If ABC and DEF are similar triangles such that ∠A = 57° and ∠E = 73°,

GIVEN: There are two similar triangles ΔABC and ΔDEF.

$\angle \mathrm{A}=57^{\circ} \cdot \angle \mathrm{E}=73^{\circ}$

TO FIND: measure of $\angle \mathrm{C}$

SAS Similarity Criterion: If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then two triangles are similar.

In ΔABC and ΔDEF if

$\frac{\mathrm{AB}}{\mathrm{DE}}=\frac{\mathrm{AC}}{\mathrm{DF}}$ and

$\angle \mathrm{A}=\angle \mathrm{D}$

Then, $\triangle \mathrm{ABC} \sim \triangle \mathrm{DEF}$

So,

$\angle \mathrm{A}=\angle \mathrm{D}$

$\angle \mathrm{D}=57^{\circ}$.....(1)

Similarly

$\angle \mathrm{B}=\angle \mathrm{E}$

$\angle \mathrm{B}=73^{\circ}$.....(2)

Now we know that sum of all angles of a triangle is equal to 180°,

$\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$

$57^{\circ}+73^{\circ}+\angle \mathrm{C}=180^{\circ}$

$130^{\circ}+\angle \mathrm{C}=180^{\circ}$

$\angle \mathrm{C}=50^{\circ}$