Let's begin by analyzing the given information. We are given that BD bisects ∠ B and that BD ⊥ AC. This is how we will begin our proof!
Statement1:& BD bisects ∠ B, & BD ⊥ AC Reason1:& Given
By the definition of an angle bisector, we can conclude that BD divides ∠ B into two congruent angles, ∠ ABD and ∠ CBD.
Statement2:& ∠ ABD ≅ ∠ CBD Reason2:& Definition of an angle bisector
We are also given that BD ⊥ AC. By the definition of perpendicular lines, we can conclude that ∠ ADB and ∠ CDB are right angles.
Statement3:& ∠ ADB and ∠ CDB are & right angles Reason3:& Definition of perpendicular lines
Next, we can tell that ∠ ADB and ∠ CDB are congruent, as all right angles are congruent. By the definition of a right angle we can say that since both ∠ ADB and ∠ CDB have a measure of 90,
they are congruent.
Statement4:& ∠ ADB ≅ ∠ CDB Reason4:& All right angles are congruent.
Now we know that ∠ ABD ≅ ∠ CBD and ∠ ADB ≅ ∠ CDB. Two angles of △ ADB are congruent to two angles of △ CDB. Therefore, by the Third Angles Theorem we can conclude that the third angles are congruent, ∠ A ≅ ∠ C. This is what we wanted to prove!
Statement5:& ∠ A ≅ ∠ C Reason5:& Third Angles Theorem
Great work so far. We have one step left, and that is to complete our two-column table!
