Similarity & Proportionality

Equal angles force proportional sides. That single fact measures buildings, mountains and stars.

Draw a triangle, then draw a line across it parallel to the base.

The line cuts the other two sides. Does it cut them in the same ratio?

a line drawn parallel to one side 6 4 9 ? parallel it cuts both sides — but does it cut them in the same ratio?

Measure and you find it does: if it cuts one side 4 to 6, it cuts the other 6 to 9 — both 2:3.

The reason is that the small triangle at the top has the same angles as the whole one.

Parallel lines gave you corresponding angles, and the apex angle is shared.

Two matching angles are enough — that is AA similarity. And similar triangles have proportional sides, which is why one measurement plus one shadow can give you the height of a tree.

a line drawn parallel to one side 6 4 9 ? parallel it cuts both sides — but does it cut them in the same ratio?

Why two angles are enough

A triangle has three angles that add to 180°. Fix two of them and the third has no choice.

So two matching angles guarantee all three match, and the triangles are similar. That is AA, and it is the test you will use most.

Note the contrast with congruence: AAA is useless there, because it says nothing about size. For similarity, size is exactly what we are not fixing.

Two triangles of different sizes with angles of 50 and 60 degrees; the third angle is forced to be 70 degrees in both, so they are similar (AA) fix two angles and the third one has no choice 50°60°70°50°60°70° 50° + 60° = 110° 180° − 110° = 70° the third angle isforced in both AA → similar AA fixes the shape,not the size

Where the angles come from

Parallel lines hand you corresponding or alternate angles. A shared angle is congruent to itself. Those two sources supply almost every AA proof.

The side-based tests

SSS similarity: all three pairs of sides in the same ratio. Compute all three and check they agree.

SAS similarity: two pairs in ratio, with the angles between them congruent. The word "included" matters exactly as it did for congruence.

Use these when the question gives lengths rather than angles.

The triangle proportionality theorem

A line parallel to one side of a triangle divides the other two sides proportionally.

If the line cuts one side into a and b, and the other into c and d, then ab = cd.

Its converse is also true: if a line cuts two sides proportionally, it must be parallel to the third.

The triangle proportionality theorem: a line parallel to one side cuts the other two sides into pieces a, b and c, d with a/b = c/d a line parallel to one side cuts the other two sides in the same ratio ab cd ab=cd top piece over bottom piece,on both sides or whole over part on both: a + ba=c + dc converse: if the ratios match, the line must be parallel to the third side

Match the pieces carefully

Top piece over bottom piece on both sides — or whole over part on both. What you must not do is mix the two, which is where most errors here come from.

Indirect measurement

A person 5 ft tall casts a 4 ft shadow; a tree casts a 32 ft shadow at the same moment.

The sun's rays arrive parallel, so the two triangles have the same angles, so they are similar.

54 = h32 gives h = 40 ft. You measured a tree with a tape measure and no ladder.

Indirect measurement: a 5 ft person casts a 4 ft shadow and a tree casts a 32 ft shadow; the sun's parallel rays make similar triangles, so the tree is 40 ft tall the sun’s rays arrive parallel, so the two triangles are similar h 5 ft4 ft32 ft same sun, same angle,so the triangles are similar 54=h32 h = 32 × 5 ÷ 4 h = 40 ft

Perimeter, area and volume

If the scale factor is k, then perimeters are in the ratio k, areas in the ratio k2, and volumes in the ratio k3.

Two similar triangles with sides in ratio 2:3 have areas in ratio 4:9.

Going backwards, areas in ratio 9:16 means sides in ratio 3:4 — take the square root.

Worked examples

AA similarity

In △ ABC, DE is parallel to BC with D on AB and E on AC. Prove △ ADE ∼ △ ABC.

  1. ∠ A ≅ ∠ A — reflexive The apex angle is shared by both triangles.
  2. ∠ ADE ≅ ∠ ABC — corresponding angles DE ∥ BC cut by transversal AB.
  3. Two pairs of congruent angles. That is enough.
  4. △ ADE ∼ △ ABC by AA The third pair follows automatically.

Similar by AA

Triangle proportionality

A line parallel to the base cuts one side into 4 (top) and 6 (bottom), and the other side into 6 (top) and x (bottom). Find x.

  1. Top over bottom on both sides: 46 = 6x Match the pieces the same way on each side.
  2. Cross-multiply: 4x = 36 Clear the fractions.
  3. x = 9 Check: 46 = 69, both 23 ✓

x = 9

Area from a scale factor

Two similar pentagons have corresponding sides 6 cm and 15 cm. The smaller has area 48 cm2. Find the larger area.

  1. Scale factor: k = 156 = 2.5 New over old.
  2. Areas scale by k2 = 6.25 Both dimensions grew.
  3. 48 × 6.25 = 300 The larger area.
  4. Sense check: 2.5 times longer should be far more than 2.5 times the area ✓ And 300 is over six times 48.

300 cm2

Words to know

Similar
Same shape, possibly different size. Written △ ABC ∼ △ DEF.
AA similarity
Two pairs of congruent angles prove two triangles similar.
SSS similarity
All three pairs of sides in the same ratio.
SAS similarity
Two pairs of sides in ratio with the included angles congruent.
Geometric mean
For a, x, b in proportion, x = √ab.
Scale factor
The multiplier k between corresponding lengths.

Mistakes to watch for

Using AAA for congruence. Equal angles give similar triangles only — the sizes are still free. For congruence you need at least one side, because something has to fix the scale.

Mixing part-to-part with whole-to-part in a proportion. Use top-over-bottom on both sides, or whole-over-part on both; switching halfway sets up an equation that is simply not the theorem.

Scaling area by k instead of k2. An area is a length times a length and both lengths scaled, so a factor of 3 multiplies area by 9. Volumes go by k3 for the same reason.

Pairing sides by appearance rather than by position. Corresponding sides are the ones between corresponding angles — mark the equal angles first and the pairing follows.

Eratosthenes measured the circumference of the Earth around 240 BC using two sticks and their shadows, in cities about 800 km apart.

Similar triangles gave him the angle between them: about one fiftieth of a circle. Multiply the distance by fifty and you have the Earth.

His answer was within a few percent of the modern figure — obtained with no instrument more advanced than a stick.

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