Exercise 25 Page 379

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. 3. Segment Addition Postulate AB+BE=AE and AC+CD=AD Using the statement from the second step and the Addition Property of Equality, we can write the fourth step. 4. Addition Property of Equality AB+BE=AC+CD Next, we will combine the third and fourth step using the Substitution Property. 5. Substitution Property AE=AD Using the definition of congruence one more time, we will write next step. 6. Definition of Congruency AE≅ AD Finally, we can complete our proof using the Isosceles Triangle Theorem. 7. Isosceles Triangle Theorem △ AED is isosceles. Combining these steps, let's construct the two column proof.

2. Isosceles Triangle Theorem ∠ ABC ≅ ∠ ACB

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. 3. Definiton of Congruent Angles m∠ ABC = m∠ ACB Using the given information BC∥ ED, we can conclude that ∠ ABC and ∠ AED, and ∠ ACB and ∠ ADE are corresponding angles. Using the Corresponding Angles Theorem, let's write the fourth step of our proof. 4. Corresponding Angles Theorem ∠ ABC≅ ∠ AED and ∠ ACB≅ ∠ ADE

By the definition of Congruent Angles, we can write the next step. 5. Definition of Congruent Angles m∠ ABC=m∠ AED and m∠ ACB=m∠ ADE Based on step three and five, we we will use the Substitution Property and substitute m∠ ACB for m∠ ABC. 6. Substitution Property m∠ AED = m∠ ACB Then, we will substitute m∠ ADE for m∠ ACB. 7. Substitution Property m∠ AED = m∠ ADE Now, let's use the definition of Congruent Angles. 8. Definition of Congruent Angles ∠ AED ≅ ∠ ADE Now let us look at the Converse of Isosceles Triangle Theorem. If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Using the theorem, we can write the ninth step. 9. Converse of Isosceles Triangle Theorem AD≅ AE Finally, we can complete the proof by the Definition of an Equilateral Triangle since AD≅ AE≅ ED. 10. Definition of Equilateral Triangle △ ADE is equilateral. Combining these steps, we will construct the two column proof.

AB≅ JK or AB≅ JL AC≅ JK or AC≅ JL Because the base angles of an isosceles triangle are congruent, if we know that ∠ K ≅ ∠ B, then we know that ∠ K ≅ ∠ L, ∠ B ≅ ∠ C and ∠ C ≅ ∠ L. Thus, one pair of congruent corresponding angles is also needed.