Exercise 26 Page 380

Let's start with showing the given information on the diagram.

To show that BD bisects ∠ ABC, let's write a two column proof. As always, we will start with stating the given statements and the statement that we will prove. Given: BD⊥ AC and △ ABC is an isosceles with base AC Prove: BD bisects ∠ ABC Using the given information BD⊥ AC, we will write the second step by the Definition of a Right Angle. 2. Definition of a Right Angle ∠ BDA and ∠ BDC are right angles. Since we know that all right angles are congruent, we can write the third step. 3. All right angles are congruent ∠ BDA ≅ ∠ BDC Considering the given information △ ABC is isosceles with base AC, let's write the next step by the Definition of an Isosceles Triangle. 4. Definition of an Isosceles Triangle AB ≅ BC The Isosceles Triangle Theorem states the following. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. In this case, the fifth step of the proof can be written by this theorem. 5. Isosceles Triangle Theorem ∠ BAD ≅ ∠ BCD

Looking at the diagram, we see that two angles and the nonincluded side of △ BAD are congruent to two corresponding angles and the nonincluded side of △ BCD. Thus, the next step can be written by the Angle-Angle-Side Theorem. 6. Angle-Angle-Side Theorem △ BAD ≅ △ BCD Since the congruent parts of the congruent triangles are congruent, let's write the seventh step. 7. CPCTC ∠ ABD ≅ ∠ CBD Finally, using the Definition of Angle Bisector, we can complete the proof. 8. Definition of Angle Bisector BD bisects ∠ ABC Combining these steps, let's construct the two column proof.