Question

points P,Q, + S are collinear. what is m<SQR
PQR= (3m+1)°
PQS= (2m+4)°

Answer 1

Given that points \( P, Q, \) and \( S \) are collinear, angle \( SQR \) can be expressed using the other angles \( PQR \) and \( PQS \). The angles around point \( Q \) must add up to \( 180^\circ \).

Given:
- \( m \angle PQR = (3m + 1)^\circ \)
- \( m \angle PQS = (2m + 4)^\circ \)

Since the angles are collinear, we have:

\[
m \angle PQR + m \angle PQS + m \angle SQR = 180^\circ
\]

Substituting the expressions into the equation:

\[
(3m + 1) + (2m + 4) + m \angle SQR = 180
\]

Combining like terms:

\[
(3m + 2m) + (1 + 4) + m \angle SQR = 180
\]

\[
5m + 5 + m \angle SQR = 180
\]

Now isolate \( m \angle SQR \):

\[
m \angle SQR = 180 - 5m - 5
\]

\[
m \angle SQR = 175 - 5m
\]

Thus, \( m \angle SQR = 175 - 5m \) is the expression for angle \( SQR \). If you need to find a numerical value for \( m \angle SQR \), you will need the value of \( m \).

If you want to find \( m \) assuming the angles are valid (i.e., \( m \) leads to angles between \( 0 \) and \( 180 \) degrees), you must ensure that both \( PQR \) and \( PQS \) yield positive angles.

To ensure both angles remain positive:

1. \( 3m + 1 > 0 \) leads to \( m > -\frac{1}{3} \).
2. \( 2m + 4 > 0 \) leads to \( m > -2 \).

Since the more restrictive requirement comes from \( 3m + 1 > 0\), the relevant value is \( m > -\frac{1}{3} \) to ensure \( \angle SQR \) is also valid.

If more information about \( m \) is provided, we could calculate exact values for angles.