Question

statement reason
1 \\(\overline{ih} \cong \overline{gh}\\) given
2 \\(\overline{fh} \cong \overline{fh}\\) line segments are congruent to themselves.
3 \\(\triangle fgh \cong \triangle fih\\) side - side congruence
what is the first error holden made in his proof?
choose 1 answer:
a holden used an invalid reason to justify the congruence of a pair of sides or angles.
b holden only established some of the necessary conditions for a congruence criterion.
c holden established all necessary conditions, but then used an inappropriate congruence criterion.
d holden used a criterion that does not guarantee congruence.

Explanation:

To determine the first error in Holden's proof, we analyze the congruence criteria for triangles. The Side - Side - Side (SSS) congruence criterion requires three pairs of congruent sides. In Holden's proof, he has two pairs of congruent sides ($\overline{IH}\cong\overline{GH}$ and $\overline{FH}\cong\overline{FH}$), but he is missing the third pair of sides needed for SSS. Also, there is no such thing as "side - side" congruence criterion (the correct ones are SSS, SAS, ASA, AAS, HL for right triangles). However, the main error here is that the congruence criterion he used (claimed as "side - side") is not a valid triangle congruence criterion, and also, even if we consider the sides, he only established two pairs of sides (but the criterion name is wrong). But looking at the options, option D says he used a criterion that does not guarantee congruence. The "side - side" (only two sides) is not a valid congruence criterion for triangles (we need three sides for SSS, or two sides and included angle for SAS etc.). So Holden used a criterion (side - side) that does not guarantee triangle congruence.

Answer:

D. Holden used a criterion that does not guarantee congruence.