How to Find if Triangles are Similar

2 triangles are similar if they have:

  • all their angles equal
  • respective sides are in the same ratio

But nosotros don't need to know all three sides and all three angles ...two or three out of the six is usually plenty.

There are iii ways to find if 2 triangles are similar: AA, SAS and SSS:

AA

AA stands for "angle, angle" and means that the triangles take ii of their angles equal.

If two triangles take two of their angles equal, the triangles are similar.

Case: these two triangles are similar:

triangles similar both have angles 72 and 35

If two of their angles are equal, so the third angle must also exist equal, considering angles of a triangle ever add to make 180°.

In this case the missing angle is 180° − (72° + 35°) = 73°

So AA could too be called AAA (because when two angles are equal, all iii angles must be equal).

SAS

SAS stands for "side, angle, side" and means that nosotros take two triangles where:

  • the ratio betwixt two sides is the same as the ratio between some other ii sides
  • and we we also know the included angles are equal.

If ii triangles accept two pairs of sides in the aforementioned ratio and the included angles are too equal, then the triangles are similar.

Case:

triangles similar both have angle 75 but sides (15,21,a) and (10,14,x)

In this example nosotros can run across that:

  • i pair of sides is in the ratio of 21 : 14 = 3 : 2
  • another pair of sides is in the ratio of xv : x = three : 2
  • there is a matching bending of 75° in between them

And so there is plenty information to tell us that the two triangles are similar.

Using Trigonometry

Nosotros could also use Trigonometry to calculate the other two sides using the Police of Cosines:

Example Continued

In Triangle ABC:

  • aii = btwo + ctwo - 2bc cos A
  • aii = 212 + 15two - ii × 21 × xv × Cos75°
  • atwo = 441 + 225 - 630 × 0.2588...
  • atwo = 666 - 163.055...
  • aii = 502.944...
  • So a = √502.94 = 22.426...

In Triangle XYZ:

  • ten2 = y2 + z2 - 2yz cos X
  • x2 = fourteen2 + 10ii - ii × fourteen × 10 × Cos75°
  • x2 = 196 + 100 - 280 × 0.2588...
  • x2 = 296 - 72.469...
  • tenii = 223.530...
  • And then x = √223.530... = 14.950...

Now permit us check the ratio of those 2 sides:

a : x = 22.426... : fourteen.950... = iii : 2

the same ratio as earlier!

Notation: we can also use the Police of Sines to bear witness that the other two angles are equal.

SSS

SSS stands for "side, side, side" and means that nosotros accept ii triangles with all three pairs of corresponding sides in the aforementioned ratio.

If two triangles accept 3 pairs of sides in the same ratio, then the triangles are like.

Case:

triangles (4,6,8) and (5,7.5,10)

In this example, the ratios of sides are:

  • a : ten = 6 : 7.5 = 12 : xv = 4 : 5
  • b : y = 8 : 10 = 4 : 5
  • c : z = 4 : five

These ratios are all equal, so the 2 triangles are similar.

Using Trigonometry

Using Trigonometry we can show that the ii triangles have equal angles by using the Law of Cosines in each triangle:

In Triangle ABC:

  • cos A = (bii + c2 - a2)/2bc
  • cos A = (82 + 4ii - half-dozen2)/(2× 8 × 4)
  • cos A = (64 + xvi - 36)/64
  • cos A = 44/64
  • cos A = 0.6875
  • So Angle A = 46.6°

In Triangle XYZ:

  • cos X = (ytwo + z2 - xii)/2yz
  • cos X = (x2 + 52 - 7.five2)/(2× x × 5)
  • cos 10 = (100 + 25 - 56.25)/100
  • cos 10 = 68.75/100
  • cos X = 0.6875
  • So Angle 10 = 46.6°

So angles A and X are equal!

Similarly we can evidence that angles B and Y are equal, and angles C and Z are equal.

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