Let us find out if the given two triangles ABC and DEF are similar triangles or not.
Triangle ABC is a right-angled triangle so we can apply the Pythagorean theorem to find the missing side.
[tex]a^2+b^2=c^2[/tex]
Where a and b are the shorter sides and c is the longest side (hypotenuse)
[tex]\begin{gathered} 20^2+21^2=c^2 \\ 400+441=c^2 \\ 841=c^2 \\ \sqrt[]{841}=c \\ 29=c \\ c=29 \end{gathered}[/tex]
Similarly, we can apply the Pythagorean theorem to triangle DEF to find the missing side.
[tex]\begin{gathered} d^2+e^2=f^2 \\ 40^2+e^2=58^2 \\ e^2=58^2-40^2 \\ e^2=3364-1600 \\ e^2=1764 \\ e=\sqrt[]{1764} \\ e=42 \end{gathered}[/tex]
Now, recall that two triangles are similar if the ratio of the corresponding sides is equal.
The corresponding sides are
AB = DE
BC = EF
AC = DF
[tex]\begin{gathered} \frac{DE}{AB}=\frac{EF}{BC}=\frac{DF}{AC} \\ \frac{40}{20}=\frac{42}{21}=\frac{58}{29} \\ \frac{2}{1}=\frac{2}{1}=\frac{2}{1} \end{gathered}[/tex]
As you can see, the ratio of the corresponding sides of the two triangles is equal.
Hence, the triangles ABC and DEF are similar.
