The theorem that justifies why triangles ABC and DEF are congruent is: HL.
What is the Hypotenuse-Leg Congruence Theorem (HL)?
The hypotenuse-leg congruence theorem (HL) states that if two triangles that are right triangles have a pair of congruent hypotenuse and a pair of corresponding congruent legs, then both right triangles are proven to be congruent to each other. This means both right triangles are have the same shape and size.
Given that:
angle ABC = angle DEF = 90° (congruent right angles, this means both triangles are right triangles)
AC = FD (a pair of congruent hypotenuse)
AB = DE (a pair of congruent legs)
Thus, we can conclude that since triangle ABC and triangle DEF have a pair of congruent hypotenuses and a pair of congruent legs, therefore, the two right triangles, triangle ABC and triangle DEF are congruent to each other by the HL congruent theorem.
Thus, the theorem that justifies why both triangles are congruent is: HL.
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