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Hi Welcome to MooMooMath. Today we are going to talk about how to solve similar triangles. First let’s talk about what are similar triangles? In Math similar looks is more than just looking like, they actually have corresponding angles. Notice this triangle is marked with one arc and this triangle (points to the triangle below) is also marked with an arc. This means the two angles are congruent to each other, and these two angles are marked with a two (points to the top angle in both triangles) so those angles are congruent. Finally these two angles are marked with a three so they are congruent. So basically we have taken this triangle and put it on a photocopier and enlarged it. Same triangle, just larger. So know we are going to learn how to solve for these two missing sides. Let’s stop and look at the rules on how to do that. The rules are as follows. You set up a proportion of the matching or corresponding sides. Then you just solve for that proportion by cross multiplying and set them equal to each other and divide by the coefficient to solve it. So basically we set up proportions of corresponding sides. Let’s walk through these two examples and I will show you how to do that. So this side three and this 9 are corresponding sides because they fall between the same corresponding angles. So I will set those two sides up as a proportion. 3 over 9 I will place my smaller triangle on top and the larger triangle on the bottom. I don’t know the side of X but I know it corresponds with 18 so since this is part of the small triangle I will put an X over 18. Now all you have to do is a cross product so you can take X times 9 and get 9X and three times 18 is 30 is 54 and divide both sides by the coefficient of 9 and I get X is 6 so that means this side up here is 6 ( points to the top side) so that is how you solve for X Now let’s solve it for Y I’m going to use the same pairing, the three and the nine because I was given these two sides 3 over 9. Smaller triangle over the larger triangle and I will solve for Y this time. The Y corresponds with the 5 but the 5 is part of the small triangle so it goes on top and Y is part of the larger triangle so it goes on the bottom. I do a cross product again 3 times Y is 3Y set equal to 9 times 5 is forty five now we divide by the coefficient of three so Y is 15 because 45 divided by 3 is 15. So that is how you set up and solve for similar triangles. 

Definition of Similar Triangles

angle side angle similar triangle
angle side angle similar triangle

Similar Triangles-Geometry Help

Definition of Similar Triangles

similar triangles
Definition of Similar Triangles Solving for a side
Practice with similar triangles:

1. Always set up two ratios (or fractions set to each other)

2. Put the smaller triangle measurements in the same place in the ratio, either both on top or both on the bottom. I like to put the smaller one on the top (numerator) and the larger on the bottom (denominator).

3. Make sure the sides that are in the same fraction “correspond,” or are on the related side. Those related sides have to fall between the same angle measures. For example, the side that measures 3 fall between the angles of 110 and 50, so the side that measures 6 is the corresponding side from the other triangle because it falls between the same two angles.

4. Cross multiply to check to see if the triangles are similar or to solve for the unknown side.
Similar triangles are triangles in which the only difference is size. Similar triangles  have the same angles and proportional sides.

Triangles are similar if any one of the following conditions are met.

All three corresponding angles of the triangles are the same.

The proportions of all three corresponding sides of the similar triangles are the same.

The included angle of the triangles and two pairs of sides are proportional.
side side side similar triangle
side side side similar triangle
side angle side similar triangles
side angle side similar triangles
Similar Triangles
70 70 40 degree triangle

If two angles are congruent, then the 3rd angle must also be congruent!  The sides of these two triangles have a ratio of 12/4 or 3/1.  The 3rd angle must be 65 degrees 

Angle Side Angle "ASA"

Side-Side-Side "SSS"

Side-Angle-Side "SAS"

Angle-Angle-Angle "AAA" ​