Question

copy the triangle and highlight congruent parts. mark and highlight additional congruent parts that can be proved congruent on your paper. select what additional information is required in order to know that the triangles are congruent by hl. ∠g ≅ ∠r hg ≅ rq ∠h ≅ ∠q fh ≅ sq

Explanation:

Step1: Recall HL Congruence Criterion

HL (Hypotenuse - Leg) congruence criterion for right - angled triangles states that if the hypotenuse and one leg of a right - angled triangle are congruent to the hypotenuse and one leg of another right - angled triangle, then the two triangles are congruent.
In right - triangle \( \triangle FHG\) (\(\angle F = 90^{\circ}\)) and right - triangle \( \triangle SRQ\) (\(\angle S=90^{\circ}\)), we already know that one pair of legs is congruent (from the given markings in the triangles).

Step2: Analyze the options

• For \( \angle G\cong\angle R\): This is an angle - angle - side (AAS) type of information (but we need HL which is a side - side - angle (SSA) but for right - angled triangles). This does not help with HL.
• For \( \overline{HG}\cong\overline{RQ}\): The hypotenuse of \( \triangle FHG\) is \(HG\) and the hypotenuse of \( \triangle SRQ\) is \(RQ\). If the hypotenuses are congruent and we already have one pair of legs congruent (from the given markings in the triangles), then by HL, \( \triangle FHG\cong\triangle SRQ\).
• For \( \angle H\cong\angle Q\): This is an angle - angle - side (AAS) type of information (not relevant for HL).
• For \( \overline{FH}\cong\overline{SQ}\): We already have one pair of legs congruent (from the given markings in the triangles). But we need the hypotenuse for HL, not another leg.

Answer:

\(\overline{HG}\cong\overline{RQ}\)