Question

7 multiple choice 1 point state what additional information is needed to prove the triangles are congruent using asa: lk ≅ tu ∠k ≅ ∠u mk ≅ su ∠l ≅ ∠t

Explanation:

Step1: Recall ASA (Angle - Side - Angle) congruence criterion

ASA states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Step2: Analyze the given triangles

We are given that \(\angle M\cong\angle S\) (marked with one arc). For ASA, we need another pair of angles and the included side.
If \(MK\cong SU\) (the side between the angles), and assume we can get another pair of angles (but in the options, the key is the side part for ASA when considering the structure with the given angle). Wait, actually, looking at the two triangles \(\triangle MLK\) and \(\triangle STU\), we know \(\angle M\cong\angle S\). If \(MK\cong SU\) (the side between the angles in the two triangles), then if we also have another pair of angles (but in the options, for ASA, the side is crucial. Wait, no - actually, ASA is two angles and the included side. If we assume that the side \(MK\) in \(\triangle MLK\) and \(SU\) in \(\triangle STU\) is the included side between the known \(\angle M\) and \(\angle S\) and another pair of angles. But actually, more precisely, ASA: in \(\triangle MLK\) and \(\triangle STU\), if \(\angle M\cong\angle S\), \(MK\cong SU\), and \(\angle K\cong\angle U\) (but \(\angle K\cong\angle U\) is an option. Wait no - wait the ASA formula is \(A - S - A\). So if we have \(\angle M\cong\angle S\), need the side between \(\angle M\) and another angle in \(\triangle MLK\) and \(\angle S\) and another angle in \(\triangle STU\). So the side \(MK\) (between \(\angle M\) and \(\angle K\) in \(\triangle MLK\)) and \(SU\) (between \(\angle S\) and \(\angle U\) in \(\triangle STU\)). If \(MK\cong SU\), then if \(\angle K\cong\angle U\), we can apply ASA. But looking at the options:

• \(LK\cong TU\): not the included side for the known angles.
• \(\angle K\cong\angle U\): is an angle, but we need the included side. Wait no - wait ASA is two angles and the included side. Wait actually, if we assume that in the two triangles, we have \(\angle M\cong\angle S\) (given as marked). For ASA, we need the side between \(\angle M\) and another angle in \(\triangle MLK\) and \(\angle S\) and another angle in \(\triangle STU\). So \(MK\) (side between \(\angle M\) and \(\angle K\)) and \(SU\) (side between \(\angle S\) and \(\angle U\)). If \(MK\cong SU\), then if we also have \(\angle K\cong\angle U\), we can use ASA. But among the options, the side \(MK\cong SU\) is present. Because ASA is two angles and the included side. If we assume that the two angles are \(\angle M\) and (say \(\angle K\)) in one triangle and \(\angle S\) and \(\angle U\) in the other, the included side between \(\angle M\) and \(\angle K\) is \(MK\), and between \(\angle S\) and \(\angle U\) is \(SU\). So \(MK\cong SU\) is the needed side for ASA (assuming the other angles \(\angle K\) and \(\angle U\) can be part of the ASA - but since the problem is about what additional info is needed, and ASA requires the side between the two angles (one of which is given as congruent). So \(MK\cong SU\) is the included side.

Answer:

\(MK\cong SU\)