Question

\delta srt and \delta ghi are two triangles. \overline{rt} \cong \overline{hi}. which statement will prove \delta srt is always congruent to \delta ghi?

select one:
a. a sequence of rigid motions that maps \angle r to \angle h and \angle t to \angle i.
b. a sequence of rigid motions that maps \angle s to \angle g and \overline{st} to \overline{gi}.
c. a sequence of rigid motions that maps point s to point g and point r to point h.
d. a sequence of rigid motions that maps \angle t to \angle i and \overline{sr} to \overline{gh}.

Explanation:

⚡ Using what you learned: congruence criteria for triangles (sss, sas, asa, aas, hl)

Step 1: Analyze the Given Information

We are given two triangles, \(\Delta SRT\) and \(\Delta GHI\), with one pair of congruent corresponding sides:

$$ \overline{RT} \cong \overline{HI} $$

We need to find which additional condition, mapped via rigid motions (which preserve angle measures and segment lengths), guarantees that \(\Delta SRT \cong \Delta GHI\).

Step 2: Evaluate the Options

• Option a: Maps \(\angle R\) to \(\angle H\) and \(\angle T\) to \(\angle I\).
• This gives \(\angle R \cong \angle H\) and \(\angle T \cong \angle I\).
• Together with the given side \(\overline{RT} \cong \overline{HI}\), the congruent side is included between the two pairs of congruent angles.
• This satisfies the Angle-Side-Angle (ASA) congruence criterion:

$$ \angle R \cong \angle H \quad (\text{Angle}) $$

$$ \overline{RT} \cong \overline{HI} \quad (\text{Side}) $$

$$ \angle T \cong \angle I \quad (\text{Angle}) $$

• Therefore, this statement proves \(\Delta SRT \cong \Delta GHI\).

• Option b: Maps \(\angle S\) to \(\angle G\) and \(\overline{ST}\) to \(\overline{GI}\).
• This gives \(\angle S \cong \angle G\) and \(\overline{ST} \cong \overline{GI}\).
• Along with \(\overline{RT} \cong \overline{HI}\), this corresponds to Side-Side-Angle (SSA), which is not a valid congruence criterion.

• Option c: Maps Point \(S\) to Point \(G\) and Point \(R\) to Point \(H\).
• This only establishes correspondence of vertices and does not provide enough congruent parts to satisfy any congruence criteria.

• Option d: Maps \(\angle T\) to \(\angle I\) and \(\overline{SR}\) to \(\overline{GH}\).
• This gives \(\angle T \cong \angle I\) and \(\overline{SR} \cong \overline{GH}\).
• Along with \(\overline{RT} \cong \overline{HI}\), this corresponds to Side-Angle-Side (SAS) but with the angle not properly included between the sides for both triangles in the correct correspondence order.

Answer:

a. A sequence of rigid motions that maps \(\angle R\) to \(\angle H\) and \(\angle T\) to \(\angle I\).