Question

An illustration shows three rays passing through three points from a common origin point at upper S. Point upper T is to the horizontal left of upper S. A ray from upper S joins upper T with an arrow near upper T. Point upper V is on the top inclined to the right of point upper S. A ray from upper S joins upper V with an arrow near upper V. Point upper R is to top right of point upper S, below point upper V. A ray from upper S joins upper R with an arrow near upper R. Angle upper T upper S upper R is labeled as 157 degrees. Angle upper V upper S upper R as 54 degrees.
What is the measure of ∠TSV
?

Answer 1

To find the measure of angle \( \angle TSV \), we can use the known angles \( \angle TSR \) and \( \angle VSR \).

We know:
- \( \angle TSR = 157^\circ \)
- \( \angle VSR = 54^\circ \)

Using the fact that the sum of angles around point \( S \) must equal \( 360^\circ \):

\[
\angle TSV + \angle TSR + \angle VSR + \angle RST = 360^\circ
\]

However, we can also express \( \angle TSV \) in terms of the other angles we have:

\[
\angle TSV = 360^\circ - \angle TSR - \angle VSR - \angle RST
\]

Since the angles TSR and VSR describe angles with respect to ray SR, we recognize that \( \angle RST = 180^\circ - (157^\circ + 54^\circ) = 180^\circ - 211^\circ = -31^\circ \), which tells us that angles TSR and VSR should be subtracted from 360 degrees instead.

Then, we can find:
\[
\angle TSV = 360^\circ - (157^\circ + 54^\circ)
\]
\[
\angle TSV = 360^\circ - 211^\circ
\]
\[
\angle TSV = 149^\circ
\]

Therefore, the measure of \( \angle TSV \) is \( 149^\circ \).