Question

$\angle rut \cong \angle rst$ and $\angle sru \cong \angle stu$. complete the proof that $\overline{ru} \parallel \overline{st}$.

statement	reason
2 $\angle sru \cong \angle stu$	given
3 $m\angle sru + m\angle rst + m\angle stu + m\angle rut = 360^\circ$
4 $m\angle sru + m\angle rst + m\angle sru + m\angle rst = 360^\circ$
5 $m\angle sru + m\angle rst = 180^\circ$	properties of addition, subtraction, multiplication, and division
6 $\overline{ru} \parallel \overline{st}$

Explanation:

Step1: Identify reason for Statement3

The sum of interior angles of a quadrilateral is $360^\circ$. Quadrilateral SRUT has interior angles $\angle SRU$, $\angle RST$, $\angle STU$, $\angle RUT$, so their measures add to $360^\circ$.

Step2: Identify reason for Statement4

Substitute congruent angles: since $\angle STU \cong \angle SRU$ and $\angle RUT \cong \angle RST$, replace $m\angle STU$ with $m\angle SRU$ and $m\angle RUT$ with $m\angle RST$ in the equation from Step1. This uses the Substitution Property of Equality.

Step3: Identify reason for Statement6

We know $m\angle SRU + m\angle RST = 180^\circ$. These are consecutive interior angles formed by transversal $\overline{SR}$ intersecting $\overline{RU}$ and $\overline{ST}$. If consecutive interior angles are supplementary, the lines are parallel.

Answer:

1. (Given, already provided)
2. (Given, already provided)
3. Reason: Sum of quadrilateral interior angles is $360^\circ$
4. Reason: Substitution Property of Equality
5. (Reason already provided)
6. Reason: Consecutive Interior Angles Converse Theorem