Question

Math

Posted 2 months ago

```
Holden tried to prove that $\triangle F G H \cong \triangle F I H$.
\begin{tabular}{lll}
& Statement & Reason \\
\hline 1 & $\overline{I H} \cong \overline{G H}$ & Given \\
2 & $\overline{F H} \cong \overline{F H}$ & Line segments are congruent to themselves. \\
3 & $\triangle F G H \cong \triangle F I H$ & Side-side congruence
\end{tabular}
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.
```

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 2

The first statement is that $\overline{IH} \cong \overline{GH}$, with the reason given as "Given". This is a valid statement and reason

step 3

The second statement is that $\overline{FH} \cong \overline{FH}$, with the reason "Line segments are congruent to themselves". This is also a valid statement and reason

step 4

The third statement claims that $\triangle FGH \cong \triangle FIH$ based on "Side-side congruence". However, there is no such congruence criterion as "Side-side congruence". The correct criteria are Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL) for right triangles

step 5

Since Holden has only shown two sides to be congruent and has not established the congruence of the third side or any angles, he has not satisfied any of the valid congruence criteria

1 Answer

D

Key Concept

Triangle Congruence Criteria

Explanation

To prove two triangles congruent, one must satisfy one of the valid congruence criteria: SSS, SAS, ASA, AAS, or HL for right triangles. "Side-side congruence" is not a recognized criterion.

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