Given: $\overline{AS}\parallel \overline{BT};m\angle 4=m\angle 5$; $\overrightarrow{SB}$bisects $\angle AST$

Find the measure of$\angle 1$

Line segments $\overline{SB}\text{and}\overline{RT}$are transversal to parallel lines

$\angle 1$and $\angle 4$forms a pair of corresponding angles

Using Postulate 10: If two parallel lines are cut by transversal, then corresponding angles are congruent.

$\angle 1\cong \angle 4$

$m\angle 1=m\angle 4\left(1\right)$

$\angle 2$and $\angle 5$forms a pair of alternate interior angles

Using theorem 3-2: If two parallel lines are cut by transversal, then alternate interior angles are congruent.

$\angle 2\cong \angle 5$

$m\angle 2=m\angle 5\left(2\right)$

From statement $\overrightarrow{SB}$bisects$\angle AST$

$m\angle 2=m\angle 3$

Substitute $m\angle 5$for $m\angle 2$in above equation

$m\angle 5=m\angle 3\left(3\right)$

Since, sum of angles of a triangle is 180

Use equation (3) and given equation

$\begin{array}{c}m\angle 3+m\angle 4+m\angle 5=180\\ m\angle 5+m\angle 5+m\angle 5=180\end{array}$

$3m\angle 5=180$

$m\angle 5=60$

Substitute 60 for $m\angle 5$and role="math" localid="1637818811573" $m\angle 1$for $m\angle 4$(from equation (1)) in given equation

$m\angle 1=60$

Hence, measure of $\angle 1$is 60.