Geometry · Semester A TEKS 6A-6E
HardDiagram
Right triangles ABC and DEF each have a right angle at A and D respectively, with AB = DE = 12. It is also given that AC = 8. Which additional single piece of information, if true, is SUFFICIENT to prove △ABC ≅ △DEF?
ABC = 17 with EF left unmeasured on the second triangle (an equal side pair needs BOTH lengths pinned down, not just one triangle's side)
BThe triangles have the same perimeter (equal perimeter does not guarantee congruent corresponding parts)
CThe triangles have the same area (equal area does not guarantee congruent corresponding parts)
DDF = 8 (this completes SAS: two legs and the included right angle, matching AB=DE and AC=DF with the right angle between them)
Explanation
We already have AB ≅ DE and the right angles at A and D (both 90°, so ∠A ≅ ∠D). If DF = 8 = AC, then two legs and the included right angle match: SAS gives △ABC ≅ △DEF. Knowing only one triangle's third side, or only that areas or perimeters match, does not pin down a full congruent side/angle correspondence.
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