Question

complete the proof that ( mangle p + mangle t + mangle pqt = 180^circ ) without using the triangle angle sum theorem.

(image of a triangle ( pqt ) with a line ( qs ) parallel to ( pt ) and a ray ( qr ) extending from ( q ))

statement	reason
2. ( angle p cong angle rqs )
3. ( angle t cong angle sqt )
4. ( mangle rqt = mangle rqs + mangle sqt )
5. ( mangle rqt = mangle p + mangle t )
6. ( mangle rqt + mangle pqt = 180^circ )
7. ( mangle p + mangle t + mangle pqt = 180^circ )

Explanation:

Step1: Identify Parallel Lines Reason

$\overline{QS} \parallel \overline{PT}$ is given (or from the diagram's markings, as the arrows indicate parallelism). So the reason for statement 1 is "Given (or Definition of Parallel Lines from Diagram Markings)".

Step2: Corresponding Angles for $\angle P \cong \angle RQS$

$\angle P$ and $\angle RQS$ are corresponding angles because $\overline{QS} \parallel \overline{PT}$ and $\overline{PR}$ is a transversal. So the reason is "Corresponding Angles Postulate (If two parallel lines are cut by a transversal, corresponding angles are congruent)".

Step3: Corresponding Angles for $\angle T \cong \angle SQT$

$\angle T$ and $\angle SQT$ are alternate interior angles (or corresponding, depending on the transversal) because $\overline{QS} \parallel \overline{PT}$ and $\overline{QT}$ is a transversal. The reason is "Alternate Interior Angles Theorem (If two parallel lines are cut by a transversal, alternate interior angles are congruent)".

Step4: Angle Addition Postulate for $m\angle RQT = m\angle RQS + m\angle SQT$

The Angle Addition Postulate states that if a point lies in the interior of an angle, the sum of the two smaller angles equals the larger angle. So the reason is "Angle Addition Postulate".

Step5: Substitution for $m\angle RQT = m\angle P + m\angle T$

We substitute $\angle RQS$ with $\angle P$ (from step 2) and $\angle SQT$ with $\angle T$ (from step 3) into the equation from step 4. So the reason is "Substitution Property (replacing $\angle RQS$ with $\angle P$ and $\angle SQT$ with $\angle T$)".

Step6: Linear Pair Postulate for $m\angle RQT + m\angle PQT = 180^\circ$

$\angle RQT$ and $\angle PQT$ form a linear pair (they are adjacent and form a straight line), so their measures sum to $180^\circ$. The reason is "Linear Pair Postulate (If two angles form a linear pair, they are supplementary)".

Step7: Substitution for $m\angle P + m\angle T + m\angle PQT = 180^\circ$

Substitute $m\angle RQT$ with $m\angle P + m\angle T$ (from step 5) into the equation from step 6. The reason is "Substitution Property (replacing $m\angle RQT$ with $m\angle P + m\angle T$)".

Answer:

The proof is completed by using the given parallel lines, corresponding/alternate interior angles theorems, angle addition postulate, linear pair postulate, and substitution properties as explained in each step. The final conclusion $m\angle P + m\angle T + m\angle PQT = 180^\circ$ is proven.