Let's begin by analyzing the given information and the desired outcome of our proof. We want to show that △ ABD is congruent to △ CDB. Recall that by the definition of congruent figures, we want to show that the sides and the angles of these triangles are congruent.
Congruent sides
Congruent angles
AB ≅ DC
∠ A ≅ ∠ C
AD ≅ BC
∠ ABD ≅ ∠ BDC
BD ≅ DB
∠ ADB ≅ ∠ CBD
As we can see, we are given that AB ≅ DC and AD ≅ BC. Therefore, we need to prove that BD ≅ DB and that the angles are congruent.
Notice that the triangles share the side BD, and by the Reflexive Property of Congruence we know that BD ≅ DB.
Statement 1)& BD ≅ DB Reason 1)& Reflexive Property & of Congruence
We are also given that AB ⊥ AD and AD ⊥ BC. Let's list this as the next step in our proof, as we will use this to make a conclusion about the angles.
Statement 2)& AB ⊥ AD and AD ⊥ BC Reason 2)& Given
Recall that the definition of perpendicular lines tells us that two perpendicular lines form a right angle. Therefore, we can conclude that ∠ A and ∠ C, which are between the sides AB, AD and AD, BC respectively, are right angles. This means that they are congruent.
Statement 3)& ∠ A ≅ ∠ C Reason 3)& Definition of perpendicular lines
Next, we are given that AD ∥ BC. Therefore, if we draw the lines passing through these sides, then we will get two parallel lines a and b. If we also draw a line passing through the side DB, then we will get a transversal t that intersects these parallel lines.
Notice that ∠ ADB and ∠ CBD are alternate interior angles. Lines a and b are parallel, thus by the Alternate Interior Angles Theorem ∠ ADB ≅ ∠ CBD.
Statement 4)& ∠ ADB ≅ ∠ CBD Reason 4)& Alternate Interior Angles & Theorem
Now, we know that ∠ A ≅ ∠ C and ∠ ADB ≅ ∠ CBD. Two angles of the triangle △ ABD are congruent to two angles of the triangle △ CDB. Therefore, by the Third Angle Theorem we can conclude that the third angles are congruent, ∠ ABD ≅ ∠ BDC.
Statement 5)& ∠ ABD ≅ ∠ BDC Reason 5)& Third Angles Theorem
We have shown that all the sides and the angles in the triangles are congruent!
Congruent sides
Congruent angles
AB ≅ DC
∠ A ≅ ∠ C
AD ≅ BC
∠ ABD ≅ ∠ BDC
BD ≅ DB
∠ ADB ≅ ∠ CBD
Therefore, by the definition of congruent figures, △ ABD ≅ △ CDB.
