Question

Given: \overline{AB} \cong \overline{BC} AB ≅ BC and DD is the midpoint of \overline{AC}. AC . Prove: \overline{BD} BD bisects \angle ABC∠ABC. Note: quadrilateral properties are not permitted in this proof. Step Statement Reason 1 \overline{AB} \cong \overline{BC} AB ≅ BC DD is the midpoint of \overline{AC} AC Given A B C D Note: the segment ACAC is a straight segment.

Answer 1

The bisector of angle ∠ABC, divides the angle into two congruent angles.

ΔABD is congruent to ΔCBD, and by CPCTC, ∠ABD ≅ ∠CBD, therefore;

• $\overline{BD}$ is bisects ∠ABC

Reasons:

The two column proof is presented as follows;

Statement ${}$                                         Reason

1. $\overline{AB}$ ≅ $\overline{BC}$ ${}$                                       Given

D is the midpoint of $\overline{AC}$

2. $\overline{AD}$ ≅ $\overline{DC}$   ${}$                                    Definition of midpoint

3. $\overline{BD}$ ≅ Reflexive property of congruency

4. ΔABD ≅ ΔCBD   ${}$                           Side-Side-Side, Congruency Postulate

5. ∠ABD ≅ ∠CBD   ${}$                         CPCTC

6. ∠ABD = ∠CBD     ${}$                         Definition of congruency

7. ∠ABC = ∠ABD + ∠CBD    ${}$             Angle addition postulate

8. $\overline{BD}$ is bisects ∠ABC     ${}$                 Definition of angle bisector

Reason 2.; The midpoint of the line $\mathbf{\overline{AC}}$ is the middle of the line that is equidistant from the points A, and C, such that $\overline{AD}$ = $\mathbf{\overline{DC}}$, therefore;

$\overline{AD}$ ≅ $\overline{DC}$  

Reason 3. $\mathbf{\overline{BD}}$ is congruent to $\overline{BD}$ (

Reasons 4; The Side-Side-Side, SSS, congruency postulate states that if three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent

Reason 5; CPCTC is the acronym for Congruent Parts of Congruent Triangles are Congruent

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