Question

what additional information is needed in order to prove triangles cde and cdq are congruent by asa? (1 point)
∠dce ≅ ∠dcq
de ≅ dq
∠e ≅ ∠q
ce ≅ cq

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 two triangles are congruent.
In \(\triangle CDE\) and \(\triangle CDQ\), we already have \(CD = CD\) (common side). We also have \(\angle CDE=\angle CDQ\) (given by the marked angles).

Step2: Analyze the Options

• For \(\angle DCE\cong\angle DCQ\):

If \(\angle DCE\cong\angle DCQ\), then in \(\triangle CDE\) and \(\triangle CDQ\), we have \(\angle CDE=\angle CDQ\) (given), \(CD = CD\) (common side), and \(\angle DCE\cong\angle DCQ\). By ASA (\(\angle - side-\angle\)), \(\triangle CDE\cong\triangle CDQ\).

• For \(DE\cong DQ\): This would be a side - angle - side (SAS) if we had the included angle, but we are looking for ASA.
• For \(\angle E\cong\angle Q\): This would be a side - side - angle (SSA) situation which is not a valid congruence criterion.
• For \(CE\cong CQ\): This would also not fit the ASA pattern as we need two angles and the included side.

Answer:

\(\angle DCE\cong\angle DCQ\)