How do you tell if two triangles are similar? I know that the angles must be congruent, and the sides must be proportional. How do you determine if the sides are proportional?
Here is the problem:
Are any of the three triangles similar? If so, write the appropriate similarity statement.
1st triangle: CAB; CA=4; CB=4; AB=5; Angles A and B are congruent.
2nd triangle: PRQ; PR=8; PQ=10; RQ not listed
3rd triangle: XYZ; XZ=8; ZY=8; XY not listed
*On triangles PRQ and XYZ, angles P and Z are congruent.
First, I agree with Prove It, that you don't have enough information here to say whether any of these triangles are similar. It is possible that triangles ABC and PQR are similar; and it's also possible that triangles ABC and XYZ are similar; but it's not possible for PQR and XYZ to be similar.
Let me try to explain how I know this, and in doing so, I may answer the first questions you asked: How can we tell if two triangles are similar? and what does it mean for the sides to be proportional?
Two triangles are similar - essentially - if they are the same shape, but not (necessarily) the same size. This means, as you said, that their angles will be congruent. So, for instance, if we were to say that triangle DEF is similar to triangle NML (with their 'vertices' - their angles - written in that order) we would mean that
- angle D = angle N
- angle E = angle M
- angle F = angle L
But having the same shape means more than simply having pairs of angles congruent: it means that we can enlarge (or reduce) one of the triangles in size to make it exactly congruent - identical - to the other. For instance, we may multiply all the lengths of sides in one triangle by 2 to make them equal to the lengths of the sides in the other triangle.
So as well as having angles that correspond (like D and N, E and M, F and L in the example above) similar triangles will have sides that correspond as well. The longest side in one triangle will always correspond to the longest side in the other; the two shortest sides will correspond, and so, of course, will the two middle-sized sides. And - here's an important fact - corresponding sides will always be opposite equal angles. So in the example above
- EF corresponds to ML
- DF corresponds to NL
- and DE corresponds to NM
So what do we mean by sides being 'proportional'? It means that if the longest side in one triangle, for instance, has to be multiplied by 2 to get the longest side in the other, then all the sides of that triangle will have to be multiplied by 2 to get the corresponding sides in the other.
And we can say more than that. If, for instance, the longest side in one triangle is 1.5 times as long as the shortest side in the same triangle (in other words, they are in the ratio 3:2), then the other triangle's longest side is also 1.5 times as long as its shortest side (so these sides are also in the ratio 3:2).
That's what it means, then, to say that the sides are proportional: that pairs of corresponding sides are in the same ratio.
Now in the triangles in your question AC = BC = 4cm, and AB = 5cm. So in triangle ABC we have two equal sides, with the third side being the longest. Could triangle PQR be the same shape? Yes, if RQ = 8cm, because all of its sides would then be twice as long as the sides in triangle ABC.
Could ABC and XYZ be similar? Yes, if we make XY = 10 cm. Again, we'd have its sides being twice as long as the sides in triangle ABC.
Could triangle XYZ be the same shape as triangle PQR? No. Why not? Well, the explanation here is a bit more complicated, and goes as follows: In triangle XYZ, XZ = YZ = 8cm; and the angle Z lies between these equal sides. But angle Z = angle P in triangle PQR, and this doesn't lie between two equal sides. We could, of course, make QR equal in length to either PR or PQ, but in neither case would you make an angle the same size as angle P, to be therefore set equal to angle Z in the other triangle. So the two triangles couldn't be the same shape. Draw it out, and you'll see what I mean.
I hope that you might understand these things a bit better now.