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 PTS and RTQ are shown congruent. Which fact is used to prove that angle PTS is always equal to angle RTQ? A. Sum of the measures of angles RTQ and QTS is 180°. B. Lines PQ and RS intersect at an angle greater than a right angle. C.Line segments PQ and RS do not have a fixed length. D. Sum of the measures of angles PTR and QTS is 180°.

Answer 1

we know that

m∠PTS=m∠RTQ -----> by vertical angles

Step 1

m∠RTQ+m∠QTS=$180\°$ -------> by supplementary angles

solve for m∠QTS

m∠QTS=$180\°$-m∠RTQ -------> equation $1$

Step 2

m∠PTS+m∠QTS=$180\°$ -------> by supplementary angles

solve for m∠QTS

m∠QTS=$180\°$-m∠PTS -------> equation $2$

Step 3

equate equation $1$ and equation $2$

$180\°$-m∠RTQ=$180\°$-m∠PTS

-m∠RTQ=-m∠PTS

so

m∠RTQ=m∠PTS

therefore

the answer is the option A

Sum of the measures of angles RTQ and QTS is 180°

Answer 2

The answer would be A. Sum of the measure of angles RTQ and QTS is 180.