The addition of the three interior angles of a triangle is equal to 180°.
Considering triangle JMR, we get:
[tex]m\angle J+m\angle M+m\angle R=180\degree[/tex]
Substituting with the data provided in the diagram:
[tex]\begin{gathered} 90\degree+m\angle M+67\degree=180\degree \\ m\angle M=180\degree-90\degree-67\degree \\ m\angle M=23\degree \end{gathered}[/tex]
Considering triangle KNP, we get:
[tex]m\angle K+m\angle N+m\angle P=180\degree[/tex]
Substituting with the data provided in the diagram:
[tex]\begin{gathered} m\angle K+90\degree+23\degree=180\degree \\ m\angle K=180\degree-90\degree-23\degree \\ m\angle K=67\degree \end{gathered}[/tex]
In consequence, the next angles are congruent:
[tex]\begin{gathered} \angle J\cong\angle N \\ \angle M\cong\angle P \\ \angle R\cong\angle K \end{gathered}[/tex]
Applying the AAA similarity theorem, then triangles JMR and NPK are similar.
A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons.
The similarity ratio of triangle JMR to triangle NPK is:
[tex]\begin{gathered} \frac{JM}{NP}=\frac{24}{36}=\frac{2}{3} \\ \frac{MR}{PK}=\frac{26}{39}=\frac{2}{3} \\ \frac{JR}{NK}=\frac{10}{15}=\frac{2}{3} \end{gathered}[/tex]
The similarity ratio of triangle NPK to triangle JMR is:
[tex]\frac{NP}{JM}=\frac{36}{24}=\frac{3}{2}[/tex]
