Class 10 Maths Notes Chapter 6 — Triangles#

Similar figures#

• Similar figures have the same shape but not necessarily the same size.
• Congruent figures have the same shape and the same size.
• All congruent figures are similar, but similar figures need not be congruent.
• All circles are similar. All squares are similar. All equilateral triangles are similar.

Two polygons with the same number of sides are similar if both:

1. their corresponding angles are equal, and
2. their corresponding sides are in the same ratio (proportional).

One condition alone is not enough. A square and a rectangle have equal angles but are not similar, and a square and a rhombus have proportional sides but are not similar.

Similar triangles#

If $\triangle ABC \sim \triangle DEF$, with the vertices matched in that order:
$\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$

The order of letters tells you which parts correspond.

Basic Proportionality Theorem (BPT)#

Theorem 6.1 (BPT / Thales’ theorem): 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.

In $\triangle ABC$, if $DE \parallel BC$, then $\dfrac{AD}{DB} = \dfrac{AE}{EC}$.

Theorem 6.2 (Converse of BPT): If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

• From BPT you can also write $\dfrac{AD}{AB} = \dfrac{AE}{AC}$ and $\dfrac{DB}{AB} = \dfrac{EC}{AC}$.
• A line through the mid-point of one side, parallel to another side, bisects the third side.
• The line joining the mid-points of two sides is parallel to the third side.

Criteria for similarity of triangles#

• AA works because the third angles are then equal too (angle sum = 180°).
• Once triangles are proved similar, use corresponding sides in the same ratio to find lengths or prove ratios such as $CA^2 = CB \cdot CD$.
• Shadow problems: at the same time, the sun’s rays make equal angles with the ground, so the pole and its shadow and the tower and its shadow form similar right triangles.

Common mistakes#

• Writing the vertices in the wrong order, for example $\triangle ABC \sim \triangle QRP$ when $A$ matches $P$.
• In SAS, using an angle that is not between the two proportional sides.
• In BPT, comparing $\frac{AD}{DB}$ with $\frac{AE}{AC}$. The two ratios must be of the same kind (part : part or part : whole).
• Treating “angles equal” alone as similarity for polygons other than triangles.
• Confusing similar ($\sim$) with congruent ($\cong$).