Question

PQ and RS are two lines that intersect at point T. Which fact is used to prove that angle PTS is always equal to angle RTQ?

A.) The sum of the measures of angles RTQ and QTS is equal to the sum of the measures of angles QTS and PTS.
B.) Angle RTQ and angle QTS are complementary angles.
C.) The sum of the measures of angles RTQ and PTS is equal to the sum of the measures of angles RTP and QTS.
D.) Angle RTQ and angle PTS are supplementary angles.

Answer 1

Let PQ and RS be two lines that intersect at point T. Angles:

• ∠PTS and ∠PTR are supplementary, then
  `m\angle PTS+m\angle PTR=180^(\circ);`
• ∠PTR and ∠RTQ are supplementary, then
  `m\angle PTR+m\angle RTQ=180^(\circ).`

Subtract from first equation second equation:

`m\angle PTS+m\angle PTR-(m\angle PTR+m\angle RTQ)=180^(\circ)-180^(\circ),\\ \\m\angle PTS+m\angle PTR-m\angle PTR-m\angle RTQ=0,\\ \\m\angle PTS-m\angle RTQ=0,\\ \\m\angle PTS=m\angle RTQ.`

As you can see, here you use option A - the sum of the measures of angles RTQ and QTS is equal to the sum of the measures of angles QTS and PTS.

Answer: correct choice is A.

Answer 2

"The sum of the measures of angles RTQ and QTS is equal to the sum of the measures of angles QTS and PTS" is the one fact among the following is used to prove that angle PTS is always equal to angle RTQ. The correct option among all the options that are given in the question is the first option or option "A".