Year 11 General Shape And Measurement

Similar Triangles

10 practice questions 2 video lessons Theory + worked examples

Theory

Two triangles are similar when they have the same shape (same three angles) but possibly different sizes. This page covers the AA, SAS and SSS similarity tests, the scale factor, parallel-line setups, and worked examples on ladders and mirror reflections.

Two triangles are similar when they have the same shape — the same three angles — but possibly different sizes. Their matching sides are all in the same ratio, called the scale factor $k$ .

To prove similarity, you only need one of three tests. AA (Angle-Angle) is the most-used: two pairs of matching angles being equal forces the third pair to be equal too. SAS (Side-Angle-Side) checks two pairs of matching sides in the same ratio with the included angles equal. SSS (Side-Side-Side) checks all three side pairs in the same ratio.

Once similarity is established, set up a ratio of corresponding sides to find unknown lengths. The most common setup in problems is a line parallel to one side of a triangle — this automatically creates a smaller triangle similar to the original.

Two similar triangles. Same three angles, sides in the ratio

1 : 2

A line parallel to one side creates a smaller similar triangle (AA).

Once two triangles are known to be similar, all matching sides are in the same ratio:

\frac{side in big triangle}{matching side in small triangle} = k

\frac{big}{small} = k

The three similarity tests:

Test	What it checks
AA	Two pairs of matching angles equal
SAS	Two pairs of matching sides in the same ratio, with the included angle equal
SSS	All three pairs of matching sides in the same ratio

Vertex notation. If $△ A B C \sim △ D E F$ , the order of vertices tells you the correspondence: $A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ . So $A B$ matches $D E$ , $B C$ matches $E F$ , and $A C$ matches $D F$ .

How to solve any similar-triangles problem

Prove the triangles are similar using AA, SAS or SSS. In parallel-line and shadow setups, AA is almost always the right choice.
Match corresponding sides: the side opposite the same angle in each triangle pairs together; the longest of each pairs with the longest.
Set up a ratio of matching sides and solve for the unknown — keep the same triangle on top throughout the equation.

Spotting hidden similar triangles. Look for shared angles (one angle common to both triangles), parallel lines (equal corresponding angles), or right-angle marks (one pair already equal). Mirror reflections and shadows are classic AA setups.

Example 1 — Find an Unknown Side

Two similar triangles have matching sides in the ratio

3 : 5

. The smaller triangle has a side of

12

cm. Find the matching side in the larger triangle.

Solution

Set up the ratio with the bigger triangle on top of each fraction.

$\frac{x}{12}$	$=$	$\frac{5}{3}$
$x$	$=$	$12 \times \frac{5}{3}$
$x$	$=$	$20 cm$

x = 20 cm

Example 2 — Parallel Line in a Triangle

△ A B C

, the line

D E

is parallel to

B C

with

D

A B

and

E

A C

. Given

A D = 6

D B = 4

A E = 9

. Find

E C

Solution

$△ A D E \sim △ A B C$ (AA: shared angle at $A$ , parallel lines give equal corresponding angles).

$\frac{A E}{A C}$	$=$	$\frac{A D}{A B}$
$\frac{9}{9 + E C}$	$=$	$\frac{6}{10}$
$9 \times 10$	$=$	$6 (9 + E C)$
$90$	$=$	$54 + 6 \cdot E C$
$E C$	$=$	$6 cm$

EC = 6 cm

Example 3 — Ladder Problem

6

m ladder leans on a wall and reaches

4.8

m up. A second ladder,

9

m long, leans at the same angle on the same wall. How high does it reach?

Solution

Same angle plus shared right angle gives AA — the two right triangles are similar.

$\frac{h}{4.8}$	$=$	$\frac{9}{6}$
$h$	$=$	$4.8 \times 1.5$
$h$	$=$	$7.2 m$

h = 7.2 m

Example 4 — Mirror Problem

A student

1.5

m tall stands

2

m from a small mirror on the ground. The top of a tree,

8

m on the other side of the mirror, is just visible in the mirror. Find the height of the tree.

Solution

The angle of incidence equals the angle of reflection, so the two right triangles share equal angles — they are similar by AA.

$\frac{tree}{1.5}$	$=$	$\frac{8}{2}$
tree	$=$	$1.5 \times 4$
tree	$=$	$6 m$

tree = 6 m

Common pitfalls

Mismatched sides. Always pair corresponding sides — the longest side of one triangle with the longest of the other, the side opposite the equal angle in one with the side opposite the equal angle in the other.

Ratio flipped on one side. Keep the same triangle on top throughout:

\frac{big}{big} = \frac{small}{small}

, never mixed.

Reading the vertex notation wrong.

△ A B C \sim △ D E F

means

A

pairs with

D

B

with

E

C

with

F

— in that exact order. Don't pair by position on the page.

Confusing similar with congruent. Similar triangles can be different sizes (scale factor any positive number). Congruent triangles are identical (scale factor

1

Frequently asked questions

What does it mean for two triangles to be similar?

Two triangles are similar when they have the same shape — the same three angles — but possibly different sizes. Their matching sides are all in the same ratio, called the scale factor.

What are the three similarity tests?

AA (angle-angle): two pairs of matching angles are equal. SAS (side-angle-side): two pairs of matching sides are in the same ratio and the included angles are equal. SSS (side-side-side): all three pairs of matching sides are in the same ratio. Any one of these is enough to prove similarity.

What is the difference between similar and congruent triangles?

Similar triangles have the same shape but possibly different sizes — matching sides are in a constant ratio. Congruent triangles have the same shape AND the same size, so matching sides are equal. Congruence is the special case of similarity where the scale factor equals 1.

How do I find an unknown side in similar triangles?

Set up a ratio of corresponding sides. Pair the unknown side with its matching side in the other triangle, and pair it with another known matching pair. Then solve the proportion. Always match smaller with smaller (or larger with larger) on both sides of the equation.

Why does a line parallel to one side of a triangle create similar triangles?

Because the parallel line creates equal corresponding angles where it meets the two other sides, and the original angle at the apex is shared by both triangles. Two pairs of equal angles means AA similarity.

What does the notation triangle ABC similar to triangle DEF tell me?

The order of the vertices tells you the correspondence: A matches D, B matches E, C matches F. So side AB matches side DE, BC matches EF, and AC matches DF. Always pair sides by their corresponding vertex labels.