Option A: [tex]\frac{S R}{B C}=\frac{R T}{C A}[/tex]
Option C: [tex]\angle R\cong \angle C[/tex]
Solution:
Given ΔRST similar to ΔABC.
To determine which statements are true for the given similarity triangles.
Option A: [tex]\frac{S R}{B C}=\frac{R T}{C A}[/tex]
By the similarity theorem,
If two triangles are similar, then the corresponding angles are equal and the corresponding sides are in the same ratio or proportion.
Therefore, [tex]\frac{S R}{B C}=\frac{R T}{C A}[/tex]
It is true.
Option B: [tex]\angle S\cong \angle A[/tex]
By the similarity theorem, corresponding angles are equal.
∠S is corresponding to ∠B.
So, [tex]\angle S\cong \angle B[/tex].
That means ∠S is not corresponding to ∠A.
Therefore, it is false.
Option C: [tex]\angle R\cong \angle C[/tex]
By the similarity theorem, corresponding angles are equal.
∠R is corresponding to ∠C.
Therefore [tex]\angle R\cong \angle C[/tex].
It is true.
Option D: [tex]\frac{S R}{B C}=\frac{R T}{AB}[/tex]
We already, proved in option A that [tex]\frac{S R}{B C}=\frac{R T}{C A}.[/tex]
Therefore, [tex]\frac{S R}{B C}\neq \frac{R T}{AB}[/tex].
It is false.
Option A and Option C are true.
Hence [tex]\frac{S R}{B C}=\frac{R T}{C A}[/tex] and [tex]\angle R\cong \angle C[/tex].
