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

∠tcd ≅ ∠ecd
∠dec ≅ ∠dtc
∠edc ≅ ∠dtc
overline{de} ≅ overline{dt}
no additional information is required.

Explanation:

Step1: Recall AAS Congruence Criterion

The Angle - Angle - Side (AAS) congruence criterion 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.
In the given figure, we already have \(\angle EDC\cong\angle TDC\) (the marked angles). If we have \(\angle TCD\cong\angle ECD\), and the side \(CD = CD\) (common side).
For \(\triangle EDC\) and \(\triangle TDC\), we can use the AAS congruence criterion.
If \(\angle TCD\cong\angle ECD\), \(\angle EDC\cong\angle TDC\) (given as the two angles) and \(CD\) is the non - included side (common side) for both triangles.

Step2: Analyze other options

• For \(\angle DEC\cong\angle DTC\): We do not have enough information to relate this angle with the existing angles and sides to form AAS.
• For \(\angle EDC\cong\angle DTC\): This is not the correct pair of angles for AAS with the given figure.
• For \(\overline{DE}\cong\overline{DT}\): This is a side - side - angle (SSA) situation which is not a valid congruence criterion.
• Since we need \(\angle TCD\cong\angle ECD\) for AAS, the option “No additional information is required” is incorrect.

Answer:

\(\angle TCD\cong\angle ECD\)