Exercise 25 Page 290

The definition of congruence says that two segments are congruent if and only if their lengths are the same. Using this definition, let's write the second step of the proof. Using this statement, let's draw the diagram to write the third step. Looking at the diagram, we can use the Segment Addition Postulate to write the next step. Using the statement from the second step and the Addition Property of Equality, we can write the fourth step. Next, we will combine the third and fourth step using the Substitution Property. Using the definition of congruence one more time, we will write next step. Finally, we can complete our proof using the Isosceles Triangle Theorem. Combining these steps, let's construct the two column proof.

Statements	Reasons
$\overline{AB}\cong \overline{AC}$ and $\overline{BE}\cong \overline{CD}$	Given
$AB=AC$ and $BE=CD$	Definition of Congruence
$AB+BE=AE$ and $AC+CD=AD$	Segment Addition Postulate
$AB+BE=AC+CD$	Addition Property of Equality
$AE=AD$	Substitution Property
$\overline{AE}\cong \overline{AD}$	Definition of Congruence
$△AED$ is isosceles.	Isosceles Triangle Theorem

Using the given information $\overline{ED}\cong \overline{AD},$ if we prove either $\overline{AE}\cong \overline{ED}$ or $\overline{AE}\cong \overline{AD},$ our proof will be done. Based on the given information $\overline{AB}\cong \overline{AC},$ we will first use Isosceles Triangle Theorem. In this case, we can write the congruent angles using the theorem. Next, we will use the definiton of Congruent Angles. The definition says that two angles are congruent if and only if their angle measures are the same. Using the given information $\overline{BC}\parallel \overline{ED},$ we can conclude that $\angle ABC$ and $\angle AED,$ and $\angle ACB$ and $\angle ADE$ are corresponding angles. Using the Corresponding Angles Theorem, let's write the fourth step of our proof. By the definition of Congruent Angles, we can write the next step. Based on step three and five, we we will use the Substitution Property and substitute $m\angle ACB$ for $m\angle ABC.$ Then, we will substitute $m\angle ADE$ for $m\angle ACB.$ Now, let's use the definition of Congruent Angles. Now let us look at the Converse of Isosceles Triangle Theorem. Using the theorem, we can write the ninth step. Finally, we can complete the proof by the Definition of an Equilateral Triangle since $\overline{AD}\cong \overline{AE}\cong \overline{ED}.$ Combining these steps, we will construct the two column proof.

Statements	Reasons
$\overline{AB}\cong \overline{AC},\overline{ED}\cong \overline{AD}$ and $\overline{BC}\parallel \overline{ED}$	Given
$\angle ABC\cong \angle ACB$	Isosceles Triangle Theorem
$m\angle ABC=m\angle ACB$	Definition of Congruent Angles
$\angle ABC\cong \angle AED$ and $\angle ACB\cong \angle ADE$	Corresponding Angles Theorem
$m\angle ABC=m\angle AED$ and $m\angle ACB=m\angle ADE$	Definition of Congruent Angles
$m\angle AED=m\angle ACB$	Substitution Property
$m\angle AED=m\angle ADE$	Substitution Property
$\angle AED\cong \angle ADE$	Definition Congruent Angles
$\overline{AD}\cong \overline{AE}$	Converse of Isosceles Triangle Theorem
$△ADE$ is equilateral.	Definition of Equilateral Triangle

Because the base angles of an isosceles triangle are congruent, if we know that $\angle K\cong \angle B,$ then we know that $\angle K\cong \angle L,$ $\angle B\cong \angle C$ and $\angle C\cong \angle L.$ Thus, one pair of congruent corresponding angles is also needed.