Question

1. state what additional information is required in order to know that the triangles are congruent by angle side angle. be specific by naming the sides or angles needed.

Explanation:

Step1: Recall ASA Congruence

ASA (Angle - Side - Angle) congruence criterion 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.
In the given figure, we have triangles \(\triangle DUT\) and \(\triangle SUT\) (assuming the two triangles are \(\triangle DUT\) and \(\triangle SUT\) with common side \(UT\)). We already have one pair of angles (the marked angles at \(T\)) congruent. The common side \(UT\) is a side for both triangles.

Step2: Identify Missing Angle

To apply ASA, we need another pair of angles that are adjacent to the common side \(UT\). So we need either \(\angle DUT\cong\angle SUT\) (the angles at \(U\) formed by side \(UT\)) or we can think in terms of the other angles. Alternatively, if we consider the triangles \(\triangle DUT\) and \(\triangle SUT\), we know that \(UT\) is common, \(\angle DTU\cong\angle STU\) (the marked angles). So we need \(\angle DUT\cong\angle SUT\) (the angles at \(U\)) or \(\angle UDT\cong\angle UST\) (but for ASA, the side should be included between the two angles). Wait, more precisely, for \(\triangle DUT\) and \(\triangle SUT\), with side \(UT\) common and \(\angle DTU\cong\angle STU\), we need \(\angle DUT\cong\angle SUT\) (so that the two angles \(\angle DUT\), \(\angle DTU\) and included side \(UT\) in \(\triangle DUT\) are congruent to \(\angle SUT\), \(\angle STU\) and included side \(UT\) in \(\triangle SUT\)) or we can also say that we need \(\angle UDT\cong\angle UST\) but that would be AAS. Wait, no, for ASA, the side must be included between the two angles. So the included side between the two angles should be congruent. We have side \(UT\) common. We have one angle at \(T\) congruent. So we need the angle at \(U\) (i.e., \(\angle DUT\) and \(\angle SUT\)) to be congruent. Alternatively, we can also say that we need \(\angle UDT\cong\angle UST\) but that is AAS. Wait, no, let's re - examine.
The two triangles are \(\triangle DUT\) and \(\triangle SUT\). The side \(UT\) is shared. We have \(\angle DTU=\angle STU\) (marked angles). For ASA, we need another pair of angles such that the side \(UT\) is between them. So the angles at \(U\): \(\angle DUT\) and \(\angle SUT\) should be congruent. Or, if we consider the other way, we can also say that we need \(\angle UDT\) and \(\angle UST\) to be congruent, but that is AAS. Wait, no, the correct missing information for ASA is that \(\angle DUT\cong\angle SUT\) (the angles at vertex \(U\) formed by side \(UT\)) or equivalently, we can say that we need to know that \(\angle UDT\cong\angle UST\) but that is not ASA. Wait, let's start over.
ASA: Two angles and the included side. So in \(\triangle DUT\), let's say the angles are \(\angle D\), \(\angle DUT\), \(\angle DTU\) with included side between \(\angle DUT\) and \(\angle DTU\) is \(UT\). In \(\triangle SUT\), the angles are \(\angle S\), \(\angle SUT\), \(\angle STU\) with included side between \(\angle SUT\) and \(\angle STU\) is \(UT\). We know \(\angle DTU = \angle STU\) and \(UT = UT\) (common side). So we need \(\angle DUT=\angle SUT\) (so that \(\angle DUT\), \(UT\), \(\angle DTU\) in \(\triangle DUT\) are congruent to \(\angle SUT\), \(UT\), \(\angle STU\) in \(\triangle SUT\) by ASA).

Answer:

To prove the triangles are congruent by ASA, we need \(\angle DUT\cong\angle SUT\) (the angles at vertex \(U\) formed by side \(UT\)) or equivalently, we can also say that we need to know that \(\angle UDT\cong\angle UST\) is incorrect. The correct additional information is that \(\angle DUT\) is congruent to \(\angle SUT\) (or the angles at \(U\) are congruent) so that the two angles (\(\angle DUT\) and \(\angle DTU\)) and the included side \(UT\) of \(\triangle DUT\) are congruent to the two angles (\(\angle SUT\) and \(\angle STU\)) and the included side \(UT\) of \(\triangle SUT\) by ASA. A more precise answer is that we need \(\angle DUT\cong\angle SUT\) (the angles at \(U\) adjacent to side \(UT\)) to apply the ASA congruence criterion.