Question

use the figure.
if \\( \overline{ae} \cong \overline{de} \\), then \\( \angle \underline{\quad\quad} \cong \angle \underline{\quad\quad}
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: Identify the Theorem

The problem involves congruent segments \( \overline{AE} \cong \overline{DE} \) and we need to find congruent angles. The Base Angles Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. Wait, actually, the Converse of the Base Angles Theorem? No, here we have \( AE = DE \), so in triangle \( AED \) (or looking at triangles \( AEB \) and \( DEB \)? Wait, the figure has points A, B, C, D, E. Wait, the first blank: we need to find the angle congruent to, say, \( \angle BEA \) or \( \angle CED \)? Wait, the options for the theorem: Base Angles Theorem, Converse, Triangle Sum, Third Angle. Wait, the Base Angles Theorem (Isosceles Triangle Theorem) says that in an isosceles triangle, the base angles are congruent. Since \( AE \cong DE \), triangle \( AED \) is isosceles with \( AE = DE \), so the angles opposite? Wait, no, maybe looking at triangles \( AEB \) and \( DEC \)? Wait, the first blank: the angle, let's see the options. The options for the angle are A, D, ABE, BEA, CED, DCE. Wait, \( \overline{AE} \cong \overline{DE} \), and maybe \( BE = CE \)? Wait, the figure: A---B---C---D, with E connected to A, B, C, D? Wait, no, the figure is a triangle with vertex E, and base A---B---C---D? Wait, maybe E is the apex, and AB and DC are segments. Wait, the problem says "If \( \overline{AE} \cong \overline{DE} \), then \( \angle \underline{\quad} \cong \angle \underline{\quad} \)". The theorem used is the Base Angles Theorem (Isosceles Triangle Theorem), which states that in an isosceles triangle, the angles opposite the congruent sides are congruent. So if \( AE = DE \), then in triangle \( AED \), the angles opposite? Wait, no, maybe triangle \( AEB \) and \( DEC \)? Wait, the options for the theorem: Base Angles Theorem. Then the angles: \( \angle BEA \cong \angle CED \)? Wait, the options for angles: BEA and CED. So first, the theorem: Base Angles Theorem? Wait, no, the Converse? Wait, no, the Base Angles Theorem (Isosceles Triangle Theorem) is if two sides are congruent, the base angles are congruent. So if \( AE = DE \), and maybe \( BE = CE \), but the problem states \( AE \cong DE \). Wait, maybe the triangles are isosceles with \( AE = DE \), so the angles at B and C? No, the angles should be \( \angle BEA \) and \( \angle CED \)? Wait, the options for the angle: BEA and CED. So the theorem is the Base Angles Theorem? Wait, no, the Converse? Wait, the problem is to select the theorem: the options are Base Angles Theorem, Converse of Base Angles Theorem, Triangle Sum Theorem, Third Angle Theorem. The Base Angles Theorem (Isosceles Triangle Theorem) is that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. So if \( AE = DE \), then in triangle \( AED \), but maybe the triangles are \( AEB \) and \( DEC \). Wait, maybe the correct theorem is the Base Angles Theorem, and the angles are \( \angle BEA \cong \angle CED \)? Wait, the options for the angle: BEA and CED. So the first blank: BEA, second: CED, and the theorem: Base Angles Theorem? Wait, no, the Converse? Wait, no, let's re-examine. The Base Angles Theorem: In an isosceles triangle, the angles opposite the equal sides are equal. So if \( AE = DE \), then the angles opposite (if triangle is AED) would be... Wait, maybe the figure has E connected to B and C, so triangles EAB and EDC? Wait, maybe the correct theorem is the Base Angles Theorem, and the angles are \( \angle BEA \) and \( \angle CED \). So the theorem is Base Angles Theorem, and the angles are \( \angle BEA \cong \angle CED \).

Step2: Select the Theorem and Angles

The theorem used is the Base Angles Theorem (since we have congruent sides \( AE \cong DE \), leading to congruent angles). The angles congruent would be \( \angle BEA \) and \( \angle CED \) (from the options). Wait, the options for the theorem: "Base Angles Theorem" is an option. So the theorem is Base Angles Theorem, and the angles are \( \angle BEA \cong \angle CED \).

Answer:

Theorem: Base Angles Theorem; Angles: \( \angle BEA \cong \angle CED \) (So the first blank: BEA, second: CED, theorem: Base Angles Theorem)