Similar Triangles
Theory
Two triangles are similar if they pass one of four tests for similar triangles: SSS, SAS, AA or RHS. This Year 9 maths page shows how to choose the right test from what is marked, find equal angles hidden in a figure, and then use the scale factor to find an unknown length, including heights found from shadows and ladders.
To prove two triangles are similar, you do not need to check every angle and every side. One of four tests is enough.
SSS: all three pairs of sides are in the same ratio.
SAS: two pairs of sides are in the same ratio, and the included angles (the angles between those sides) are equal.
AA: two pairs of angles are equal. The third pair must then be equal too, because the angles in a triangle add to \(180^\circ\). Such triangles are called equiangular.
RHS: both triangles are right-angled, and the hypotenuses and one other pair of sides are in the same ratio.
Once two triangles are proved similar, write a similarity statement with the vertices in matching order, find the scale factor, and use it to find unknown sides.
| Test | What you need to show |
|---|---|
| SSS | \(\dfrac{a'}{a} = \dfrac{b'}{b} = \dfrac{c'}{c}\) |
| SAS | two side ratios equal, and the angles between those sides equal |
| AA | two pairs of equal angles |
| RHS | a right angle in each, and \(\dfrac{\text{hyp}'}{\text{hyp}} = \dfrac{\text{side}'}{\text{side}}\) |
How to prove triangles similar and find a length
- Look at what is marked or given: sides, angles, right angles, parallel lines.
- Choose the test: three sides (SSS), two sides and the angle between them (SAS), two angles (AA), or a right angle with the hypotenuse and a side (RHS).
- Show the test holds: divide matching sides (matched by size), or state each pair of equal angles with a reason.
- Write the similarity statement with the vertices in matching order.
- Use the scale factor to find the unknown side.
Real problems
- Shadows: at the same moment the sun makes the same angle for every object, and both stand at right angles to the ground, so the triangles are similar (AA). Height matches height and shadow matches shadow.
- Ladders, cables and ramps at the same angle to the ground make similar right-angled triangles with the wall or pole.
- Write the unknown on top: \(\dfrac{h}{\text{its match}} = \dfrac{\text{known}}{\text{its match}}\). A \(1.2\) m post with a \(0.9\) m shadow and a building with a \(15\) m shadow give \(\dfrac{h}{1.2} = \dfrac{15}{0.9}\), so \(h = 20\) m.
- Not similar? One ratio that differs is enough to show it. Two equal ratios are not enough for SSS: check the third pair too.
Put both sets in order, pair smallest with smallest, and divide.
| \(\dfrac{12}{6}\) | \(=\) | \(2\) |
| \(\dfrac{14}{7}\) | \(=\) | \(2\) |
| \(\dfrac{18}{9}\) | \(=\) | \(2\) |
All three ratios are equal, so the triangles are similar by SSS. No angle was needed.
Pair the sides by size and divide. Then check the angle is the one between the two sides in both triangles.
| \(\dfrac{9}{6}\) | \(=\) | \(1.5\) |
| \(\dfrac{15}{10}\) | \(=\) | \(1.5\) |
Two pairs of sides are in the same ratio and the included angles are both \(70^\circ\). The triangles are similar by SAS.
Compare hypotenuse with hypotenuse, then the second pair of sides.
| \(\dfrac{18}{30}\) | \(=\) | \(\dfrac{3}{5}\) |
| \(\dfrac{15}{25}\) | \(=\) | \(\dfrac{3}{5}\) |
Both triangles have a right angle, and the two ratios agree. The test is RHS.
Find two pairs of equal angles, giving a reason for each.
| \(\angle ACB\) | \(=\) | \(\angle ECD\) (vertically opposite) |
| \(\angle BAC\) | \(=\) | \(\angle DEC\) (alternate, \(AB \parallel DE\)) |
So \(\triangle ABC \sim \triangle EDC\) by AA. \(AC\) matches \(EC\), and \(AB\) matches \(ED\).
| \(k\) | \(=\) | \(\dfrac{CE}{CA} = \dfrac{10}{4} = 2.5\) |
| \(DE\) | \(=\) | \(6 \times 2.5\) |
| \(\) | \(=\) | \(\mathbf{15}\) |
Common pitfalls
Frequently asked questions
What are the four tests for similar triangles?
SSS: three pairs of sides in the same ratio. SAS: two pairs of sides in the same ratio with equal included angles. AA: two pairs of equal angles. RHS: right-angled triangles with the hypotenuses and one other pair of sides in the same ratio.
What is the AA similarity test?
If two angles of one triangle equal two angles of another, the triangles are similar. The third angles are then equal too, because each triangle's angles add to \(180^\circ\).
What is the difference between SAS for similar and congruent triangles?
For congruent triangles the two pairs of sides must be equal. For similar triangles they only need to be in the same ratio. In both, the angle between the sides must be equal.
How do you know which similarity test to use?
Use what the diagram gives you. Three sides: SSS. Two sides and the angle between them: SAS. Two angles: AA. A right angle with the hypotenuse and another side: RHS.
How do you find a missing side once triangles are similar?
Write the similarity statement in matching order, find the scale factor from a pair of matching sides you know, and multiply the matching side by it.