The complete proofs are:
Proof 1
- a || b; c || d ⇒ Given
- ∠5 is supplementary to ∠4 ⇒ ∠5 + ∠4 = 180
- ∠2 = ∠4 ⇒ Definition of corresponding angles
- ∠5 + ∠2 = 180 ⇒ Substituting property of equality
- ∠5 + ∠2 = 180 ⇒ Definition of supplementary angles
Proof 2
- BD bisects ∠ABC; AD || BC; AB || CD ⇒ Given
- ∠3 ≅ ∠4 ⇒ Definition of congruent angles
- ∠1 ≅ ∠3 ⇒ Definition of corresponding angles
- ∠1 ≅ ∠4 ⇒ Definition of corresponding angles
How to complete the proofs?
Proof 1
Lines a & b and c and d are given as parallel lines.
So, we have
a || b; c || d ⇒ Given
From the question, we have:
∠5 is supplementary to ∠4
So, we have
∠5 + ∠4 = 180
This is so because supplementary angles add up to 180
Corresponding angles are equal.
So, we have:
∠2 = ∠4
By the substituting ∠2 for ∠4 in ∠5 + ∠4 = 180, we have
∠5 + ∠2 = 180
Hence, angles ∠5 and ∠2 are supplementary angles
Proof 2
Line BD bisects angle ABC; lines AD & BC and AB & CD are given as parallel lines.
So, we have
BD bisects ∠ABC; AD || BC; AB || CD ⇒ Given
From the question, we have:
∠3 ≅ ∠4 ⇒ Definition of congruent angles
∠1 ≅ ∠3 ⇒ Definition of corresponding angles
Substitute ∠1 ≅ ∠3 in ∠3 ≅ ∠4
∠1 ≅ ∠4
Hence, the angles 1 and 4 are congruent
Read more about supplementary angles at:
https://brainly.com/question/2046046
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