ANSWER
Initially, it was established that line p is parallel to line r.
To begin with the proof:
As stated in the prompt:
∠1 ≈∠5
∠1 and ∠5 are corresponding angles.
Utilizing the property of corresponding angles,
if two lines are intersected by a transversal such that the corresponding angles are
congruent, then those lines must be parallel.
In the diagram, q acts as the transversal.
Thus, based on this characteristic,
line p is parallel to line r.
Proof of 1(a)
REASON
Vertically opposite angle
When two lines intersect, the angles formed are referred to as vertically opposite angles.
Therefore,
∠4 and ∠1 are vertically opposite angles,
hence,
∠4 ≈∠1
Proof of 2(b)
REASON
Alternate interior angle
The angles situated on either side of the transversal, within the two lines, are termed alternate interior angles. When two parallel lines are crossed by a transversal, the alternate interior angles produced will be congruent.
Since line p is parallel to line r (as proven above)
and q is the transversal,
then
∠4 ≈∠5
Thus, it is established.
Proof of 3 (c)
Given that ∠4 ≈ ∠5 (as demonstrated above)
REASON
If two lines are intersected by a transversal such that the alternate interior angles are congruent, then those lines are parallel.
Thus, based on the previously mentioned property,
line p is parallel to line r.
Therefore, it is proven.
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