Chapter 4.4, Problem 11WE

To determine

To prove: the given theorem two columns method.

Explanation of Solution

Given information:

Given theorem 4-1,

Two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

Calculation:

The Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

An auxiliary line is a line added to a diagram to help in a proof. It is shown as dashed line in a diagram.

Given: $\overline{AB}\cong \overline{AC}$

Prove: $\angle B\cong \angle C$

Draw an auxiliary line $AD$ such that it is the bisector of $\angle A$ .

Proof:

  $\overline{AB}\cong \overline{AC}$

Reason: given

$AD$ such that it is the bisector of $\angle A$

Reason: because of auxiliary line

  $\angle BAD\cong \angle CAD$

Reason: definition of bisector of an angle

  $\overline{AD}\cong \overline{AD}$

Reason: reflexive property

  $\Delta BAD\cong \Delta CAD$

Reason: SAS postulate

  $\angle B\cong \angle C$

Reason: corresponding parts of $\cong$ triangles are $\cong$