Question

The proof diagram to complete the question state the missing reason in the proof for the letter given

Answer 1

The required statements and proofs is given as follows:

     Statement                               Proof

     ∠1 ≈ ∠5                                  Given
     ∠4 ≈ ∠1                                  Vertical opposite angles
     ∠4 ≈ ∠5                                  Transitive property
        p ║ r                                    Parallel lines intersected by the transverse line q.

Answer 2

ANSWER

First proved that line p is parallel to line r

to proof

As given in the question

∠1 ≈∠5

∠1 and ∠5 are corresponding angles

by using the property of the corresponding angles

 two lines are cut by a transversal so that the corresponding angles are

congruent, then these lines are parallel.

As shown in diagram q is transversal line.

Thus by using the above property

line p is parallel to line r.

proof of 1(a)

REASON

Vertically opposite angle

The pair of angles formed when two lines intersect each other are called vertically opposite angles.

Thus

∠4 and ∠1 are vertically opposite angle

thus

∠4 ≈∠ 1

proof of 2(b)

REASON

Alternate interior angle

the pairs of angles on either side of the transversal and inside the two lines are called the alternate interior angles . If two parallel lines are cut by a transversal, then the alternate interior angles formed are congruent .

as line p is parallel to line r (proof above)

q is transversal

thus

∠4 ≈∠ 5

Hence proved

proof of 3 (c)

As ∠4 ≈∠5 (proof above)  

REASON

If two lines are cut by a transversal so that the alternate interior angles

are congruent, then these lines are parallel.

Thus by above property

line p is parallel to line r

Hence proved