Exercise 26 Page 291

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

To show that $\overline{BD}$ bisects $\angle ABC,$ let's write a two column proof. As always, we will start with stating the given statements and the statement that we will prove. Using the given information $\overline{BD}\perp \overline{AC},$ we will write the second step by the Definition of a Right Angle. Since we know that all right angles are congruent, we can write the third step. Considering the given information $△ABC$ is isosceles with base $\overline{AC},$ let's write the next step by the Definition of an Isosceles Triangle. The Isosceles Triangle Theorem states the following. In this case, the fifth step of the proof can be written by this theorem. 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. Since the congruent parts of the congruent triangles are congruent, let's write the seventh step. Finally, using the Definition of Angle Bisector, we can complete the proof. Combining these steps, let's construct the two column proof.

0. Statements	0. Reasons
1. $\overline{BD}\perp \overline{AC}$ and $△ABC$ is an isosceles with base $\overline{AC}$	1. Given
2. $\angle BDA$ and $\angle BDC$ are right angles.	2. Definition of Right Angle
3. $\angle BDA\cong \angle BDC$	3. All right angles are congruent
4. $\overline{AB}\cong \overline{BC}$	4. Definition of Isosceles Triangle
5. $\angle BAD\cong \angle BCD$	5. Isosceles Triangle Theorem
6. $△BAD\cong △BCD$	6. Angle-Angle-Side Theorem
7. $\angle ABD\cong \angle CBD$	7. Congruent parts of the congruent triangles are congruent
8. $\overline{BD}$ bisects $\angle ABC$	8. Definition of Angle Bisector