Let's begin by analyzing the given information. We are given that one side of △ ABD is congruent to one side of △ CBD. This is how we will begin our proof!
Statement1)& BA ≅ BC Reason1)& Given
We are also given that BD bisects ∠ ABC. Therefore, by the definition of an angle bisector, it divides ∠ ABC into two congruent angles ∠ ABD and ∠ CBD.
Statement2)& ∠ ABD ≅ ∠ CBD Reason2)& Definition of an angle bisector
From the diagram we can tell that the triangles share the side BD. By the Reflexive Property of Congruence we know that BD ≅ DB.
Statement 3)& BD ≅ DB Reason 3)& Reflexive Property & of Congruence
Now, we know that two sides and the included angle of △ ABD are congruent to two sides and the included angle of △ CBD. Therefore, by the Side-Angle-Side (SAS) Congruence Theorem, △ ABD ≅ △ CBD.
Statement4)& △ ABD ≅ △ CBD Reason4)& SAS Theorem
Corresponding parts of congruent triangles are congruent. Thus, the corresponding parts of △ ABC and △ CBD are congruent, and hence ∠ ADB ≅ ∠ CDB.
Statement5)& ∠ ADB ≅ ∠ CDB Reason5)& Corresponding parts of & congruent triangles are & congruent.
Now, notice that ∠ ADB and ∠ CDB are supplementary angles, as they form a linear pair . If two angles are congruent and supplementary, then each is a right angle.
Statement6)& ∠ ADB and ∠ CDB are & right angles Reason6)& If two angles are congruent & and supplementary, then each & is a right angle.
We know that BD and AC intersect at a right angle. Thus, by the definition of perpendicular segments, BD ⊥ AC.
Statement7)& BD ⊥ AC Reason7)& Definition of perpendicular & segments.
Lastly, we want to show that BD bisects AC. Recall that we showed that △ ABD ≅ △ CBD. Corresponding parts of congruent triangles are congruent. Therefore, AD ≅ CD.
Statement8)& AD ≅ CD Reason8)& Corresponding parts of & congruent triangles are & congruent.
Since AD is congruent to CD, we can conclude that D is the midpoint of AC. Hence, by the definition of a segment bisector, BD bisects AC.
Statement9)& BD bisects AC Reason9)& Definition of a segment bisector
Completed Proof
Statements
Reasons
1.
BA ≅ BC
1.
Given
2.
∠ ABD ≅ ∠ CBD
2.
Definition of an angle bisector
3.
BD ≅ DB
3.
Reflexive Property of Congruence
4.
△ ABD ≅ △ CBD
4.
SAS Theorem
5.
∠ ADB ≅ ∠ CDB
5.
Corresponding parts of congruent triangles are congruent.
6.
∠ ADB and ∠ CDB are right angles
6.
If two angles are congruent and supplementary, then each is a right angle.
7.
BD ⊥ AC
7.
Definition of perpendicular segments
8.
AD ≅ CD
8.
Corresponding parts of congruent triangles are congruent.
