Question

in exercises 3–6, copy and complete the statement. state which theorem you used. (see example 1.)

1. if \\(\overline{ae} \cong \overline{de}\\), then \\(\angle\\_\\_ \cong \angle\\_\\_.
2. if \\(\overline{ab} \cong \overline{eb}\\), then \\(\angle\\_\\_ \cong \angle\\_\\_.
3. if \\(\angle d \cong \angle ced\\), then \\(\\_\\_ \cong \\_\\_.
4. if \\(\angle ebc \cong \angle ecb\\), then \\(\\_\\_ \cong \\_\\_.

in exercises 7–10, find the

Explanation:

Step1: Recall the Isosceles Triangle Theorem

The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Step2: Analyze Problem 3

In triangle \( AED \), we are given \( \overline{AE} \cong \overline{DE} \). The sides \( AE \) and \( DE \) are the two congruent sides. The angles opposite these sides are \( \angle A \) (opposite \( DE \)) and \( \angle D \) (opposite \( AE \)). So by the Isosceles Triangle Theorem, \( \angle A \cong \angle D \).

Step3: Analyze Problem 4

In triangle \( ABE \), we are given \( \overline{AB} \cong \overline{EB} \). The sides \( AB \) and \( EB \) are congruent. The angles opposite these sides are \( \angle AEB \) (opposite \( AB \)) and \( \angle A \) (opposite \( EB \)). So by the Isosceles Triangle Theorem, \( \angle A \cong \angle AEB \).

Step4: Analyze Problem 5

We are given \( \angle D \cong \angle CED \) in triangle \( CED \). By the converse of the Isosceles Triangle Theorem (if two angles of a triangle are congruent, then the sides opposite those angles are congruent), the sides opposite \( \angle D \) and \( \angle CED \) are \( \overline{CE} \) and \( \overline{CD} \) respectively. So \( \overline{CE} \cong \overline{CD} \).

Step5: Analyze Problem 6

We are given \( \angle EBC \cong \angle ECB \) in triangle \( EBC \). By the converse of the Isosceles Triangle Theorem, the sides opposite these angles (\( \overline{EB} \) opposite \( \angle ECB \) and \( \overline{EC} \) opposite \( \angle EBC \)) are congruent. So \( \overline{EB} \cong \overline{EC} \).

Answer:

s:

1. If \( \overline{AE} \cong \overline{DE} \), then \( \angle A \cong \angle D \) (Theorem: Isosceles Triangle Theorem)
2. If \( \overline{AB} \cong \overline{EB} \), then \( \angle A \cong \angle AEB \) (Theorem: Isosceles Triangle Theorem)
3. If \( \angle D \cong \angle CED \), then \( \overline{CE} \cong \overline{CD} \) (Theorem: Converse of Isosceles Triangle Theorem)
4. If \( \angle EBC \cong \angle ECB \), then \( \overline{EB} \cong \overline{EC} \) (Theorem: Converse of Isosceles Triangle Theorem)