Prove that the two triangles are congruent using the SAS congruence criteria. Show your work.
Triangle DEF: D(3,2), E(5,3), F(4,5)
Triangle FGH: F(4,5), G(3,7), H(5,8)
Given: ∠DFE and ∠GFH are vertical angles
To prove that the two triangles are congruent using the SAS congruence criteria, we need to show that two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of the other triangle.
1. Side DE is congruent to side FG:
DE = sqrt((5-3)^2 + (3-2)^2) = sqrt(5)
FG = sqrt((3-4)^2 + (7-5)^2) = sqrt(5)
Therefore, DE = FG.
2. Side EF is congruent to side GH:
EF = sqrt((4-5)^2 + (5-3)^2) = sqrt(5)
GH = sqrt((5-3)^2 + (8-7)^2) = sqrt(5)
Therefore, EF = GH.
3. Angle DFE is congruent to angle GFH:
These angles are vertical angles, which are always congruent.
Therefore, by the SAS congruence criteria, triangle DEF is congruent to triangle FGH.