Tests for Similar Triangles
Two triangles are similar when their matching angles are equal (AA), all three pairs of sides are in the same ratio (SSS), or two pairs of sides are in the same ratio with equal included angles (SAS). This NSW Year 9 Advanced maths subtopic uses these tests (NESA outcome MA5‑GEO‑P‑01).
You will name the test for a pair of triangles, write the similarity statement and use the scale factor to find sides. You will find heights from shadows, ladders and cables, the width of a river and the length of a ramp, decide whether two sides in proportion are enough, prove that \(AB^2 = AD \times AC\), and find a tree's height by looking over a fence. It is part of Year 9 Advanced Congruence and Similar Figures.
Every question with a fully worked solution.
- Tests for Similar Triangles - Video - Tests For Similar Triangles SSS Watch
- Tests for Similar Triangles - Video - Tests For Similar Triangles SAS Watch
- Tests for Similar Triangles - Video - Tests For Similar Triangles AAA Watch
- Tests for Similar Triangles - Video - Tests For Similar Triangles RHS Watch
Theory
Two triangles are similar when one of four tests holds: AA (equal angles), SSS, SAS or RHS with sides in the same ratio. Once triangles are proved similar, their matching sides give the unknown lengths. This Year 9 Advanced maths page shows how. It comes up if you are heading for Extension maths.
Two triangles are similar if one is an enlargement of the other. For triangles, you do not need to check every angle and every side: any one of these similarity tests is enough.
AA (also written AAA or equiangular): two pairs of equal angles. The third angles are then equal too, because the angle sum is \(180^\circ\).
SSS: all three pairs of matching sides in the same ratio. SAS: two pairs of sides in the same ratio and the included angles equal. RHS: both right-angled, with the hypotenuses and one other pair of sides in the same ratio.
Equal angles often come from a common angle, vertically opposite angles, parallel lines (alternate or corresponding angles) or right angles.
SSS: all three ratios equal.
SAS: two ratios equal and the angle between those sides equal.
How to decide if two triangles are similar
- Look for angles first: a common angle, right angles, vertically opposite angles, parallel lines. Two pairs of equal angles is AA.
- Otherwise use sides: pair the sides shortest with shortest. All three ratios equal is SSS.
- One angle known: check it is between the two sides, then compare those two ratios (SAS). For right-angled triangles, compare hypotenuses and one other side (RHS).
- Write the statement with matching vertices in order and the test in brackets.
Using similar triangles
- Find the scale factor from one matching pair, then multiply or divide.
- Overlapping triangles: redraw them separately; remember whole sides such as \(AC = AD + DC\).
- Not similar? One ratio that differs is enough to show it. Two equal ratios is not enough without the included angle.
- Real problems (shadows, ladders, mirrors, lines of sight): find two triangles with a right angle and another equal angle. For a line of sight, measure heights from eye level.
(i) Pair the sides by size: \(\dfrac{12.5}{5} = \dfrac{17.5}{7} = \dfrac{20}{8} = 2.5\). SSS.
(ii) The \(72^\circ\) angle is between the sides in both, and \(\dfrac{12}{9} = \dfrac{8}{6} = \dfrac{4}{3}\). SAS.
(iii) Right angles, and hypotenuse with hypotenuse, side with side: \(\dfrac{15}{10} = \dfrac{9}{6} = 1.5\). RHS.
(i) Prove that \(\triangle DEC \sim \triangle ABC\).
(ii) Find \(DE\) and \(DC\).
(i) In \(\triangle DEC\) and \(\triangle ABC\):
\(\angle DEC = \angle ABC = 90^\circ\) (given)
\(\angle C\) is common
\(\therefore\ \triangle DEC \sim \triangle ABC\) (AA)
(ii) \(EC\) matches \(BC\), so \(k = \dfrac{4}{12} = \dfrac{1}{3}\). \(DE\) matches \(AB\); \(DC\) matches \(AC\).
| \(DE\) | \(=\) | \(9 \times \dfrac{1}{3} = \mathbf{3}\text{ cm}\) |
| \(AC\) | \(=\) | \(\sqrt{9^2 + 12^2} = 15\) |
| \(DC\) | \(=\) | \(15 \times \dfrac{1}{3} = \mathbf{5}\text{ cm}\) |
(i) Are the triangles similar?
(ii) What length of \(YZ\) would make them similar?
(i) The \(40^\circ\) angles are both between the given sides, so test SAS: the two ratios must be equal.
| \(\dfrac{XY}{PQ}\) | \(=\) | \(\dfrac{12}{8} = 1.5\) |
| \(\dfrac{YZ}{QR}\) | \(=\) | \(\dfrac{16}{10} = 1.6\) |
\(1.5 \ne 1.6\), so the triangles are not similar.
(ii) For SAS, \(YZ\) must use the same scale factor as \(XY\).
| \(YZ\) | \(=\) | \(1.5 \times 10 = \mathbf{15}\text{ cm}\) |
(The diagram is not to scale.) Both triangles have a right angle at the ground, and the angles at the mirror are equal. So the triangles are similar (AA).
| \(\dfrac{h}{1.5}\) | \(=\) | \(\dfrac{18}{2.4}\) |
| \(h\) | \(=\) | \(1.5 \times 7.5 = \mathbf{11.25}\text{ m}\) |
Common pitfalls
Frequently asked questions
What are the tests for similar triangles?
The four tests are AA (two pairs of equal angles), SSS (three pairs of sides in the same ratio), SAS (two pairs of sides in the same ratio with equal included angles) and RHS (right angle, with hypotenuse and one side in the same ratio).
Is AAA the same as AA?
Yes. If two pairs of angles are equal, the third pair must be equal too because each triangle's angles add to \(180^\circ\). Some books write AAA or equiangular.
What is the difference between congruence tests and similarity tests?
Congruence tests need equal sides. Similarity tests need sides in the same ratio. AA is a similarity test but not a congruence test, because equal angles do not fix the size.
How do you prove two triangles are similar?
Name the two triangles, list the equal angles or the equal ratios of sides with reasons, then write the similarity statement in matching order with the test in brackets.
How do you use similar triangles to find a height?
Find two triangles with a right angle and another equal angle, such as from shadows or a mirror. Then use the ratio of matching sides to find the unknown height.