proportionality Theorem.

Takeaway. You are asked to APPLY them, not prove them.

Where a mark lives

Name the criterion

Finish with 'by AA similarity' or 'by SAS similarity'.

Takeaway. State it, do not just imply it.

A Class IX confusion

Not the congruence rules

Congruence uses SSS, SAS, ASA and RHS with EQUAL sides.

Takeaway. Equal for congruence, proportional for similarity.

RHS similarity

A fourth criterion

If the hypotenuse and one side of one right triangle are proportional to those of another, the triangles are similar.

Takeaway. The textbook notes it; it shortens some proofs.

AA in the open air

A tree without a ladder

A tree and its shadow, and a stick and its shadow, form two similar triangles by AA — the sun's rays give the equal angle.

Takeaway. Measure the shadows, compute the height.