Question

identifying missing components of a paragraph proof
given: ( l parallel m ); ( angle 1 cong angle 3 )
prove: ( p parallel q )
complete the missing parts of the paragraph proof.
we know that angle 1 is congruent to angle 3 and that line ( l ) is parallel to line ( m ) because (\boxed{\text{it is given}}). we see that (\boxed{\text{by the alternate reason}}), angle 1 is congruent by some property. so, we can conclude ( p parallel q ) by the (\boxed{\text{reason}}).

Explanation:

Step1: Analyze the first blank

The statement "line \( l \) is parallel to line \( m \)" and "\( \angle 1\cong\angle 3 \)" are given in the problem's "Given" section. So the reason we know these is "it is given".

Step2: Analyze the angle - relationship step

Since \( l\parallel m \), by the alternate interior angles theorem, \( \angle 1\cong\angle 2 \) (if we consider the transversal for \( l \) and \( m \) and the lines \( p \) and \( q \)). But we know \( \angle 1\cong\angle 3 \), so by the transitive property \( \angle 2\cong\angle 3 \). Wait, actually, when we have \( l\parallel m \), the alternate interior angles (for the transversal that cuts \( l \) and \( m \) and the lines related to \( \angle 1 \) and \( \angle 2 \) or \( \angle 3 \)) - but the key here is that the first blank is filled with "it is given" because the information about \( l\parallel m \) and \( \angle 1\cong\angle 3 \) is given. Then, for the angle congruence, since \( l\parallel m \), \( \angle 1\cong\angle 2 \) (alternate interior angles), and since \( \angle 1\cong\angle 3 \), \( \angle 2\cong\angle 3 \) (transitive property). But the first dropdown (the first blank) is for why we know \( l\parallel m \) and \( \angle 1\cong\angle 3 \), so the answer for the first blank is "it is given".

Step3: Analyze the last blank (for proving \( p\parallel q \))

If \( \angle 2\cong\angle 3 \), then by the converse of the alternate interior angles theorem, \( p\parallel q \). But the options in the dropdown - the first part (the first blank) is "it is given" because the given information is \( l\parallel m \) and \( \angle 1\cong\angle 3 \).

Answer:

For the first blank (the reason we know \( l\parallel m \) and \( \angle 1\cong\angle 3 \)): it is given.