Question

Write proofs in two–column form.

Theorem 4–1.

Short answer

Statement	Reason
$\overline{\mathrm{AB}}\cong \overline{\mathrm{AC}}$	Given
$\angle \mathbf{BAD}\cong \angle \mathbf{CAD}$	Angle bisector theorem
$\overline{\mathrm{AD}}\cong \overline{\mathrm{AD}}$	Reflexive property
$\mathrm{\Delta }\mathbf{BAD}\cong \mathrm{\Delta }\mathbf{CAD}$	SAS postulate
$\angle \mathbf{B}\cong \angle \mathbf{C}$	Congruent angles of congruent triangles

Step-by-step solution

Step 1. Draw an auxiliary line through angle A.

Step 2. Apply angle bisector theorem.

Angle bisector divides an angle into two equal parts.

From the figure in step 1 it can be observed that$\overline{AD}$ bisects $\angle A$.

Therefore, $\angle BAD\cong \angle CAD$.

Step 3. Reflexive property.

Reflexive property states that $\overline{PQ}\cong \overline{PQ}$.

Therefore, $\overline{AD}\cong \overline{AD}$.

Step 4. SAS postulate.

If two sides and included angle of one triangle are congruent to two sides and included angle of other triangle then the triangles are said to be congruent.

Here $\overline{AB}\cong \overline{AC}$,$\overline{AD}\cong \overline{AD}$ and $\angle BAD\cong \angle CAD$. Therefore,$\mathrm{\Delta }BAD\cong \mathrm{\Delta }CAD$ which implies that $\angle B\cong \angle C$.

It is proved that if the two sides of a triangle are congruent then the angles opposite to those sides are congruent.

Step 5. Two column proof.

Write a two-column proof based on above explanation.

Statement	Reason
$\overline{AB}\cong \overline{AC}$	Given
$\angle BAD\cong \angle CAD$	Angle bisector theorem
$\overline{AD}\cong \overline{AD}$	Reflexive property
$\Delta BAD\cong \Delta CAD$	SAS postulate
$\angle B\cong \angle C$	Congruent angles of congruent triangles