Look at this diagram:

Question

N
O
P
Q
R
S
T
U




























If
OQ
and
RT
are parallel lines and
m
TSP
= 138°, what is
m
QPS
?

Answer 1

To determine the measure of angle \( m\angle QPS \), we can use the properties of parallel lines and the angles formed by a transversal.

Since \( OQ \) and \( RT \) are parallel lines and \( PS \) is a transversal that intersects them at points \( S \) and \( Q \), we can analyze the angles.

Given that \( m\angle TSP = 138^\circ \):

1. \( m\angle TSP \) and \( m\angle QPS \) are same-side interior angles because they are both located between the parallel lines \( OQ \) and \( RT \).

2. The sum of the measures of same-side interior angles is \( 180^\circ \).

Thus, we can find \( m\angle QPS \) as follows:

\[ m\angle QPS + m\angle TSP = 180^\circ \]

Substituting the value of \( m\angle TSP \):

\[ m\angle QPS + 138^\circ = 180^\circ \]

Now, isolate \( m\angle QPS \):

\[ m\angle QPS = 180^\circ - 138^\circ \] \[ m\angle QPS = 42^\circ \]

Therefore, \( m\angle QPS = 42^\circ\).