Answer:
For completing the proof we need to understand the following definitions:
Similar triangles: If two triangles are similar then their corresponding angles are equal.
By the transitive property of equality if a = b, and b= c then a = c.
AA postulate of similarity states that when two corresponding angles of two triangles are equal then they are called similar to each other.
Now, the complete proof is mentioned below,
Given : [tex]\triangle ABC\sim \triangle RST[/tex]
[tex]\triangle D EF\sim \triangle RST[/tex]
To Prove : [tex]\triangle ABC\sim \triangle D EF[/tex]
[tex]\triangle ABC\sim \triangle RST[/tex] ( Given )
[tex]\triangle D EF\sim \triangle RST[/tex]
[tex]\angle A = \angle R[/tex], [tex]\angle D = \angle R[/tex] ( By the Definition of similar triangles)
[tex]\angle C = \angle T[/tex], [tex]\angle F = \angle T[/tex]
[tex]\angle A = \angle C[/tex], [tex]\angle D = \angle F[/tex] (By the transitive property of equality)
[tex]\triangle ABC\sim\triangle D EF[/tex] ( By AA similarity postulate)
Hence proved.
