What is the ratio of corresponding angles?

Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around).

These triangles are all similar:

What is the ratio of corresponding angles?

(Equal angles have been marked with the same number of arcs)

Some of them have different sizes and some of them have been turned or flipped.

For similar triangles:

What is the ratio of corresponding angles?

All corresponding angles are equal

and

What is the ratio of corresponding angles?

All corresponding sides have the same ratio

Also notice that the corresponding sides face the corresponding angles. For example the sides that face the angles with two arcs are corresponding.

Corresponding Sides

In similar triangles, corresponding sides are always in the same ratio.

For example:

What is the ratio of corresponding angles?

Triangles R and S are similar. The equal angles are marked with the same numbers of arcs.

What are the corresponding lengths?

  • The lengths 7 and a are corresponding (they face the angle marked with one arc)
  • The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs)
  • The lengths 6 and b are corresponding (they face the angle marked with three arcs)

Calculating the Lengths of Corresponding Sides

We can sometimes calculate lengths we don't know yet.

  • Step 1: Find the ratio of corresponding sides
  • Step 2: Use that ratio to find the unknown lengths

What is the ratio of corresponding angles?

Step 1: Find the ratio

We know all the sides in Triangle R, and
We know the side 6.4 in Triangle S

The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle R.

So we can match 6.4 with 8, and so the ratio of sides in triangle S to triangle R is:

6.4 to 8

Now we know that the lengths of sides in triangle S are all 6.4/8 times the lengths of sides in triangle R.

Step 2: Use the ratio

a faces the angle with one arc as does the side of length 7 in triangle R.

a = (6.4/8) × 7 = 5.6

b faces the angle with three arcs as does the side of length 6 in triangle R.

b = (6.4/8) × 6 = 4.8

Done!

What is the ratio of corresponding angles?

Did You Know?

Similar triangles can help you estimate distances.

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If we have two similar triangles, here we have triangle abc is similar to triangle def. Then we can say that the corresponding altitudes, medians, and angle bisectors are all proportional.So let's say that I drew in an angle bisector in triangle abc. And I call that angle bisector e, and actually I have e over here so I'm not going to use e. I'm going to use g. and then if I went over to the other triangle and I drew in an angle bisector from the corresponding vertex. I'm going to call that h. Now remember there's going to be 3 angle bisectors in each of these triangles. There's going to be 3 altitudes and 3 medians. So by making this statement we're saying that 9 different segments that correspond are all going to be proportional.So getting back to what I was talking about, we can say that the ratio of g:h is going to be equal to the ratio of corresponding sides. So that will be equal to the ratio of c:f which would be the same thing as a:d and finally of b:e and make it an e.So this is going to be true for the 3 altitudes, the 3 medians and for the 3 angle bisectors which you're going to use to find missing lengths and a problem on a quiz or a test.