U and V are points on line RS and Line RT respectively. Line SU and line TV intersect at W. Under which of the following conditions can we conclude the triangle RUS is congruent to triangle RVT?
A) RS=RT and SU=TV
B) RS=RT and RU=RV
C) VW=UW and RU=RV
D) VW=UW and SW=TW
E) SV=TU and SW=TW
F) SV=TU and SU=TV
The answer is F: SV=TU and SU=TV.
According to the 5 conditions for congruent triangles, we can say that two triangles are congruent if:
Side-Side-Side (SSS): All three sides of one triangle are equal to the corresponding sides of the other triangle.
Side-Angle-Side (SAS): Two sides and the included angle of one triangle are equal to the corresponding sides and included angle of the other triangle.
Angle-Side-Angle (ASA): Two angles and the included side of one triangle are equal to the corresponding angles and included side of the other triangle.
Angle-Angle-Side (AAS): Two angles and any side of one triangle are equal to the corresponding angles and the same side of the other triangle.
Right Angle-Hypotenuse-Side (RHS): The right angle and the hypotenuse of one triangle are equal to the right angle and the hypotenuse of the other triangle.
In this case, we are given that U and V are points on line RS and Line RT respectively, and line SU and line TV intersect at W. This means that we know the values of SV, TU, SU, and TV.
If SV=TU and SU=TV, then we can see that triangles RUS and RVT are congruent by the Side-Side-Side (SSS) condition.
The other conditions are not sufficient to conclude that the triangles are congruent. For example, if RS=RT and RU=RV, then we only know that the corresponding sides of the triangles are equal, but we do not know that the included angles are equal. Similarly, if VW=UW and RU=RV, then we only know that the included angle and the corresponding side of one triangle are equal to the corresponding angle and side of the other triangle, but we do not know that the other two sides are equal.
Therefore, the correct answer is F: SV=TU and SU=TV.
