Question

in the diagram below, the measure of ∠pts is 151°. what is the measure of ∠rtq ?

Explanation:

Step1: Identify Angle Relationship

∠PTS and ∠RTQ are adjacent supplementary angles? No, wait, ∠PTS and ∠RTQ: Wait, actually, ∠PTS and ∠RTQ—wait, looking at the diagram, lines PQ and RS intersect at T. So ∠PTS and ∠RTQ: Wait, no, ∠PTS and ∠RTQ—wait, ∠PTS and ∠RTQ: Wait, actually, ∠PTS and ∠RTQ are vertical angles? No, wait, no. Wait, ∠PTS and ∠RTQ: Wait, no, the angle ∠PTS is 151°, and ∠RTQ—wait, actually, ∠PTS and ∠RTQ: Wait, no, let's see. The lines PQ and RS intersect at T. So ∠PTS and ∠RTQ: Wait, ∠PTS and ∠RTQ—wait, no, ∠PTS and ∠RTQ: Wait, actually, ∠PTS and ∠RTQ are supplementary? Wait, no, wait. Wait, ∠PTS and ∠RTQ: Wait, the angle ∠PTS is 151°, and ∠RTQ—wait, no, actually, ∠PTS and ∠RTQ: Wait, no, let's think again. The angle ∠PTS and ∠RTQ: Wait, when two lines intersect, vertical angles are equal, and adjacent angles are supplementary. Wait, ∠PTS and ∠RTQ: Wait, no, ∠PTS and ∠RTQ—wait, maybe I made a mistake. Wait, ∠PTS is 151°, and ∠RTQ: Wait, actually, ∠PTS and ∠RTQ are supplementary? No, wait, no. Wait, the sum of ∠PTS and ∠RTQ: Wait, no, looking at the diagram, ∠PTS and ∠RTQ—wait, maybe ∠PTS and ∠RTQ are adjacent and form a linear pair? Wait, no, ∠PTS is 151°, so the adjacent angle (supplementary) would be 180° - 151° = 29°? Wait, no, wait. Wait, ∠PTS and ∠RTQ: Wait, maybe ∠PTS and ∠RTQ are vertical angles? No, that can't be. Wait, no, let's check the diagram again. The lines PQ (with points P, T, Q) and RS (with points R, T, S) intersect at T. So ∠PTS and ∠RTQ: Wait, ∠PTS is at T, between P, T, S. ∠RTQ is at T, between R, T, Q. So these two angles: are they supplementary? Wait, no, actually, ∠PTS and ∠RTQ—wait, maybe ∠PTS and ∠RTQ are equal? No, that doesn't make sense. Wait, no, wait. Wait, ∠PTS is 151°, so the angle adjacent to it (on a straight line) would be 180° - 151° = 29°. Wait, but ∠RTQ—wait, maybe ∠RTQ is equal to that supplementary angle? Wait, no, let's see. Wait, ∠PTS and ∠RTQ: Wait, maybe I got the angles wrong. Wait, the blue arc is between R, T, Q. So ∠RTQ is the blue arc. And ∠PTS is 151°. Wait, no, ∠PTS is at P, T, S. So ∠PTS and ∠RTQ: Wait, are they vertical angles? No, vertical angles would be ∠PTS and ∠RTQ? Wait, no, vertical angles are opposite each other. So ∠PTS and ∠RTQ—wait, no, ∠PTS and ∠RTQ: Wait, maybe ∠PTS and ∠RTQ are supplementary? Wait, no, 151° + 29° = 180°, so if ∠PTS is 151°, then the angle supplementary to it is 29°, and ∠RTQ is that angle? Wait, maybe. Let's calculate: 180° - 151° = 29°. So ∠RTQ is 29°? Wait, no, wait. Wait, ∠PTS and ∠RTQ: Wait, maybe ∠RTQ is equal to ∠PTS? No, that would be 151°, but that doesn't make sense. Wait, no, let's re-express. The sum of angles on a straight line is 180°. So ∠PTS + ∠RTQ = 180°? Wait, no, ∠PTS and ∠RTQ: Wait, maybe ∠PTS and ∠RTQ are adjacent and form a linear pair? Wait, no, ∠PTS is at P, T, S, and ∠RTQ is at R, T, Q. So the lines PQ and RS intersect at T, so ∠PTS and ∠RTQ—wait, actually, ∠PTS and ∠RTQ are vertical angles? No, vertical angles are ∠PTS and ∠RTQ? Wait, no, vertical angles are ∠PTS and ∠RTQ? Wait, no, vertical angles would be ∠PTS and ∠RTQ? Wait, no, vertical angles are opposite each other when two lines intersect. So ∠PTS and ∠RTQ: Wait, ∠PTS is between P, T, S, and ∠RTQ is between R, T, Q. So these two angles: are they vertical angles? No, because vertical angles are ∠PTS and ∠RTQ? Wait, no, ∠PTS and ∠RTQ—wait, maybe I'm confusing the labels. Let's label the points: P---T---Q (line PQ), R---T---S (line RS). So the intersection is at T. So the angles at T: ∠PTS (between P, T, S), ∠STQ (between S, T, Q), ∠QTR (between Q, T, R), ∠RTP (between R, T, P). So ∠PTS and ∠QTR are vertical angles? No, ∠PTS and ∠RTQ: Wait, ∠RTQ is ∠R-T-Q, which is the same as ∠QTR? Wait, no, ∠RTQ is R-T-Q, so that's the angle between R, T, Q. And ∠PTS is P-T-S. So these two angles: ∠PTS and ∠RTQ—are they supplementary? Wait, ∠PTS is 151°, so ∠RTQ would be 180° - 151° = 29°? Wait, because they are adjacent and form a linear pair? Wait, no, ∠PTS and ∠RTQ—wait, maybe the blue arc is ∠RTQ, and ∠PTS is 151°, so they are supplementary? Wait, yes, because a straight line is 180°, so ∠PTS + ∠RTQ = 180°? Wait, no, that can't be, because ∠PTS is 151°, so 180 - 151 = 29. So ∠RTQ is 29°? Wait, maybe. Let's confirm. If two angles are adjacent and form a linear pair (they are on a straight line), their sum is 180°. So ∠PTS and ∠RTQ: are they on a straight line? Wait, line PQ: P---T---Q, line RS: R---T---S. So ∠PTS is on line RS (R-T-S) and line PQ (P-T-Q). Wait, no, ∠PTS is between P, T, S. ∠RTQ is between R, T, Q. So the straight line here is R-T-S? No, R-T-S is a straight line, so ∠R-T-S is 180°. Wait, no, ∠PTS is part of that? Wait, I think I messed up the angle labels. Let's start over. The measure of ∠PTS is 151°. We need to find ∠RTQ. Looking at the diagram, ∠PTS and ∠RTQ are adjacent angles that form a linear pair? Wait, no, ∠PTS and ∠RTQ—wait, maybe ∠PTS and ∠RTQ are supplementary because they are on a straight line (line PQ or RS)? Wait, no, line PQ is straight, so ∠PTS + ∠STQ = 180°, but ∠STQ is vertical to ∠RTP. Wait, maybe ∠RTQ is equal to the supplementary angle of ∠PTS. So 180° - 151° = 29°. So ∠RTQ is 29°.

Step2: Calculate ∠RTQ

Since ∠PTS and ∠RTQ are supplementary (they form a linear pair, sum to 180°), we use the formula:
$\angle RTQ = 180^\circ - \angle PTS$
Substitute $\angle PTS = 151^\circ$:
$\angle RTQ = 180^\circ - 151^\circ = 29^\circ$

Answer:

29°