Question

PQ and RS are two lines that intersect at point T, as shown below: Two lines PQ and RS intersect at point T. Angles PTR and STQ are shown congruent. Which statement is used to prove that angle PTR is always equal to angle STQ?

 Lines PQ and RS do not have a fixed length.
 Angle PTR and angle PTS are supplementary angles. 
Lines PQ and RS intersect at an angle less than a right angle. 
Angle PTR and angle PTS are complementary angles.

Answer 1

The logic is a bit circuitous/confusing but the best answer is:
"Angle PTR and angle PTS are supplementary angles.".

This means that PQ and RS are straight lines, and that their angles follow the properties or supplementary angles, and opposite angles. 

This implies the more relevant detail that PTR and STQ are opposite angles, and opposite angles are always equal. 

Answer 2

Answer:

B. Angle PTR and angle PTS are supplementary angles.

Step-by-step explanation:

As R, T, S are collinear and PQ intersects RS at T, so ∠RTP and ∠STP are supplementary angles.

$\Rightarrow m\angle RTP+m\angle PTS=180^{\circ}$  -------1

As P, T, Q are collinear and RQ intersects PQ at T, so ∠PTS and ∠QTS are supplementary angles.

$\Rightarrow m\angle PTS+m\angle QTS=180^{\circ}$  ------2

Subtracting equation 1 and 2,

$\Rightarrow m\angle RTP+m\angle PTS-m\angle PTS-m\angle QTS=180^{\circ}-180^{\circ}$

$\Rightarrow m\angle RTP-m\angle QTS=0$

$\Rightarrow m\angle RTP=m\angle QTS$

$\Rightarrow m\angle PTR=m\angle STQ$

Therefore, to prove that angle PTR is always equal to angle STQ the statement needed is angle PTR and angle PTS are supplementary angles.