
How do we prove triangles are similar?

Answer: Use any of the 3 theorems : AA, SAS, SSS .
I. AA Theorem
III. SSS Theorem
II. SAS Theorem
Practice Problems
Problem 1) Can you use one of the theorems on this page to prove that the triangles are similar?
Problem 2)
Two different versions of triangles HYZ and HIJ. (Assume scale is not consistent). Use the side side side theorem to determine which pair is similar.
The SSS theorem requires that 3 pairs of sides that are proportional.
In pair 1, all 3 sides have a ratio of $$ \frac{1}{2}$$ so the triangles are similar.
In pair 2, two pairs of sides have a ratio of $$ \frac{1}{2}$$ , but the ratio of $$ \frac{HZ}{HJ} $$ is the problem.
First off, you need to realize that ZJ is only part of the triangle side, and that HJ = 6 + 2 =8 .
Once you recognize that, you can look at the ratio $$ \frac{HZ}{HJ} $$ :
$
\frac{6}{6+2}
\\
\frac{6}{8}
\\
\frac{3}{4}
$
And since these side lengths do not have the same proportion as the other two pairs, they are not similar by SSS.
Problem 3)
Can you use one of the theorems on this page to prove that the triangles are similar?
Yes, since two pairs of corresponding sides have a $$ \frac{1}{3}$$ ratio and the included angle is congruent, these two triangles
are similar by the Side Angle Side theorem.
Problem 4)
Can you use one of the theorems on this page to prove that the triangles are similar?
No, although the included angle is equal, the sides do not have a constant ratio $$ \frac{5}{15} $$ and $$ \frac{7}{20} $$.
Problem 4)
Can you use one of the theorems on this page to prove that the triangles are similar?
No, although we have 2 pairs of sides with an the same $$\frac{1}{3}$$ ratio, we do not have the included angle so we can not
state that the triangles are similar.
