The angles ∠1 ≅ ∠4 since the line BD bisects ∠ABC, AD║ BC, and AB ║ CD. This is obtained by using the angle bisector theorem, alternate interior angles, and the transitive property of congruence.
What is the transitive property of congruence?
Transitive property:
If ∠a and ∠b are congruent and ∠b and ∠c are congruent, then ∠a and ∠c are also congruent.
I.e., If ∠a ≅ ∠b and ∠b ≅ ∠c, then ∠a ≅ ∠c.
What does the angle bisector theorem state?
The angles bisector theorem states that the ray or line which bisects the angle divides the angle into two equal parts.
I.e., If line BD bisects the angle ∠ABC, then ∠B = ∠B1 + ∠B2
(where ∠B1 = ∠B2)
Given:
The line BD bisects ∠ABC, AD║BC, and AB║CD
Proof:
The line BD bisects ∠ABC. So,
∠B = ∠3 + ∠4
According to the angle bisector theorem, ∠3 ≅ ∠4.
Since AD║BC and AB║CD, the alternate angle are congruent.
I.e., ∠3 ≅ ∠1
Thus, by the transitive property of congruence,
∠1 ≅ ∠4
Hence proved.
Learn more about the transitive property of congruence here:
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