Question

use the figure.
when ∠d ≅ ∠ced, then blank ≅ blank.
select the theorem used to find the answer.
base angles theorem
converse of the base angles theorem
triangle sum theorem
third angles theorem

Explanation:

Step1: Recall Theorem Definitions

• Base Angles Theorem: In an isosceles triangle, the angles opposite the equal sides are equal.
• Converse of Base Angles Theorem: If two angles of a triangle are equal, then the sides opposite those angles are equal.
• Triangle Sum Theorem: The sum of interior angles of a triangle is \(180^\circ\).
• Third Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, the third angles are congruent.

Step2: Analyze Given Information

We know \(\angle D \cong \angle CED\). We need to find which segments are congruent. In \(\triangle CED\), \(\angle D\) and \(\angle CED\) are angles, so by the Converse of the Base Angles Theorem (if two angles are congruent, the sides opposite them are congruent), the side opposite \(\angle D\) is \(CE\) and the side opposite \(\angle CED\) is \(CD\)? Wait, no—wait, looking at the figure (implied: a triangle with angles \(\angle D\) and \(\angle CED\) in \(\triangle CED\)), actually, if \(\angle D \cong \angle CED\), then the sides opposite: side opposite \(\angle D\) is \(CE\), side opposite \(\angle CED\) is \(CD\)? Wait, no, maybe the triangle is \(\triangle CDE\), with \(\angle D\) and \(\angle CED\) (i.e., \(\angle CED\) is at \(E\)). Wait, the key is: Converse of Base Angles Theorem applies here because we have two congruent angles in a triangle, so the sides opposite them are congruent. So the theorem needed is the Converse of the Base Angles Theorem.

Answer:

Converse of the Base Angles Theorem