Question

given the figure of triangles below with sides and angles as marked.
select the additional information that is required in order to know the triangles are congruent by angle angle side congruency
$\bigcircangle deccongangle dtc$
$\bigcircoverline{de}congoverline{dt}$
$\bigcircangle tcdcongangle ecd$
$\bigcircangle edccongangle dtc$
$\bigcirc$ no additional information is required.

Explanation:

Step1: Recall AAS congruence

Angle - Angle - Side (AAS) congruence states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Step2: Analyze the given figure

We already have a pair of congruent angles (the marked angles at \(D\)). We need another pair of angles and a non - included side.
If \(\overline{DE}\cong\overline{DT}\), and we have the pair of congruent angles at \(D\) and if we can show another pair of angles (by vertical angles or other angle relationships in the triangle) congruent, we can apply AAS. But let's check each option:

• For \(\angle DEC\cong\angle DTC\): This is just an angle, we still lack a side.
• For \(\overline{DE}\cong\overline{DT}\): If we assume this side congruence, along with the already given congruent angles (the two angles at \(D\) which are supplementary and form a linear pair, but if we consider the triangles \(\triangle DEC\) and \(\triangle DTC\), and assume \(\overline{DE}\cong\overline{DT}\), and we know \(DC = DC\) (common side) and if we can use the AAS (if we have two angles). But wait, actually, if we consider the two triangles \(\triangle DEC\) and \(\triangle DTC\), we have \(\angle EDC\) and \(\angle TDC\) (the marked angles) are supplementary. But if we assume \(\overline{DE}\cong\overline{DT}\), and \(DC\) is common. Also, if we consider the fact that \(\angle DEC\) and \(\angle DTC\) (if we can get them from other angle - side relationships). But actually, for AAS, we need two angles and a non - included side.
• For \(\angle TCD\cong\angle ECD\): This is just an angle, we still lack a side.
• For \(\angle EDC\cong\angle DTC\): This is just an angle, we still lack a side.

Answer:

\(\overline{DE}\cong\overline{DT}\)